A mapping method based on double discrete memristor

Abstract

The application discloses a mapping method based on double discrete memristor, and relates to the technical field of memristor, two new double-memristor hyperchaotic mapping models of isomorphic sine-sine discrete memristor SSDM mapping and heterogeneous sine-cosine discrete memristor SCDM mapping are constructed by using two periodic discrete memristors; the mapping has a plane fixed point, and stability distribution is divided according to the second characteristic root in the parameter and initial value plane; bifurcation behavior depending on coupling strength, coexistence extreme multistability controlled by the initial value of the memristor are studied by using a bifurcation diagram, Lyapunov exponent, phase diagram and local attraction basin; under the control of the memristor, coexistence infinite heterogeneous attractors are self-replicated along a straight line and a plane; numerical results show that the SSDM and SCDM mappings can not only generate highly complex hyperchaotic attractors, but also exhibit extreme multistability behavior controlled by the initial condition of the memristor distributed on a straight line and a plane.

Description

Technical Field

[0001] This invention relates to the field of memristor technology, specifically a mapping method based on dual discrete memristors. Background Technology

[0002] Since their introduction and successful fabrication, memristors have attracted widespread and sustained attention from scholars due to their unique memory effect and strong nonlinear characteristics. Therefore, memristors hold significant research value for the development of nonlinear circuits / systems, artificial neural networks, and neuromorphic computing. In particular, memristors are often used to construct memristor chaotic systems or circuits to generate rich dynamic behaviors. For example, a sixth-order memristor hyperchaotic circuit exhibiting multiple types of vortex hyperchaotic attractors was designed using three memristors. Specifically, an oscillating circuit based on locally active memristors was constructed, and a large number of neuromorphic behaviors were observed at the edge of the chaotic domain. Furthermore, by using memristors to simulate synaptic and electromagnetic induction effects, various memristor neurons and neural networks exhibited near-realistic neural electrical activity. Interestingly, by coupling multiple segments of linear memristors to a conventional SprottA system, any number of hidden attractors can be generated. However, in practice, the models of these continuous memristor systems are extremely complex and difficult to realize, especially the memristor elements themselves. For example, at least four dimensions are required to generate hyperchaotic behavior in continuous systems, while only two dimensions are needed for discrete mappings. Therefore, considering that discrete memristors have similar inherent properties to continuous memristors, developing new and simple discrete memristor mappings to replace continuous memristor systems is of great significance and specific value for practical engineering applications, but it has not yet received enough attention.

[0003] In existing technologies, some mappings based on discrete memristors have been modeled, resulting in complex hyperchaotic phenomena and coexistence behaviors. Specifically, by discretizing an ideal continuous memristor using the forward Euler difference method, four 2D hyperchaotic mappings consisting only of single memristors have been developed, and the bistable phenomenon of coexistence of hyperchaos and stable point attractors can be non-destructively controlled by initial conditions. Simultaneously, by coupling discrete memristors with traditional mappings, such as Logistic, Gaussian, and various hyperchaotic mappings, complex hyperchaotic attractors and coexisting bistable behaviors controlled by initial conditions have been successfully applied to pseudo-random number generators, image encryption, secure communication, and reservoir computing. Recently, to enrich the dynamic behavior related to memristor initial conditions and enhance its application value, more complex multistability and extreme multistability phenomena in memristor mappings have increasingly become the research frontier. By coupling discrete memristors with a 2D mapping with a sinusoidal function and adding a constant term, a coexistence attractor with multistability is exhibited in a novel 3D memristor hyperchaotic mapping. Furthermore, by characterizing the electromagnetic induction effect of Rulkov neurons using discrete memristors, the mode transition behavior dependent on the initial state of the memristor exhibits extreme multistability.

[0004] However, the difference equations for these memristor maps are quite complex to achieve sophisticated extreme multistability. Therefore, through parallel and composite operations, maps constructed solely from discrete memristors achieve coexistence behavior with extreme multistability in a simpler way. Unfortunately, these maps fail to exhibit the extreme multistability with memristor initial value control found in continuous memristor systems.

[0005] In summary, there is currently a lack of a simple method to obtain discrete mappings of hyperchaos and extreme multistability based on memristors. Summary of the Invention

[0006] To address the problem in existing technologies of lacking a simple method for obtaining hyperchaotic and extremely multi-stability discrete mappings based on memristors, this invention provides a mapping method based on dual discrete memristors, comprising the following steps:

[0007] Construct sinusoidal discrete memristors (S-DM) and cosine discrete memristors (C-DM);

[0008] A three-dimensional dual discrete memristor hyperchaotic mapping model is constructed by parallel coupling of a sinusoidal discrete memristor (S-DM) and a cosine discrete memristor (C-DM) and connecting them through self-feedback. The three-dimensional dual discrete memristor hyperchaotic mapping model includes a three-dimensional sinusoidal-sinusoidal discrete memristor (SSDM) hyperchaotic mapping model and a three-dimensional sinusoidal-cosine discrete memristor (SCDM) hyperchaotic mapping model.

[0009] A three-dimensional dual discrete memristor hyperchaotic mapping model is written into the digital circuit of a microcontroller. The microcontroller converts the generated digital signal into an analog voltage corresponding to the state variable sequence. The analog voltage is input to an oscilloscope for capture, thus obtaining a discrete mapping of hyperchaos and extreme multistability.

[0010] Furthermore, the mapping method based on dual discrete memristors provided by this invention also includes:

[0011] Verification of the mapping dynamics and extreme multistability of the three-dimensional dual discrete memristor hyperchaotic mapping model, specifically including:

[0012] Determine the stability distribution of the three-dimensional dual discrete memristor hyperchaotic mapping model;

[0013] The coupling strength-dependent dynamic behavior distribution and local behavior and attractor evaluation of the three-dimensional dual-discrete memristor hyperchaotic mapping model are determined, and the parameter-dependent hyperchaotic dynamic evolution mechanism of the mapping of the three-dimensional dual-discrete memristor hyperchaotic mapping model is determined.

[0014] The mechanism of coexistence behavior of initial value regulation, initial value-dependent extreme multistability, coexistence parameter bifurcation of initial value regulation, coexistence line attractor of initial value regulation, and planar coexistence behavior of initial value regulation in a three-dimensional dual discrete memristor hyperchaotic mapping model are determined, thereby determining the bifurcation and coexistence extreme multistability of initial value regulation in the three-dimensional dual discrete memristor hyperchaotic mapping model.

[0015] Furthermore, the discrete memristor model is as follows:

[0016]

[0017] The sinusoidal discrete memristor S-DM is:

[0018]

[0019] The cosine-type discrete memristor C-DM is:

[0020]

[0021] In the formula, sin(πq) n ) and cos(πq n ) represents the sampled value of the memristor function M(q) of the continuous memristor in the nth iteration; v n i n and q n These are the sampled values ​​of voltage v(t), current i(t), and charge q(t) in the nth iteration, respectively; and q n+1 It is the sampled value of the (n+1)th iteration.

[0022] Furthermore, the memristor function of the sinusoidal discrete memristor S-DM is M1(q n The memristor function of a cosine-type discrete memristor (C-DM) is M2(q). n When M1(q) n )=M2(q n When M1(q) is used, the three-dimensional dual discrete memristor hyperchaotic mapping model is isomorphic; when M1(q) is used, the model is isomorphic. n )≠M2(q n When ), the three-dimensional dual discrete memristor hyperchaotic mapping model is heterogeneous;

[0023] When M1(q) n )=M2(q n )=sin(πq n When ), the isomorphic three-dimensional sinusoidal-sinusoidal discrete memristor SSDM hyperchaotic mapping model is:

[0024]

[0025] When M1(q) n )=sin(πq n ) and M2(q n )=cos(πq n When the heterogeneous three-dimensional sinusoidal-cosine discrete memristor (SCDM) hyperchaotic mapping model is:

[0026]

[0027] In the formula, k1 and k2 are the coupling strengths of the two discrete memristors; x n The current i of the two discrete memristors n y n and z n The internal states q of two discrete memristors n ;x n+1 y n+1 and z n+1 x n y n and z n The value of the (n+1)th iteration.

[0028] Furthermore, determining the stability distribution of the discrete mapping of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0029] Let the fixed point of the SSDM mapping be S = (x * ,y * ,z * If ), then the algebraic structure at the fixed point S is:

[0030]

[0031] Therefore, the fixed point S is calculated as S = (0, μ, η), where μ and η correspond to the initial conditions of the discrete memristor. Since μ and η can take any constant, S is an infinite set of planar fixed points.

[0032] To quantitatively analyze the stability boundary of S, the Jacobian matrix of the SSDM mapping at S is written as:

[0033]

[0034] According to J S The characteristic equation is derived as follows:

[0035] det(λI-J S )=(λ-1)(λ-1)(λ-k1M1(μ)-k2M2(η))=0

[0036] Therefore, the eigenvalues ​​are obtained as:

[0037] λ1=1, λ2=1, λ3=k1M1(μ)+k2M2(η)

[0038] Since |λ1|=|λ2|=1 is constant and lies on the unit circle, the stability of the fixed point S can only be critically stable or unstable, uniquely determined by |λ3|. For |λ3|>1, the fixed point S is unstable, while for |λ3|≤1, it is critically stable. The value of |λ3| in the SSDM mapping is controlled by the coupling strengths k1 and k2, as well as the initial conditions μ and η, indicating that the stability of the fixed point is extremely complex. When |λ3|=1 and (μ,η) are constants, the critical stability boundary condition is defined as:

[0039] k1M1(μ)+k2M2(η)±1=0

[0040] Critically stable boundary conditions can be used to determine the stability distribution of a mapping;

[0041] For SSDM and SCDM mappings, the critical stability boundary conditions are precisely expressed as:

[0042]

[0043] Furthermore, determining the coupling strength-dependent dynamic behavior distribution of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0044] The distribution of dynamic behavior is characterized by a two-dimensional bifurcation diagram, which further distinguishes and refines different dynamic behaviors. In order to reveal the dynamic evolution mechanism of SSDM and SCDM mapping, coupling strengths k1 and k2 are selected as control parameters and the initial conditions are fixed as (x0, y0, z0) = (0.1, 0.1, 0.1). When the intervals of the control parameters are k1 ∈ [-2, 2] and k2 ∈ [-2, 2], the oscillation period and Lyapunov exponent of the trajectory corresponding to each parameter set (k1, k2) can be calculated by the period number judgment method and the method based on the Wolf Jacobi matrix, respectively, so as to obtain the two-dimensional bifurcation diagram and Lyapunov exponent spectrum about the k1-k2 plane.

[0045] In the two-dimensional bifurcation diagram, the various dynamic behaviors generated by the SSDM and SCDM mappings are distinguished according to the period number and Lyapunov exponent and filled with different color blocks: the hyperchaotic attractor with two positive LEs is labeled HCH; the chaotic attractor with only one positive LE is labeled CH; the stable point attractor labeled SP has a large distribution; the divergent behavior is labeled DI; for the periodic behavior with zero maximum LE, labeled P02, P04, P06, P08, QP and MP are defined as period 2, period 4, period 6, period 8, quasi-periodic and multi-periodic attractors, respectively, and these periodic windows are also filled with different color blocks, indicating that the two mappings enter chaos through period-doubling bifurcation paths;

[0046] Comparing the two-dimensional bifurcation diagrams, significant differences exist between the distributions of dynamic behavior, and the heterogeneous SCDM mapping has a larger and more irregular dynamic behavior parameter domain than the isomorphic SSDM mapping; moreover, the two-dimensional bifurcation diagrams have clear boundaries, consistent with the stability distribution, proving the effectiveness of the theoretical analysis;

[0047] To accurately measure different dynamic behaviors, a two-dimensional LE exponent spectrum was plotted based on the first LE value corresponding to each parameter set (k1,k2), where a darker color indicates a larger LE value. The LE exponent spectrum accurately characterizes the dynamic behavior of the two-dimensional bifurcation graph, proving that the coupling strength can induce complex dynamic effects in the SSDM and SCDM mappings. The parameter domain of the resulting dynamic behavior can be directly obtained from the two-dimensional bifurcation graph.

[0048] Furthermore, the determination of the local behavior and attractor evaluation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0049] The statistical characteristics of bifurcation points can intuitively show the evolution of dynamic behavior within the parameter range. Therefore, the one-dimensional bifurcation process controlled by a single parameter can be used to determine the evolution mechanism of coupling strength-induced attractors. Following the distribution of dynamic behavior, when the initial condition is (x0,y0,z0)=(0.1,0.1,0.1), a single-parameter bifurcation diagram is generated by k1-induced SSDM and SCDM mapping.

[0050] According to the single-parameter bifurcation diagram, when k2 = 1 is fixed, k1 induces the SSDM mapping to produce chaotic and hyperchaotic behavior through a complete period-doubling bifurcation in the interval [0.4, 0.94]. As k1 changes, the SSDM mapping transitions to a period 2 bifurcation at k1 = 0.477, and then splits into a period 4 bifurcation at k1 = 0.533. The period 8 bifurcation then occurs in the interval k1 ∈ [0.667, 0.697]. According to the LE exponent spectrum, the periodic bifurcation has a zero maximum Lyapunov exponent, indicating that the bifurcation is quasi-periodic. The chaotic behavior is located at k1 ∈ [0.697, 0.726] ∪ [0.858, 0.932], which shows a positive LE1 in the LE exponent spectrum. For k1 ∈ [0.72... [0.784]∪[0.808,0.858], hyperchaotic behavior with positive LE1 and LE2 can be found, and a narrow periodic window is distributed in k1∈[0.784,0.808]. Unlike the SSDM mapping, when k2=-1.5, the positive and negative period-doubling bifurcation processes of the SCDM mapping are distributed in k1∈[-1.3,1.3], and the bifurcation point of the variable x is symmetric, the cosine function is an even function and leads to conditional symmetry about k1. It can be observed that the SCDM mapping has similar dynamic behaviors to the SSDM mapping, such as symmetric periodicity, chaos and hyperchaos, consistent with the parameter domain of dynamic behavior. Therefore, the two mappings under consideration can produce complex coupling strength-dependent dynamic evolution.

[0051] The topology and evolution of attractors are visualized under the influence of typical parameters. When the initial conditions are (x0, y0, z0) = (0.1, 0.1, 0.1), and k2 is fixed at 1 and –1.5 for SSDM and SCDM mappings respectively, different attractors with different types, topologies, and spatial locations are generated as k1 increases. For attractor performance evaluation, LE, spectral entropy SE, permutation entropy PE, C0 complexity, and correlation dimension CorDim are calculated and compared. The maximum LE of periodic attractors is 0, chaotic attractors have one positive LE, and hyperchaotic attractors have two positive LEs. By comparing the values ​​of SE, PE, C0, and CorDim, it is found that hyperchaotic attractors have the highest randomness and complexity, while periodic attractors have the lowest dynamic performance. Therefore, the coupling strength-dependent SSDM and SCDM mappings not only exhibit complex dynamic evolution processes but also generate attractors with different dynamic performances, including high-performance hyperchaotic and chaotic attractors.

[0052] Furthermore, the mechanism for determining the coexistence behavior of initial value regulation in the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0053] Let μ = x0 and η = y0, the SSDM and SCDM mapping models are transformed according to the iterative mechanism into:

[0054]

[0055] in

[0056] The initial conditions μ and η of the two memristors are decomposed into even and even parts and rewritten as follows:

[0057]

[0058] Where m and n are arbitrary natural numbers, and μ0 and η0 are y n and z n The following constraints must be satisfied for the non-even portion extracted from the given data:

[0059]

[0060] The expression for SSDM mapping is described as follows:

[0061]

[0062] The expression for SCDM mapping is described as follows:

[0063]

[0064] Therefore, the SSDM and SCDM mappings exhibit periodic cyclic characteristics, thus verifying the performance of memristor initial value control. Specifically, the initial conditions μ and η of the sine and cosine memristors have a period of 2, resulting in non-destructive control of the offsets of the variables in the two mappings along the y and z axes. The memristor initial value control process does not affect the offset of the variable x, as demonstrated by the topological invariance of the SSDM and SCDM mappings after periodic initial value transformation. Since the offset is changed simultaneously in the yz(μ-η) plane, the SSDM and SCDM mappings can generate planar coexistence dynamics of memristor initial value control. To reveal the evolution mechanism of the coexistence dynamics, we first consider the case where a single memristor initial value is used as the control parameter.

[0065] The determination of the extreme multistability of the initial value dependence of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0066] Using the initial conditions (x0,μ,η), the two-dimensional / three-dimensional local attraction basins are used to detect the initial value attraction domain to which the long-term dynamic trajectories of two mappings belong and to characterize the extreme multistability dependent on the initial value.

[0067] Without considering the switching period of the memristor initial value, the two mappings use fixed coupling strength values ​​and the non-memristor initial conditions are restricted to the interval x0∈[-2,2]. When the initial conditions of the memristor in the SSDM mapping change in μ=η∈[-1,1], the three-dimensional local attractor basin in the μ-η-x0 space is drawn, which is obtained by LEs and the period number judgment method. The three-dimensional local attractor basin of the SCDM mapping is drawn using the same method. Different color regions fully demonstrate the dependence and sensitivity of the dynamic behavior of the two mappings on the initial conditions, indicating the occurrence of coexistence. Since there are infinitely many initial conditions in the attractor basin, the coexistence of infinitely many attractors can be confirmed.

[0068] To demonstrate the coexistence of infinitely many attractors, the SSDM mapping is used as an example, and a cross section of η = 0.1 is selected to obtain local attractor basins on the x0-μ plane. The dynamics of the SCDM mapping are robust to the non-memristor initial condition x0 because the dynamic behavior does not change with the adjustable x0 when μ is fixed. Under the induction of the memristor initial condition, the SCDM mapping successively generates a stable point, period 2, period 4, period 6, period 8, chaotic and hyperchaotic attractors, proving and visualizing the coexisting multistability phenomenon. The coexisting attractors have different types, topologies and spatial locations, therefore the infinitely many attractors in the SSDM and SCDM mappings that depend on the memristor initial condition have extreme multistability.

[0069] Furthermore, the determination of the coexistence parameter bifurcation for initial value control of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0070] The control parameters and initial conditions for the SSDM mapping are set as k1∈[0.4,0.94], k2=1 and (x0,y0,z0)=(0.1,μ,η), where μ and η satisfy μ=η=0.1+2m (m=0,±1,±2). A bifurcation diagram of coexistence parameters dependent on the coupling strength of y and z is obtained through numerical simulation, where five sets of bifurcation points correspond to the dynamic evolution of different initial values. The spatial location of the positive period-doubling bifurcation point in the SSDM mapping is determined by the initial value of the memristor. The vertical direction is raised, but the bifurcation process and structure are not affected by the initial value of the memristor; similarly, the coexistence parameter bifurcation diagram of the SCDM mapping is obtained, where k1∈[-1.3,1.3], k2=1 and the same initial conditions as the SSDM mapping are set; the symmetrical forward and reverse bifurcation structure of the SCDM mapping is accurately copied by the initial value of the memristor and is distributed in the figure with an amplitude offset of 2; it is found that the bifurcation structures with respect to y and z in the two mappings are the same, because the two discrete memristors have the same internal state;

[0071] The coexistence parameter bifurcation diagram of SSDM and SCDM mappings confirms that periodically switched memristor initial values ​​can cause both SSDM and SCDM mappings to simultaneously generate bifurcation phenomena with respect to offset control of y and z. As the memristor initial values ​​increase infinitely, the two mappings will generate coexistence extreme parameter bifurcation with initial value control, which greatly improves the flexibility of the mapping and the flexibility of the coupling strength-dependent dynamic behavior.

[0072] The determination of the initial value-dependent bifurcation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0073] The initial conditions for the two mappings are set as (x0, y0, z0) = (-0.5, μ, 0.1) and (x0, y0, z0) = (0.1, 0.1, η). When (k1, k2) = (0.84, 1) and (k1, k2) = (1.08, -1.5), the two mappings bifurcate with changes in μ and η. The extreme multistable bifurcation set (BS) within each period is represented by different color depths and is labeled as EMBS-2, EMBS-1, EMBS, EMBS+1 and EMBS+2, respectively. As μ and η change, the bubble-shaped period-doubling bifurcation generated by the SSDM mapping and the forward and reverse period-doubling bifurcation generated by the SCDM mapping exhibit an infinite number of different attractors coexisting, which is the extreme multistable state. Since the offset of the step-like bifurcation with respect to y and z and the switching period of the initial conditions are 2, it is proved that the extreme multistable bifurcation can be controlled by the initial value of the memristor.

[0074] The LEs of the two mappings switch with a period of 2. The distribution of LEs corresponding to different periods does not change, indicating that the initial value of the memristor does not affect the LEs during the period switching process. For the mean of the long-term trajectory, the mean(y) and mean(z) curves show a gradual upward trend, which is caused by the offset of the memristor initial value control state. For the mean of the uncontrolled state, it does not change throughout the initial value domain, and the mean corresponding to different dynamic behaviors is also different.

[0075] Each extreme multistability bifurcation in the SSDM and SCDM mappings can produce a fixed offset switching under memristor initial value control, and the state variables not involved in the control are unaffected; the SSDM and SCDM mappings have different bifurcation structures, resulting in differences in the coexistence extreme multistability of memristor initial value control.

[0076] Furthermore, the coexistence line attractor for determining the initial value modulation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0077] Local attractor basin maps on the μ-x0 and η-x0 planes are generated by SSDM and SCDM mappings, which can display coexisting line attractors with offset control induced by memristor initial values. These are divided into local attractor basins LBA-2, LBA-1, LBA, LB1+1, and LBA+2, based on the offset of the memristor initial value or an interval of 2 periods. The local attractor basin maps show layered attractor domains along the μ or η direction, with extreme multistability behaviors present in each LBA, rendered in different colors. Comparing the two local attractor basin maps reveals that the SSDM mapping exhibits richer extreme multistability than the SCDM mapping. This is because the initial value domains of various coexisting behaviors in the SSDM mapping are much larger than those in the SCDM mapping. After selecting typical parameters (μ, η) in the local attractor basin maps, the two mappings produce a variety of offset-controlled line coexisting attractors with an offset of 2 widths between adjacent attractors.

[0078] The planar coexistence behavior of initial value regulation for determining the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes:

[0079] Based on the critical stability condition, the stability distribution of SSDM and SCDM mappings on the μ-η plane is visualized, where the initial conditions are (x0, y0, z0) = (0.1, μ, η), μ ∈ [-3, 3] and η ∈ [-3, 3]. The yellow unstable region induced by the memristor initial value simultaneously produces a shift of width 2 in both the horizontal and vertical directions, proving that the planar stability distribution of the two mappings obeys the initial value regulation mechanism. Due to the symmetry of the sine and cosine functions, two unstable regions satisfying |λ3| > 1 can be generated within one period, indicating that a total of 2 × 3 × 3 = 18 coexisting unstable regions are generated simultaneously on the μ-η plane.

[0080] The extreme multistability of the planar coexistence of the two mappings can be characterized using local attractor basins, resulting in approximately 18 mutually separate coexisting local attractor basins, proving the correctness of the stability distribution. Each local attractor basin contains multiple coexisting dynamic behaviors, implying that each local attractor basin is extremely multistable. Each local attractor basin infinitely replicates itself on the μ and η axes with an offset of width 2, leading to the production of planar coexistence phenomena through memristor initial value manipulation of any dynamic behavior. In the local attractor basins of the SSDM, the red hyperchaotic attractor occupies a large region, while a large divergence region exists in the local attractor basins of the SCDM mapping, indicating that the extreme multistability of the SSDM mapping is richer and that the planar coexistence behaviors of the two mappings are fundamentally different. The normalized spectral entropy based on Fourier transform is estimated, and the results are visualized through a 2-D plot, showing that the more complex the dynamic behavior, the larger the value of SE and the darker the corresponding color. The distribution of SE complexity also exhibits periodic cyclic characteristics, meaning that the performance of the dynamic behavior does not change during the manipulation process, i.e., it possesses lossless control characteristics.

[0081] When μ = μ0 + 2m (m = 0, ±1) and η = η0 + 2n (n = 0, ±1), after selecting typical parameters, an attractor with coexistence of SSDM and SCDM mapping planes can be constructed to exhibit the periodic cyclic characteristics of various attractors induced by the initial value of the memristor.

[0082] Compared with existing technologies, the mapping method based on dual discrete memristors provided by this invention has the following advantages:

[0083] This invention constructs two dual-discrete memristor hyperchaotic mapping models by connecting two discrete memristors with periodic memdeterminants in parallel: isomorphic SSDM and heteromorphic SCDM mappings. When the coupling strength is used as a control parameter, a parameter bifurcation mechanism and various attractors with excellent dynamic performance are revealed. Since the considered mapping has a planar fixed point related to the memristor initial value, coexisting stable points, periods, chaos, and hyperchaotic attractors, i.e., extreme multistability phenomena, are discovered. The extreme multistability phenomenon of coexistence of lines and planes can be continuously and non-destructively controlled by the initial value of periodic discrete memristors, and an infinite number of rich coexisting attractors show flexible self-replication on lines and planes. In summary, the two dual-discrete memristor hyperchaotic mapping models provided by this invention have simple structures but can exhibit complex hyperchaotic dynamics and controllable extreme multistability. Attached Figure Description

[0084] Figure 1 The application of excitation i provided in the embodiments of the present invention n Performance analysis diagram of S-DM after A sin(ωn);

[0085] Figure 2 A diagram illustrating the construction scheme of a three-dimensional dual-discrete memristor hyperchaotic mapping provided in an embodiment of the present invention;

[0086] Figure 3 Stability distribution diagram of SSDM and SCDM mapping provided for embodiments of the present invention;

[0087] Figure 4 is a distribution diagram of the dynamic behavior on the k1-k2 plane induced by the coupling strength for (x0,y0,z0)=(0.1,0.1,0.1) provided by the embodiment of the present invention. Among them, Figure 4(a) is a two-dimensional bifurcation diagram of SSDM mapping, Figure 4(b) is a two-dimensional bifurcation diagram of SCDM mapping, Figure 4(c) is a two-dimensional maximum Lyapunov exponent spectrum of SSDM mapping, and Figure 4(d) is a two-dimensional maximum Lyapunov exponent spectrum of SCDM mapping.

[0088] Figure 5 is a single-parameter bifurcation diagram induced by k1 for (x0,y0,z0)=(0.1,0.1,0.1) provided by an embodiment of the present invention. Figure 5(a) is an SSDM mapping diagram with k2=1, and Figure 5(b) is a two-dimensional bifurcation diagram with k2=-1.5.

[0089] Figure 6 The following diagrams illustrate various dynamic behaviors induced by k1 for (x0,y0,z0)=(0.1,0.1,0.1) as provided in embodiments of the present invention.

[0090] Figure 7 The initial value-dependent extreme multistability and coexistence attractor graph provided in the embodiments of the present invention;

[0091] Figure 8 is a bifurcation diagram of coexistence parameters for memristor initial value control when (x0,y0,z0)=(0.1,μ,η) and μ=η=0.1+2m (m=0,±1,±2) according to an embodiment of the present invention. In Figure 8(a), it is the SSDM mapping diagram when k1∈[0.4,0.94] and k2=1, and Figure 8(b) is the SSDM mapping diagram when k1∈[-1.3,1.3] and k2=-1.5.

[0092] Figure 9 An extreme multistable bifurcation diagram of memristor initial value control provided in an embodiment of the present invention;

[0093] Figure 10 A local attraction basin diagram of memristor initial value control on the μ-x0 and η-x0 planes provided in an embodiment of the present invention;

[0094] Figure 11 is a coexistence line attractor diagram of memristor initial value control provided in an embodiment of the present invention. In Figure 11(a1-a2), the coexistence attractor diagrams generated by different μ-induced SSDM mappings when (k1,k2)=(0.84,1) and (x0,y0,z0)=(0.1,μ,0.1) are shown. Figure (b1-b2) are the coexistence attractor diagrams generated by different η-induced SCDM mappings when (k1,k2)=(1.08,-1.5) and (x0,y0,z0)=(0.1,0.1,η).

[0095] Figure 12 shows the stability distribution and local attraction basin diagram on the μ-η plane of the memristor initial value control when (x0,y0,z0)=(0.1,μ,η) according to the embodiment of the present invention. Among them, Figure 12(a) is the stability distribution diagram of SSDM mapping for (k1,k2)=(0.84,1), Figure 12(b) is the stability distribution diagram of SCDM mapping for (k1,k2)=(1.08,-1.5), Figure 12(c) is the local attraction basin diagram of SSDM mapping, and Figure 12(d) is the local attraction basin diagram of SCDM mapping.

[0096] Figure 13 shows various planar coexistence attractor diagrams for memristor initial value control when μ=η∈[-3,3] according to the embodiments of the present invention. Among them, Figure 13(a1-a2) is the planar coexistence attractor diagram generated by SSDM mapping when (k1,k2)=(0.84,1), and Figure 13(b1-b2) is the planar coexistence attractor diagram generated by SCDM mapping when (k1,k2)=(1.08,-1.5).

[0097] Figure 14 A diagram of a microcontroller-based digital hardware experimental platform provided in an embodiment of the present invention;

[0098] Figure 15 The diagram shows the experimental results of the microcontroller-based digital hardware circuit provided in the embodiments of the present invention. Detailed Implementation

[0099] The following is in conjunction with the appendix Figure 1-15 The following describes specific embodiments of the present invention in further detail. These embodiments are merely for illustrating the technical solutions of the present invention more clearly and should not be construed as limiting the scope of protection of the present invention.

[0100] Example 1: This invention proposes a mapping method based on dual discrete memristors, including:

[0101] 1. Dual Discrete Memristor Hyperchaotic Mapping Model

[0102] In this section, a novel three-dimensional dual-discrete memristor hyperchaotic mapping model is modeled, constructed from isomorphic periodic memristors. The infinitely many equilibrium points in the proposed mapping are accurately locked, and their stability distribution patterns in the two-dimensional parameter plane and the memristor initial value plane are discussed in detail. The related theoretical analysis provides support for the research in subsequent sections.

[0103] 1.1 Periodic Discrete Memristor

[0104] Recently, numerous discrete memristor models have been developed, including sinusoidal, cosine, quadratic, absolute value, and exponential memristors, thus expanding the memristor system and its application areas. Discrete memristors, like continuous memristors, exhibit three important characteristic fingerprints. In fact, discrete memristor models are obtained by applying backward difference theory to continuous memristor models, and an ideal charge-controlled discrete memristor model is defined as:

[0105]

[0106] Where M(q) n ) represents the sampled value of the memristor function M(q) of the continuous memristor in the nth iteration; v n i n and q n These are the sampled values ​​of voltage v(t), current i(t), and charge q(t) in the nth iteration, respectively; and q n+1 This is the sampled value from the (n+1)th iteration. Therefore, it can be seen that discrete memristors possess a memory effect and a simple iterative mechanism, which can replace continuous memristors in practical applications.

[0107] To generate diverse dynamic effects, sinusoidal and cosine discrete memristors, abbreviated as S-DM and C-DM, are considered as examples, and their models are modeled according to the literature as follows:

[0108]

[0109] and

[0110]

[0111] Since both S-DM and C-DM memristor functions are periodic trigonometric functions, S-DM is used as a representative to verify the main characteristics of discrete memristors. When a discrete bipolar periodic current excites i... n When Asin(ωn) is applied to S-DM, Figure 1 The main performance characteristics of S-DM are presented. Figure 1 (a) reflects the input current i as n changes. n and output voltage v nThe different phases and amplitudes of the iterative sequences indicate that the memory conductance of the discrete memristor exhibits nonlinearity. When i is set... n When the amplitude is A = 0.02 and the initial condition for the internal charge is q0 = 0, as the angular frequency ω gradually increases, i n -v n The plane exhibits a contracting tight hysteresis loop that depends on ω. Similarly, when ω = 0.1 and q0 = 0, the maximum current i in the amplitude-dependent tight hysteresis loop depends on A. n The values ​​were limited to 0.03, 0.02, and 0.01, respectively. More notably, the S-DM produced different q0-dependent hysteresis loops under different initial conditions, indicating that the S-DM is multistable. Therefore, the main characteristics of the memristor were confirmed through the characteristic analysis of the S-DM.

[0112] in, Figure 1 Apply incentive i n Performance analysis of S-DM after =Asin(ωn). (a) For ω=0.1, A=0.02 and q0=0, i n and v n (a) the iterative sequence; (b) the tight hysteresis loop when A = 0.02 and q0 = 0, ω = [0.1, 0.15, 0.2]; (c) the tight hysteresis loop when A = [0.01, 0.02, 0.03] and ω = 0.1 and A = 0.02, q0 = [0, -1, -2, -3]

[0113] 1.2 Dual Discrete Memristor Model

[0114] In memristor systems or mappings, the memristor's memristor / memderivative function introduces significant nonlinear characteristics, and the internal state charge and magnetic flux are highly sensitive to initial conditions, which is key to generating complex dynamic behaviors.

[0115] The charge / magnetic flux changes exhibiting memory effects are highly sensitive to initial conditions, leading to the coexistence of attractors. Leveraging these inherent advantages of memristors, a novel three-dimensional dual-discrete memristor hyperchaotic mapping construction scheme has been developed. Figure 2 The description describes a method that couples two discrete memristors in parallel and establishes a discrete relationship through a self-feedback connection. M1(q) n ) and M2(q n ) are the memristor functions of two discrete memristors, and when M1(q) n )=M2(q n When M1(q) is an isomorphism, the mapping is isomorphic; conversely, when M1(q) is an isomorphism, the mapping is isomorphic. n )≠M2(q nWhen the mapping is heterogeneous, the mapping schemes under consideration can produce complex and diverse coexistence multistability or even extreme multistability because they generate a set of surface equilibrium points related to the initial conditions of the two memristors, rather than a set of line equilibrium points or a set of fixed points.

[0116] Importantly, to ensure the proposed mapping scheme exhibits attractor self-replication properties, i.e., shift-modulation behavior, the S-DM with periodic memetic derivation from Section 1.1 is considered for modeling the mapping. When M1(q n )=M2(q n )=sin(πq n When ), the isomorphic three-dimensional sinusoidal-sinusoidal discrete memristor (SSDM) hyperchaotic mapping model is represented as:

[0117]

[0118] And when M1(q) n )=sin(πq n ) and M2(q n )=cos(πq n When the heterogeneous three-dimensional sinusoidal-cosine discrete memristor (SCDM) hyperchaotic mapping model is expressed as:

[0119]

[0120] In fact, k1 and k2 are the coupling strengths of two discrete memristors; x n The current i of the two discrete memristors n , and y n and z n Then it is defined as the internal state q of two discrete memristors. n Similarly, x n+1 y n+1 and z n+1 x n y n and z n The n+1 iteration values. According to equations (4) and (5), the dynamic evolution generated by both SSDMmap and SCDMmap is controlled by the coupling strengths k1 and k2, and is also strongly dependent on the initial conditions (x0, y0, z0) of the mapping. Therefore, discussing the stability distribution of the two mappings is crucial to revealing the dynamic mechanisms that depend on both coupling strength and initial conditions. It can be seen that the mapping composed of periodic memristors includes a variety of dynamic behaviors.

[0121] 1.3 Stability distribution of infinitely many fixed points

[0122] Unlike the stability analysis of continuous systems, the stability of discrete mappings is closely related to the fixed point to which the mapping originates. Assume the fixed point of the SSDM mapping is S = (x... * ,y * ,z * Then, the algebraic structure of equation (3) at the fixed point S can be derived as follows:

[0123]

[0124] Therefore, the fixed point S can be calculated as S = (0, μ, η), where μ and η correspond to the initial conditions of the discrete memristor. Since μ and η can take any constant values, S is an infinite set of planar fixed points.

[0125] To quantitatively analyze the stability boundary of S, the Jacobian matrix of the SSDM mapping at S can be written as:

[0126]

[0127] Next, according to J S The characteristic equation can be derived as follows:

[0128] det(λI-J S )=(λ-1)(λ-1)(λ-k1M1(μ)-k2M2(η))=0 (8)

[0129] Therefore, the eigenvalues ​​can be obtained as:

[0130] λ1=1, λ2=1, λ3=k1M1(μ)+k2M2(η) (9)

[0131] Since |λ1|=|λ2|=1 is constant and lies on the unit circle, the stability of the fixed point S can only be critically stable or unstable, uniquely determined by |λ3|. For |λ3|>1, the fixed point S is unstable, while for |λ3|≤1, it is critically stable. Clearly, the value of |λ3| in the SSDM mapping is controlled by the coupling strengths k1 and k2, as well as the initial conditions μ and η, indicating that the stability of the fixed point is extremely complex. When |λ3|=1 and (μ,η) are constants, the critically stable boundary condition can be defined as…

[0132] k1M1(μ)+k2M2(η)±1=0 (10)

[0133] It can be used to determine the stability distribution of a mapping.

[0134] For SSDM and SCDM mappings, according to equation (10), the critical stability boundary condition can be precisely expressed as:

[0135]

[0136] When the relevant parameters of the two mappings are specified as (μ,η)=(0.1,0.1), and the coupling strengths k1 and k2 vary in [-1,1] and [-1,1] respectively, the stability distribution on the k1-k2 plane can be visualized according to Equation (11), as follows: Figure 3 As shown in (a) and (b). When the SSDM mapping has (k1,k2) = (0.84,1) and the SCDM mapping has (k1,k2) = (1.08,-1.5), Figure 3 (c) and (d) give the stability distribution on the μ-η plane. Figure 3 In the gray region, the critically stable fixed points with |λ3|≤1 are distinguished, while the light-colored region indicates unstable fixed points with |λ3|>1. As can be seen, there are clear boundary conditions between different stability distributions, which corresponds perfectly to Equation (11). Observation Figure 3 It is evident that the stable distributions of coupling strength and memristor initial value differ significantly between the two mappings, implying fundamentally different dynamic behaviors related to coupling strength and memristor initial value. Therefore, the dynamic analysis of the two mappings can be performed... Figure 3 With the help of [unclear], it was carried out.

[0137] in, Figure 3 Let represent the stability distributions of the SSDM and SCDM mappings. When (μ,η)=(0.1,0.1), k1∈[-1,1] and k2∈[-1,1], (a) and (b) represent the stability distributions of the SSDM and SCDM mappings on the k1-k2 plane, respectively; when μ∈[-1,1] and η∈[-1,1], (c) and (d) represent the stability distributions of the SSDM with (k1,k2)=(0.84,1) and the SCDM with (k1,k2)=(1.08,-1.5) on the μ-η plane, respectively.

[0138] 2. Hyperchaotic Dynamics and Extreme Multistability

[0139] This section primarily investigates the parameter-dependent hyperchaotic dynamic evolution mechanism and initial-condition-dependent extreme multistability of SSDM and SCDM mappings. Simultaneously, diverse numerical simulation methods are employed to determine the parameter domain of dynamic behavior and visualize attractors, including bifurcation graphs, Lyapunov exponents, local attractor basins, phase point diagrams, and iterative sequences.

[0140] 2.1 Distribution of Coupling Strength-Dependent Dynamic Behavior

[0141] and Figure 3 Unlike the stability distribution, the dynamic behavior distribution can be characterized by a two-dimensional bifurcation diagram, which further distinguishes and refines different dynamic behaviors.

[0142] To reveal the dynamic evolution mechanism of the SSDM and SCDM mappings, coupling strengths k1 and k2 were chosen as control parameters, and the initial conditions were fixed as (x0, y0, z0) = (0.1, 0.1, 0.1). When the intervals of the control parameters are k1 ∈ [-2, 2] and k2 ∈ [-2, 2], the oscillation period and Lyapunov exponent of the trajectory corresponding to each parameter set (k1, k2) can be calculated by the period number determination method and the method based on the Wolf Jacobi matrix, respectively. Thus, we obtain the two-dimensional bifurcation diagram and Lyapunov exponent spectrum of the k1-k2 plane.

[0143] As illustrated in Figures 4(a) and (b), the various dynamic behaviors generated by the SSDM and SCDM maps are distinguished by period number and Lyapunov exponent and filled with different color blocks. Specifically, the hyperchaotic attractor with two positive LEs is labeled HCH, while the chaotic attractor with only one positive LE is labeled CH. Meanwhile, the stable point attractor labeled SP has a larger distribution, and divergent behavior is observed and labeled DI. For periodic behaviors with zero maximum LE, labeled P02, P04, P06, P08, QP, and MP are defined as period 2, period 4, period 6, period 8, quasi-periodic, and multi-periodic attractors, respectively, and these periodic windows are also filled with different color blocks, indicating that the two maps enter chaos via period-doubling bifurcation paths. Comparing Figures 4(a) and (b), there are significant differences between the distributions of dynamic behaviors, and the heterogeneous SCDM map has a larger and more irregular domain of dynamic behavior parameters than the isogeneous SSDM map. At the same time, it is obvious that the two-dimensional bifurcation diagrams shown in Figures 4(a) and (b) have clear boundaries, which is consistent with... Figure 3 The stability distributions in (a) and (b) are consistent, thus proving the validity of the theoretical analysis.

[0144] Figure 4 shows the distribution of dynamic behavior on the k1-k2 plane induced by coupling strength for (x0, y0, z0) = (0.1, 0.1, 0.1). (a) Two-dimensional bifurcation diagram of SSDM mapping; (b) Two-dimensional bifurcation diagram of SCDM mapping; (c) Two-dimensional maximum Lyapunov exponent spectrum of SSDM mapping; (d) Two-dimensional maximum Lyapunov exponent spectrum of SCDM mapping;

[0145] To accurately measure different dynamic behaviors, Figures 4(c) and (d) plot two-dimensional LE exponent spectra based on the first LE value corresponding to each parameter set (k1, k2), where darker colors indicate larger LE values. Comparing Figures 4(a, b) and (c, d), it can be seen that the LE exponent spectra accurately characterize the dynamic behavior of the two-dimensional bifurcation graph, proving that the coupling strength can induce complex dynamic effects in SSDM and SCDM mappings. Most importantly, the parameter domain of the resulting dynamic behavior can be directly obtained from Figure 4.

[0146] 2.2 Evaluation of Local Behavior and Attractors

[0147] To further understand the evolution mechanism of coupling strength-induced attractors, the one-dimensional bifurcation process controlled by a single parameter is emphasized. This is because the statistical characteristics of the bifurcation point can intuitively show the evolution of dynamic behavior within the parameter range. For this purpose, following the dynamic behavior distribution shown in Figure 4, a detailed discussion is conducted using the single-parameter bifurcation graph generated by k1-induced SSDM and SCDM mappings as an example, with the initial conditions (x0, y0, z0) = (0.1, 0.1, 0.1), as shown in Figure 5.

[0148] Figure 5 shows the single-parameter bifurcation diagram induced by k1 for (x0, y0, z0) = (0.1, 0.1, 0.1). (a) SSDM mapping with k2 = 1; (b) two-dimensional bifurcation diagram with SCDM mapping with k2 = -1.5.

[0149] When k2 = 1 is fixed, Figure 5(a) shows that k1 induces chaotic and hyperchaotic behavior in the SSDM map through complete period-doubling bifurcation in the interval [0.4, 0.94]. As k1 changes, the SSDM map transitions to a period 2 bifurcation at a stable point at k1 = 0.477, then splits into a period 4 bifurcation at k1 = 0.533, followed by a period 8 bifurcation in the interval [0.667, 0.697]. Observing the LE exponent spectrum at the top of Figure 5(a), these periodic bifurcations all have zero maximum Lyapunov exponents, indicating that the bifurcations are quasi-periodic. However, the chaotic behavior is located at k1 ∈ [0.697, 0.726] ∪ [0.858, 0.932], which shows a positive LE1 in the LE exponent spectrum. For k1∈[0.726,0.784]∪[0.808,0.858], hyperchaotic behavior with positive LE1 and LE2 can be observed, and a narrow periodic window is distributed in k1∈[0.784,0.808]. Unlike the SSDM mapping, when k2=-1.5, the positive and negative period-doubling bifurcation processes of the SCDM mapping described in Figure 5(b) are distributed in k1∈[-1.3,1.3], and the bifurcation points of the variable x are symmetric because the cosine function is an even function and leads to conditional symmetry about k1. Observations show that there are similar dynamic behaviors to the SSDM mapping, such as symmetric periodicity, chaos, and hyperchaos, which are consistent with the parameter domain of the dynamic behavior described in Figure 4(b). Therefore, the two mappings under consideration can produce complex coupling strength-dependent dynamic evolution.

[0150] According to Figure 5, the topology and evolution of the attractor under the influence of typical parameters can be... Figure 6 Visualization. With initial conditions (x0, y0, z0) = (0.1, 0.1, 0.1), and k2 fixed at 1 and –1.5 for the SSDM and SCDM mappings respectively, as k1 increases, the two mappings produce different attractors with different types, topologies, and spatial locations. For attractor performance evaluation, LE, spectral entropy (SE), permutation entropy (PE), C0 complexity, and correlation dimension (CorDim) were calculated, and the comparative results are summarized in Table 1. From the LEs in Table 1, it can be seen that the maximum LE of the periodic attractor is 0, the chaotic attractor has one positive LE, and the hyperchaotic attractor has two positive LEs. Furthermore, by comparing the values ​​of SE, PE, C0, and CorDim, it can be seen that the hyperchaotic attractor has the highest randomness and complexity, while the periodic attractor has the lowest dynamic performance. Therefore, the coupling strength-dependent SSDM and SCDM mappings not only exhibit complex dynamic evolution processes but also produce attractors with different dynamic performances, including high-performance hyperchaotic and chaotic attractors.

[0151] in, Figure 6 For (x0, y0, z0) = (0.1, 0.1, 0.1), k1 induces various dynamic behaviors, where the upper phase point represents the SSDM mapping with k2 = 1 and the lower phase point represents the SCDM mapping with k2 = -1.5. (a) k1 = 0.6 and k1 = 0.5, period 4; (b) k1 = 0.68 and k1 = 0.75, period 8; (c) k1 = 0.78 and k1 = 0.96, hyperchaos; (d) k1 = 0.8 and k1 = 0.99, period 6; (e) k1 = 0.84 and k1 = 1.08, hyperchaos; (f) k1 = 0.9 and k1 = 1.2, chaos.

[0152] Table 1 Performance metrics of SSDM and SCDM mappings for generating various attractors

[0153]

[0154] 3. Bifurcation and Coexistence of Initial Value Control in Memristors: Extreme Multistability

[0155] Sine or cosine memristors are often introduced into continuous systems to achieve initial value displacement control characteristics, enabling non-destructive control of the offset behavior in memristor-involved systems. Importantly, studying the mechanism of offset control and its coexistence multistability in memristor mappings is of significant research importance. This section mainly investigates the flexible switching dynamics of memristor initial value dependence in SSDM and SCDM mappings from theoretical and numerical simulation perspectives, including the coexistence behavior mechanism of initial value control, parametric bifurcation, parametric-free bifurcation, and extreme multistability.

[0156] 3.1 Mechanism of coexistence behavior of memristor initial value control

[0157] The SSDM and SCDM mappings constructed by paralleling two discrete memristors with periodic memleaders can generate novel coexisting extreme multistabilities. More notably, this phenomenon can be modulated by the initial value of the memristors. To further reveal the mechanism of the coexisting behavior of memristor initial value modulation, assuming μ = x0 and η = y0, the SSDM and SCDM mapping models (4) and (5) can be transformed according to the iterative mechanism.

[0158]

[0159] in and

[0160] The key point is that the two memristor initial conditions μ and η can be decomposed into even and even parts and rewritten as:

[0161]

[0162] Where m and n are arbitrary natural numbers; μ0 and η0 are y n and z n The non-even part extracted from the data must meet the following constraints:

[0163]

[0164] Furthermore, substituting equation (13) into the sine and cosine periodic derivative functions in equation (12) yields an expression for the SSDM mapping, described as follows:

[0165]

[0166] The expression for mapping to SCDM is described as follows:

[0167]

[0168] Therefore, the SSDM and SCDM mappings exhibit periodic cyclic characteristics, i.e., memristor initial value control performance, which has been verified. Specifically, Equation (13) shows that the initial conditions μ and η of the sine and cosine memristors have a period of 2, resulting in non-destructive control of the offsets of the variables in the two mappings along the y and z axes. It is worth noting that the memristor initial value control process does not affect the offset of the variable x, which can be proven by the topological invariance of the SSDM and SCDM mappings after periodic initial value transformations, as described by Equations (15) and (16). Since the offset is changed simultaneously in the yz(μ-η) plane, it can be concluded that the SSDM and SCDM mappings can produce planar coexistence dynamics of memristor initial value control. However, in order to fully and thoroughly reveal the evolution mechanism of coexistence dynamics, the case of using a single memristor initial value as the control parameter is first considered.

[0169] 3.2 Extreme Multistability Depending on Initial Values

[0170] The coupling strength-dependent hyperchaotic dynamics in Section 2 successively exhibit periodic, quasi-periodic, chaotic, and hyperchaotic behaviors. However, the initial condition-dependent coexistence behavior is also worth exploring in both SSDM and SCDM mappings because the long-term trajectories of the mappings are highly sensitive to initial conditions. For this purpose, two-dimensional / three-dimensional local attraction basins with respect to the initial conditions (x0,μ,η) are used to detect the initial value attraction domains to which the long-term dynamic trajectories of the two mappings belong and to characterize the extreme multistability dependent on the initial value.

[0171] Without considering the switching period of the memristor initial value, the two mappings adopt... Figure 7 The coupling strength values ​​used and the non-memristor initial conditions are restricted to the interval x0∈[-2,2]. When the initial conditions of the SSDM-mapped memristor vary in μ=η∈[-1,1], a three-dimensional local attraction basin in the μ-η-x0 space is plotted. Figure 7 In (a), it is also obtained by the LEs and period number determination method. Similarly, the three-dimensional local attraction basin mapped by SCDM is as follows: Figure 7 As shown in (b). As we can see, Figure 7 The different colored regions comprehensively demonstrate the dependence and sensitivity of the dynamic behavior of the two mappings to initial conditions, indicating the occurrence of coexistence. Since there are infinitely many initial conditions in the attractor basin, the coexistence of infinitely many attractors can be confirmed.

[0172] in, Figure 7 For initial value-dependent extreme multistability and coexistence attractors: (a) 3D local attractor basin of SSDM mapped on μ-η-x0 space when (k1,k2)=(0.84,1); (b) 3D local attractor basin of SCDM mapped on μ-η-x0 space when (k1,k2)=(1.08,-1.5); (c) 2D local attractor basin of SSDM mapped on x0-μ plane with η=0.1 in (a) as the cross section; (d) coexistence attractor generated by SSDM mapping depending on the memristor initial value μ with (x0,y0,z0)=(0.1,μ,0.1) and (k1,k2)=(0.84,1) fixed.

[0173] To further illustrate the coexistence of an infinite number of attractors, we take SSDM mapping as an example and select... Figure 7 In (a), the η = 0.1 section, therefore, we can obtain the local attraction basin on the x0-μ plane, such as Figure 7 As shown in (c), observations reveal that the dynamics of the SCDM mapping exhibit strong robustness to the non-memristor initial condition x0. This is because, when μ is fixed, the dynamic behavior remains unchanged with the adjustable x0. However, under the influence of the memristor initial condition, the SCDM mapping successively generates a stable point (μ = 1), period 2 (μ = 0.5), period 4 (μ = 0.4), period 6 (μ = 0.25), period 8 (μ = 0.36), chaos (μ = 0.33), and a hyperchaotic attractor (μ = 0.1), demonstrating the coexistence of multiple stability phenomena, which are... Figure 7 (d) Visualization. The same analysis can be performed on η. Importantly, the coexisting attractors have different types, topologies, and spatial locations, so it can be concluded that the infinite number of attractors dependent on the memristor initial conditions in the SSDM and SCDM maps exhibit extreme multistability.

[0174] 3.3 Bifurcation of Coexisting Parameters in Memristor Initial Value Control

[0175] As the coupling strength with specific initial values ​​changes, the SSDM and SCDM maps exhibit a process of evolution from period-doubling bifurcation into chaos and hyperchaos. However, whether the bifurcation process related to coupling strength can produce corresponding coexistence offset regulation phenomena in spatial location under the influence of multiple sets of periodic initial values ​​warrants further investigation.

[0176] Figure 8 shows the coexistence parameter bifurcation of memristor initial value control when (x0, y0, z0) = (0.1, μ, η) and μ = η = 0.1 + 2m (m = 0, ±1, ±2). (a) SSDM mapping when k1 ∈ [0.4, 0.94] and k2 = 1; (b) SSDM mapping when k1 ∈ [-1.3, 1.3] and k2 = -1.5;

[0177] For the above purposes, the control parameters and initial conditions of the SSDM mapping are set to k1∈[0.4,0.94], k2=1 and (x0,y0,z0)=(0.1,μ,η), where μ and η satisfy μ=η=0.1+2m (m=0,±1,±2). Therefore, the coexistence parameter bifurcation diagram with respect to the coupling strength of y and z can be obtained through numerical simulation, as shown in Figure 8(a), where the five bifurcation points correspond to the dynamic evolution of different initial values. Figure 8(a) clearly shows that the spatial position of the positive period-doubling bifurcation point in the SSDM mapping is lifted in the vertical direction by the initial value of the memristor, but the bifurcation process and structure are not affected by the initial value of the memristor. Similarly, the coexistence parameter bifurcation of the SCDM mapping is plotted in Figure 8(b), where k1∈[-1.3,1.3], k2=1 and the same initial conditions as in Figure 8(a). Subsequently, the symmetrical forward and reverse bifurcation structure of the SCDM mapping is precisely replicated by the initial value of the memristor and distributed in Figure 8(b) with an amplitude offset of 2. Observing Figure 8, it can be found that the bifurcation structures with respect to y and z in the two mappings are the same, because the two discrete memristors in equations (4) and (5) have the same internal state.

[0178] In summary, Figure 8 confirms that periodically switched memristor initial values ​​can cause bifurcation of offset control with respect to y and z in both SSDM and SCDM mappings, consistent with the theoretical analysis in Section 3.1. Furthermore, as the memristor initial values ​​increase infinitely, the two mappings will exhibit coexisting extreme parameter bifurcations with initial value control, greatly enhancing the flexibility of the mappings and the flexibility of the coupling strength-dependent dynamic behavior.

[0179] 3.4 Initial-value-dependent bifurcation of memristors with offset control

[0180] In fact, extreme multistability can be observed by a one-dimensional bifurcation diagram that varies with the initial conditions of memristor, and can simultaneously show the shift modulation of attractors with extreme multistability by SSDM and SCDM mappings. This is an interesting phenomenon that has not yet been studied in depth in memristor mapping.

[0181] in, Figure 9 The extreme multistable bifurcation of the memristor initial value control includes a one-dimensional bifurcation diagram, LEs spectrum, and mean curve. (a) One-dimensional bifurcation diagram of the SSDM mapping as μ varies with (k1,k2)=(0.84,1) and (x0,y0,z0)=(-0.5,μ,0.1); (b) One-dimensional bifurcation diagram of the SSDM mapping as η varies with (k1,k2)=(1.08,-1.5) and (x0,y0,z0)=(0.1,0.1,η); (c) LEs spectrum and mean curve for different μ for the SSDM mapping; (d) LEs spectrum and mean curve for different η for the SCDM mapping.

[0182] Suppose the initial conditions of two mappings are set as (x0, y0, z0) = (-0.5, μ, 0.1) and (x0, y0, z0) = (0.1, 0.1, η). When (k1, k2) = (0.84, 1) and (k1, k2) = (1.08, -1.5), the bifurcations of the two mappings as a function of μ and η are as follows: Figure 9 The top descriptions of (a) and (b) show that the extreme multistable bifurcation sets (BS) within each period are represented by different color depths, labeled EMBS-2, EMBS-1, EMBS, EMBS+1, and EMBS+2, respectively. It can be seen that as μ and η change, the bubble-shaped period-doubling bifurcation generated by the SSDM mapping and the forward and reverse period-doubling bifurcation generated by the SCDM mapping exhibit an infinite number of different attractors coexisting, i.e., extreme multistable states. Since the offset of the step-like bifurcations with respect to y and z and the switching period of the initial conditions are 2, it is proven that the extreme multistable bifurcations can be controlled by the memristor initial value, consistent with the theoretical analysis in Section 3.1. It is worth noting that from... Figure 9 In (a) and (b) and at the bottom, we can see that the extreme multistable bifurcations of other states are replicated infinitely without any shift.

[0183] in addition, Figure 9 (c) and (d) show the mean values ​​of the LEs and state trajectories for memristor initial value regulation, distributed at the top and bottom of the graphs, respectively. As can be seen, the LEs of the two maps switch with a period of 2, but the distribution of LEs corresponding to different periods remains unchanged, meaning that the memristor initial value does not affect the LEs during period switching. Regarding the mean of the long-term trajectory, Figure 9The mean(y) and mean(z) curves in (c) and (d) show a gradually increasing trend, which is caused by the offset of the memristor's initial value control state. Similarly, the mean for the uncontrolled state does not change throughout the initial value domain, and the mean is also different for different dynamic behaviors.

[0184] In summary, the above analysis shows that each extreme multistability bifurcation in the SSDM and SCDM mappings can produce a fixed offset switching under memristor initial value control, and the state variables not involved in the control remain unaffected. Meanwhile, the SSDM and SCDM mappings have different bifurcation structures, leading to differences in the coexistence extreme multistability of memristor initial value control.

[0185] 3.5 Coexistence Line Attractor for Memristor Initial Value Adjustment

[0186] To demonstrate the coexistence line attractor with offset control induced by the initial value of the memristor, Figure 10 First, the local attraction basins on the μ-x0 and η-x0 planes generated by SSDM and SCDM mappings are given, which are divided into local attraction basins LBA-2, LBA-1, LBA, LB1+1, and LBA+2 based on the offset of the memristor initial value or an interval with a period of 2. Figure 10 Hierarchical attraction domains can be observed along the μ or η direction, and extreme multistability behavior exists in each LBA, which is rendered with different color depths, which is consistent with... Figure 9 The extreme multistability bifurcations in the two local attracting basins are consistent. Comparing the two local attracting basins clearly shows that the SSDM map has richer extreme multistability than the SCDM map. This is because the initial domains of various coexisting behaviors in the SSDM map are much larger than those in the SCDM map. Next, in... Figure 10 After selecting typical parameters (μ, η), the two mappings produce a variety of offset-controlled line coexistence attractors with an offset of width 2 between adjacent attractors, as shown in Figures 11(a1-a2) and (b1-b2). The coexistence of period 2 (P02), period 4 (P04), hyperchaotic (HCH), and chaotic (CH) attractors is visualized in Figures 11(a1) and (b1), while the coexistence of the stable point (SP), period 6, and period 8 is described in Figures 11(a2) and (b2). Of course, various combinations of attractors can achieve coexistence, leading to an infinite number of coexistence modes, which has not yet been addressed in most existing research on memristor mappings.

[0187] in, Figure 10For the local attractor basins controlled by the initial memristor values ​​on the μ-x0 and η-x0 planes. (a) When (k1,k2)=(0.84,1) and (x0,y0,z0)=(0.1,μ,0.1), the local attractor basin of the SSDM mapping has μ∈[-5,5]; (b) When (k1,k2)=(1.08,-1.5) and (x0,y0,z0)=(0.1,0.1,η), the local attractor basin of the SCDM mapping has η∈[-5,5].

[0188] Figure 11 shows the coexistence line attractors for memristor initial value control, including the stable point, period 2, period 4, period 6, period 8, chaotic, and hyperchaotic attractors. (a1-a2) Coexistence attractors generated by different μ-induced SSDM mappings when (k1,k2)=(0.84,1) and (x0,y0,z0)=(0.1,μ,0.1), where μ=μ0+2m (m=0±1,±2); (b1-b2) Coexistence attractors generated by different η-induced SCDM mappings when (k1,k2)=(1.08,-1.5) and (x0,y0,z0)=(0.1,0.1,η), where η=η0+2n (n=0±1,±2).

[0189] 3.6 Planar Coexistence Behavior of Memristor Initial Value Regulation

[0190] Interestingly, the planar coexistence extreme multistability caused by the combined effect of memristor initial values ​​can be infinitely replicated on both the μ and η axes simultaneously. This differs from the research in the previous subsection and is rarely found in existing memristor mappings. To reveal this interesting planar coexistence extreme multistability, the stability distributions of the SSDM and SCDM mappings on the μ-η plane are visualized in Figures 12(a) and (b) according to the critical stability condition characterized by Equation (10), where the initial conditions are (x0, y0, z0) = (0.1, μ, η), μ ∈ [-3, 3] and η ∈ [-3, 3]. As expected, the yellow unstable region induced by the memristor initial values ​​produces a shift of width 2 in both the horizontal and vertical directions, proving that the planar stability distributions of the two mappings obey the initial value regulation mechanism. Due to the symmetry of the sine and cosine functions, two unstable regions satisfying |λ3| > 1 can be generated within one period. Figures 12(a) and (b) show that a total of 2 × 3 × 3 = 18 coexisting unstable regions are generated simultaneously on the μ-η plane.

[0191] Similarly, the extreme multistability of the planar coexistence of the two mappings can also be characterized using local attractor basins, as shown in Figures 12(c) and (d). Unsurprisingly, approximately 18 separate coexisting local attractor basins were generated, confirming the correctness of the stability distribution. Importantly, each local attractor basin contains multiple coexisting dynamic behaviors, implying that each local attractor basin is extremely multistable. Simultaneously, each local attractor basin infinitely replicates itself on the μ and η axes with an offset of width 2, resulting in any dynamic behavior being modulated by memristor initial values ​​to produce planar coexistence. Furthermore, the red hyperchaotic attractor occupies a larger region in the local attractor basin of the SSDM mapping, while a larger divergence region exists in the local attractor basin of the SCDM mapping, indicating that the extreme multistability of the SSDM mapping is richer and that the planar coexistence behaviors of the two mappings are fundamentally different. To further investigate the dynamic performance of modulating planar behavior, the normalized spectral entropy based on Fourier transform was estimated, and the results were visualized using 2-D plots, as shown in Figures 12(c) and (d). The results show that the more complex the dynamic behavior, the larger the SE value and the darker the corresponding color. For example, the hyperchaotic attractor has a large SE, while the stable point corresponds to the lowest SE. At the same time, the distribution of SE complexity also exhibits periodic cyclic characteristics, which means that the performance of the dynamic behavior does not change during the control process, i.e., it has lossless control characteristics.

[0192] Figure 12 shows the stability distribution and local attraction basin on the μ-η plane for memristor initial value control when (x0, y0, z0) = (0.1, μ, η), with μ = η ∈ [-3, 3]. (a) Stability distribution of SSDM mapping for (k1, k2) = (0.84, 1); (b) Stability distribution of SCDM mapping for (k1, k2) = (1.08, -1.5); (c) Local attraction basin of SSDM mapping; (d) Local attraction basin of SCDM mapping.

[0193] Figure 13 shows various planar coexistence attractors for memristor initial value control when μ = η ∈ [-3, 3], where (x0, y0, z0) = (0.1, μ, η), μ = μ0 + 2m (m = 0 ± 1, ± 2), and μ = μ0 + 2m (m = 0 ± 1, ± 2). (a1-a2) Planar coexistence attractor generated by SSDM mapping when (k1, k2) = (0.84, 1); (b1-b2) Planar coexistence attractor generated by SCDM mapping when (k1, k2) = (1.08, -1.5).

[0194] Therefore, when μ = μ0 + 2m (m = 0, ±1) and η = η0 + 2n (n = 0, ±1), after selecting typical parameters in Figure 12, the SSDM and SCDM mapping planar coexistence attractor shown in Figure 13 can be constructed, with μ0 and η0 of the SSDM and SCDM mappings being the same as the initial values ​​marked in Figure 11. To avoid confusion, two planar coexistence modes are given here: the coexistence of period 2, period 4, chaotic, and hyperchaotic attractors shown in Figure 13 (a1, b1), and the coexistence of stable point, period 6, and period 8 attractors shown in Figure 13 (a2, b2). Figure 13 clearly demonstrates the periodic cyclic characteristics of various attractors induced by the memristor initial value, which is similar to Figure 11.

[0195] The numerical simulations presented in this section effectively reveal the planar periodic cyclic characteristics of the memristor's initial conditions. Its control dynamics distribution, local attractor basin, SE diagram, and phase diagram show obvious dynamic components with fixed offsets. Furthermore, by adjusting μ and η, various heterogeneous attractors in the coexistence behavior were discovered, such as periodic, chaotic, and hyperchaotic attractors, which exhibit strange extreme multistability under the control of the memristor's initial conditions. Therefore, the essential characteristic of extreme multistability in memristor initial condition control in SSDM and SCDM mappings is confirmed: an infinite number of heterogeneous attractors coexist with fixed offsets.

[0196] The numerical simulations shown in Figures 12 and 13 effectively reveal the coexistence of planar extreme multistability and infinite attractors in memristor initial value modulation during SSDM and SCDM mappings. This significantly improves the flexibility of the mappings and enables lossless switching control with extreme multistability. In short, the planar coexistence behavior of initial value modulation generated by SSDM and SCDM mappings has specific research significance and practical value for mapping-based engineering applications.

[0197] 4 Hardware Implementation and Application in Pseudorandom Number Generators

[0198] This section presents an experiment demonstrating the implementation of SSDM and SCDM mappings using a microcontroller-based digital circuit. Furthermore, considering practical applications, a pseudo-random number generator is designed based on the hyperchaotic sequences generated by the two mappings and rigorously tested.

[0199] 4.1 Implementation of Digital Circuits Based on Microcontrollers

[0200] The iterative mechanism of discrete mappings is well-suited to the operation of microcontrollers, and digital circuits play a crucial role in various fields. Therefore, we developed a digital circuit experimental platform based on microcontroller-based SSDM and SCDM mappings, simultaneously verifying the effectiveness of numerical simulations and generating corresponding dynamic behaviors. First, the differential models and related parameters of the two mappings were written into C language programs and executed in the microcontroller. Second, the digital signals generated by the microcontroller were converted into analog voltages corresponding to the state variable sequence by a DAC module, which were then captured by an oscilloscope. Finally, the digital hardware experimental platform used to verify the SSDM and SCDM mappings is as follows: Figure 14 As shown.

[0201] in, Figure 15 The results are experimental findings of a microcontroller-based digital hardware circuit. (a) Attractors of SSDM and SCDM mappings, where (a1) is a periodic 8 attractor, (a2, a3) are hyperchaotic attractors, and (a4) is a chaotic attractor; (b) Initial value-controlled dynamics of SSDM and SCDM mappings, where (b1) and (b2) are linear coexistence behaviors controlled by μ and η, respectively, and (b3, b4) are planar coexistence behaviors controlled by both μ and η.

[0202] The experiments described were performed using a high-performance STM32F407VET6 development board with a 32-bit ARM Cortex-M4 processor, a 168MHz clock speed, 512KB embedded Flash, and 192KB embedded SRAM. A dual-channel 16-bit DAC8563 module was selected as the digital-to-analog converter to obtain the analog voltage of the hyperchaotic sequence, with its range set to 0–10V. A YOKOGAWADL850E oscilloscope connected via an RF cable was used to display the output voltage.

[0203] After defining the conversion scale between digital values ​​and iterative values, the state of the SSDM and SCDM mappings after performing the above experiment is displayed on the oscilloscope as follows: Figure 15 As shown.

[0204] By comparing Figure 13 and Figure 6 (a) The oscilloscope accurately captured the hyperchaotic phenomena in the four mappings, thus proving the feasibility of the dual memristor coupled mapping based on digital hardware implementation, and further demonstrating the effectiveness of the numerical method and the practicality of the proposed mapping.

[0205] 4.2 Design and Testing of Pseudo-random Number Generator

[0206] Discrete mappings, with their simple algebraic structure and low computational complexity, are widely used in various industrial fields. For example, the high-performance chaotic and hyperchaotic sequences they generate are used to design pseudo-random number generators, thereby improving the performance of secure communication. Considering that SSDM and SCDM mappings can generate complex dynamic behaviors and high-performance hyperchaotic behaviors, we designed two pseudo-random number generators to demonstrate the highly random performance and application value of these two mappings.

[0207] Based on the typical parameters in Table 1, the excellent hyperchaotic sequences generated by SSDM and SCDM mappings are defined as X, respectively. SSDM ={x(1),x(2),…,x(n),…} and X SCDM ={x(1),x(2),…,x(n),…}. First, the IEEE 754 binary floating-point standard is used to evaluate X. SSDM and X SCDM Preprocessing, in which each x(n) is converted into a standard 52-bit binary number x B (n). Next, randomly select x. B The 8 significant binary bits in (n) form the pseudo-random number. Here, we choose bits 37 to 44, thus obtaining the designed pseudo-random number generator, denoted as:

[0208]

[0209] To further test P SSDM and P SCDM The generated pseudo-random numbers exhibit high randomness. According to the NIST SP800-22 technical guidelines, the SSDM and SCDM mappings are iterated 15,000,000 times to generate a hyperchaotic sequence X. SSDM and X SCDM And according to equation (17), it is transformed into 120 groups of length 10. 6 test samples and

[0210] To evaluate the randomness performance of pseudo-random numbers, 15 rigorous tests were performed using the NIST SP800-22 test suite. For this purpose, the pass rate and p-value were measured. T These are two explicit indicators that determine whether a test sample meets the high randomness of pseudo-random numbers. The pass rate statistically reflects the number of test samples whose corresponding P-values ​​pass the randomness test, and is calculated as follows:

[0211]

[0212] in The significance level is α, typically 0.01. Additionally, the uniform distribution characteristic of random numbers is determined by the χ² value in 10 equal intervals from 0 to 1 for the p-value. 2 The distribution reflects, and χ 2 Calculated as

[0213]

[0214] Where F i Let be the number of P-values ​​in interval i. A P-value used for evaluation is represented as P-value. T =igamc(9 / 2,χ 2 / 2). Therefore, when m = 120, if the minimum pass rate calculated by equation (18) is greater than 0.9628 and the P-value T A p-value ≥ 0.0001 indicates that the generated pseudo-random numbers pass the rigorous test of NIST SP800-22 and have a uniform distribution, indicating high randomness. Similarly, according to the criteria in [reference needed], the minimum pass rate and p-value are achieved when the confidence level used to calculate the statistical error is set to 0.01 and the PRN test sample size is 120. T The values ​​are 0.9628 and 0.0001, respectively.

[0215] At this time, for P SSDM and P SCDM generated pseudo-random numbers and The test results are summarized in Table 2, where an asterisk indicates the average value of that test. As we can see, both sets of test results meet the criteria of a pass rate ≥ 0.9628 and a p-value. T ≥0.0001 indicates and The system has passed rigorous random testing. Therefore, the pseudo-random numbers corresponding to the hyperchaotic sequences generated by SSDM and SCDM mappings have high randomness and uniform distribution, which can be widely used in various industrial fields such as secure communication.

[0216] Table 2. Test results of NIST SP800-22 on two PRNGs.

[0217]

[0218] In summary, this invention constructs two novel dual-discrete memristor hyperchaotic mapping models—isomorphic SSDM and heteromorphic SCDM—by connecting two discrete memristors with periodic memristor derivatives in parallel. These models possess simple structures yet exhibit complex hyperchaotic dynamics and tunable extreme multistability. When coupling strength is used as a control parameter, parametric bifurcation mechanisms and various attractors with excellent dynamic performance are revealed. Due to the planar fixed points related to the memristor initial values ​​of the considered mappings, coexisting stable points, periodicity, chaos, and hyperchaotic attractors—that is, extreme multistability phenomena—are observed. Particularly noteworthy is that the extreme multistability phenomenon of line-plane coexistence is continuously and non-destructively tunable through the initial values ​​of periodic discrete memristors, and an infinite number of abundant coexisting attractors exhibit flexible self-replication on both the line and plane. To verify the numerical simulation results, a microcontroller-based digital circuit platform was built, accurately capturing the hyperchaotic attractors and coexistence behavior in the mappings. Meanwhile, both pseudo-random number generators designed based on the proposed mapping passed rigorous randomization tests, demonstrating that the excellent hyperchaotic sequences generated by the two mappings possess high randomness and application value. In the future, further applications of the dual memristor discrete mapping and enhancement of chaotic complexity warrant further consideration.

[0219] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0220] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A mapping method based on dual discrete memristors, characterized in that, Includes the following steps: Construct sinusoidal discrete memristors (S-DM) and cosine discrete memristors (C-DM); A three-dimensional dual discrete memristor hyperchaotic mapping model is constructed by parallel coupling of a sinusoidal discrete memristor (S-DM) and a cosine discrete memristor (C-DM) and connecting them through self-feedback. The three-dimensional dual discrete memristor hyperchaotic mapping model includes a three-dimensional sinusoidal-sinusoidal discrete memristor (SSDM) hyperchaotic mapping model and a three-dimensional sinusoidal-cosine discrete memristor (SCDM) hyperchaotic mapping model. The three-dimensional dual discrete memristor hyperchaotic mapping model is written into the digital circuit of the microcontroller. The microcontroller converts the generated digital signal into an analog voltage corresponding to the state variable sequence. The analog voltage is input into the oscilloscope for capture, and the discrete mapping of hyperchaos and extreme multistability is obtained. The method further includes: Verification of the mapping dynamics and extreme multistability of the three-dimensional dual discrete memristor hyperchaotic mapping model, specifically including: Determine the stability distribution of the three-dimensional dual discrete memristor hyperchaotic mapping model; The coupling strength-dependent dynamic behavior distribution and local behavior and attractor evaluation of the three-dimensional dual-discrete memristor hyperchaotic mapping model are determined, and the parameter-dependent hyperchaotic dynamic evolution mechanism of the mapping of the three-dimensional dual-discrete memristor hyperchaotic mapping model is determined. The mechanism of coexistence behavior of initial value regulation, initial value-dependent extreme multistability, coexistence parameter bifurcation of initial value regulation, coexistence line attractor of initial value regulation, and planar coexistence behavior of initial value regulation are determined in the three-dimensional dual discrete memristor hyperchaotic mapping model. In addition, the bifurcation and coexistence extreme multistability of initial value regulation of the three-dimensional dual discrete memristor hyperchaotic mapping model are determined. The sinusoidal discrete memristor S-DM is: The cosine-type discrete memristor C-DM is: In the formula, and Represents the memristor function of a continuous memristor In the The sampled values ​​of the next iteration; , and They are voltages Current and charge In the The sampled values ​​of the next iteration; and It is the first The sampled values ​​of the next iteration; The memristor function of the sinusoidal discrete memristor S-DM is: The memristor function of a cosine-type discrete memristor (C-DM) is: ,when When the three-dimensional dual discrete memristor hyperchaotic mapping model is isomorphic, when... At that time, the three-dimensional dual discrete memristor hyperchaotic mapping model is heterogeneous; when At that time, the isomorphic three-dimensional sinusoidal-sinusoidal discrete memristor SSDM hyperchaotic mapping model is: when and At that time, the heterogeneous three-dimensional sinusoidal-cosine discrete memristor (SCDM) hyperchaotic mapping model is as follows: In the formula, and The coupling strength between two discrete memristors; It is the current of two discrete memristors. , and The internal states of two discrete memristors ; , and They are respectively , and of The value of the next iteration.

2. The mapping method based on dual discrete memristors as described in claim 1, characterized in that, The determination of the stability distribution of the discrete mapping of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: Let the fixed point of the SSDM mapping be... Then at the fixed point The algebraic structure at the location is: Fixed point ,in and Corresponding to the initial conditions of the discrete memristor, since and It can take any constant, so It is an infinite set of fixed points in a plane; For quantitative analysis The stability boundary, SSDM mapping on The Jacobian matrix at point is written as: according to The characteristic equation is derived as follows: Therefore, the eigenvalues ​​are obtained as: because It is constant and lies on the unit circle, therefore, a fixed point. Its stability can only be either critically stable or unstable, which is determined by... The only decision; for Fixed point It is unstable, and for It is critically stable; the SSDM mapping is The value is controlled by the coupling strength and and initial conditions and This indicates that the stability of a fixed point is extremely complex; when and When the constant is constant, the critical stability boundary condition is defined as: Critically stable boundary conditions can be used to determine the stability distribution of a mapping; For SSDM and SCDM mappings, the critical stability boundary conditions are precisely expressed as: 。 3. The mapping method based on dual discrete memristors as described in claim 1, characterized in that, The determination of the coupling strength-dependent dynamic behavior distribution of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: The distribution of dynamic behaviors is characterized by a two-dimensional bifurcation diagram, which further distinguishes and refines different dynamic behaviors; in order to reveal the dynamic evolution mechanism of SSDM and SCDM mappings, coupling strength is... and Selected as a control parameter and with fixed initial conditions When the range of control parameters is and At that time, each set of parameters The oscillation period and Lyapunov exponent of the corresponding trajectory can be calculated using the period number method and the method based on the Wolf Jacobi matrix, respectively, thus obtaining information about... Two-dimensional bifurcation diagram and Lyapunov index spectrum in a plane; In the two-dimensional bifurcation diagram, the various dynamic behaviors generated by the SSDM and SCDM mappings are distinguished according to the period number and Lyapunov exponent and filled with different color blocks: the hyperchaotic attractor with two positive LEs is labeled HCH; the chaotic attractor with only one positive LE is labeled CH; the stable point attractor divergent behavior labeled SP is labeled DI; for the periodic behavior with zero maximum LE, labeled P02, P04, P06, P08, QP and MP are defined as period 2, period 4, period 6, period 8, quasi-periodic and multi-periodic attractors, respectively, and these periodic windows are also filled with different color blocks, indicating that the two mappings enter chaos through period-doubling bifurcation paths; Comparing the two-dimensional bifurcation diagrams, there are significant differences in the distribution of dynamic behavior, and the heterogeneous SCDM mapping has a larger and more irregular dynamic behavior parameter domain than the isomorphic SSDM mapping; moreover, the two-dimensional bifurcation diagrams have obvious boundaries, consistent with the stability distribution, proving the effectiveness of the theoretical analysis. To accurately measure different dynamic behaviors, based on each set of parameters The first corresponding LE value was used to plot a two-dimensional LE exponent spectrum, where darker colors indicate larger LE values. The LE exponent spectrum accurately characterizes the dynamic behavior of the two-dimensional bifurcation graph, proving that the coupling strength can induce complex dynamic effects in the SSDM and SCDM mappings. The parameter domain of the resulting dynamic behavior can be directly obtained from the two-dimensional bifurcation graph.

4. The mapping method based on dual discrete memristors as described in claim 3, characterized in that, The evaluation of the local behavior and attractors of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: The statistical characteristics of bifurcation points can intuitively show the evolution of dynamic behavior within the parameter range. Therefore, a one-dimensional bifurcation process controlled by a single parameter can be used to determine the coupling strength-induced attractor evolution mechanism; following the distribution of dynamic behavior, when the initial conditions are... At that time, with Single-parameter bifurcation graph generated by induced SSDM and SCDM mapping; According to the single-parameter bifurcation diagram, when fixed hour, In the interval Internally induced SSDM mapping generates chaotic and hyperchaotic behavior through complete period-doubling bifurcation; with Changes, SSDM mapping At the point where a stable point transitions to a period 2 bifurcation, and at... The bifurcation splits into a periodic 4-bifurcation, followed by a periodic 8-bifurcation. Within the interval; according to the LE exponent spectrum, periodic bifurcation has a zero maximum Lyapunov exponent, indicating that the bifurcation is quasi-periodic; chaotic behavior is located in It shows a positive LE1 in the LE index spectrum; for Hyperchaotic behavior with positive LE1 and LE2 can be observed, and narrow periodic windows are distributed in In the middle; unlike SSDM mapping, when At that time, the forward and reverse period-doubling bifurcation processes of the SCDM mapping are distributed in In, and variables The bifurcation point is symmetric, the cosine function is an even function and leads to the following about The condition is symmetric; SCDM mapping and SSDM mapping can produce complex coupling strength-dependent dynamic evolution; When the initial condition is And for SSDM and SCDM mapping, When fixed at 1 and –1.5 respectively, as As the coefficients increase, the two mappings produce different attractors with different types, topologies, and spatial locations. For attractor performance evaluation, the LE (Leadership), spectral entropy (SE), permutation entropy (PE), C0 complexity, and correlation dimension (CorDim) are calculated and compared. The periodic attractor has a maximum LE of 0, the chaotic attractor has one positive LE, and the hyperchaotic attractor has two positive LEs. Comparing the values ​​of SE, PE, C0, and CorDim reveals that the hyperchaotic attractor has the highest randomness and complexity, while the periodic attractor has the lowest dynamic performance. The coupling strength-dependent SSDM and SCDM mappings not only exhibit complex dynamic evolution processes but also produce attractors with different dynamic performances, including high-performance hyperchaotic and chaotic attractors.

5. The mapping method based on dual discrete memristors as described in claim 1, characterized in that, The mechanism for determining the coexistence behavior of initial value regulation in the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: set up and The SSDM and SCDM mapping models are transformed into the following based on the iterative mechanism: in , ; Initial conditions of two memristors and It is decomposed into even and even parts and rewritten as: in and For any natural number, and yes and The following constraints must be satisfied for the non-even portion extracted from the given data: The expression for SSDM mapping is described as follows: The expression for SCDM mapping is described as follows: Initial conditions for sine and cosine memristors and The period is 2, causing the offset of the variables in the two mappings to be in... and The axis is non-destructively controlled; the initial value adjustment process of the memristor does not affect the variable. The offset is proven by the topological invariance of the SSDM and SCDM mappings after the periodic initial value transformation; since simultaneously in ( The offset of the plane is changed, so the SSDM and SCDM mapping can produce planar coexistence dynamics with memristor initial value control; in order to reveal the evolution mechanism of coexistence dynamics, we first consider the case of a single memristor initial value as the control parameter. The determination of the extreme multistability of the initial value dependence of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: Using initial conditions Two-dimensional / three-dimensional local attraction basins are used to detect the initial value attraction domain to which the long-term dynamic trajectories of two maps belong and to characterize the extreme multistability dependent on the initial value. Without considering the switching period of the memristor initial value, the two mappings use a fixed coupling strength value and the non-memristor initial conditions are restricted to... Within the interval; when the initial conditions of the SSDM mapped memristor are in When changes occur, The three-dimensional local attraction basin in space was plotted using the LEs and period number determination method; the three-dimensional local attraction basin of the SCDM mapping was plotted using the same method; different color regions comprehensively demonstrate the dependence and sensitivity of the dynamic behavior of the two mappings on the initial conditions, indicating the occurrence of coexistence; since there are infinitely many initial conditions in the attraction basin, the coexistence of infinitely many attractors can be confirmed. To illustrate the coexistence of an infinite number of attractors, we will take SSDM mapping as an example and select... cross section, obtain Local attraction basins on a plane, the dynamics of SCDM mapping for non-memristor initial conditions It has strong robustness because when fixed At that time, the dynamic behavior changes with the adjustable No change was observed; under the induction of the memristor initial conditions, the SCDM mapping successively generated a stable point, period 2, period 4, period 6, period 8, chaotic and hyperchaotic attractors, proving and visualizing the coexisting multistability phenomenon; the coexisting attractors have different types, topological structures and spatial locations, so the infinite number of attractors on which the memristor initial conditions depend in the SSDM and SCDM mappings have extreme multistability.

6. The mapping method based on dual discrete memristors as described in claim 1, characterized in that, The coexistence parameter bifurcation for determining the initial value modulation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: The control parameters and initial conditions for SSDM mapping are set as follows: , and ,in and satisfy Numerical simulations yielded information about... and The coexistence parameter bifurcation diagram depends on the coupling strength, where five sets of bifurcation points correspond to the dynamic evolution of different initial values. In the SSDM mapping, the spatial position of the positive period-doubling bifurcation point is lifted vertically by the initial value of the memristor, but the bifurcation process and structure are not affected by the initial value of the memristor. Similarly, the coexistence parameter bifurcation diagram of the SCDM mapping is obtained, where... , Furthermore, the same initial conditions as the SSDM mapping are set; the symmetrical forward and reverse bifurcation structure of the SCDM mapping is precisely replicated by the memristor initial value, and distributed in the figure with an amplitude offset of 2; thus, the relationship between the two mappings is obtained. and The bifurcation structures are the same because the two discrete memristors have the same internal state; The bifurcation diagram of the coexistence parameters of SSDM and SCDM mappings confirms that the periodically switched memristor initial value enables SSDM and SCDM mappings to simultaneously generate parameters related to SSDM and SCDM mappings. and The bifurcation phenomenon of offset control; as the initial value of the memristor increases infinitely, the two mappings will produce coexisting extreme parameter bifurcation of initial value control, which greatly improves the flexibility of the mapping and the flexibility of the dynamic behavior dependent on the coupling strength. The determination of the initial value-dependent bifurcation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: Set the initial conditions for the two mappings as follows: and ,when and At that time, the two mappings follow and The variational bifurcation, where the extreme multi-stability bifurcation set (BS) within each period is represented by different color depths and labeled as EMBS-2, EMBS-1, EMBS, EMBS+1, and EMBS+2, respectively; as... and The changes show that the bubble-shaped period-doubling bifurcation generated by the SSDM mapping and the forward and reverse period-doubling bifurcation generated by the SCDM mapping exhibit an infinite number of different attractors coexisting; due to the... and The offset of the stepped bifurcation and the switching period of the initial condition are 2, which proves that the extreme multistable bifurcation can be controlled by the initial value of the memristor. The LEs of the two mappings switch with a period of 2. The distribution of LEs corresponding to different periods does not change, indicating that the initial value of the memristor does not affect the LEs during the period switching process. For the mean of the long-term trajectory, the mean(y) and mean(z) curves show a gradual upward trend, which is caused by the offset of the memristor initial value control state. For the mean of the uncontrolled state, it does not change throughout the initial value domain, and the mean corresponding to different dynamic behaviors is also different. Each extreme multistability bifurcation in the SSDM and SCDM mappings can produce a fixed offset switching under memristor initial value control, and the state variables not involved in the control are unaffected; the SSDM and SCDM mappings have different bifurcation structures, resulting in differences in the coexistence extreme multistability of memristor initial value control.

7. The mapping method based on dual discrete memristors as described in claim 1, characterized in that, The coexistence line attractor for determining the initial value modulation of the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: Generate through SSDM mapping and SCDM mapping and A planar local attractor diagram can illustrate the coexistence line attractor with offset control induced by the memristor initial value. It is divided into local attractor basins LBA-2, LBA-1, LBA, LB1+1, and LBA+2, based on the offset of the memristor initial value or an interval of 2 periods. Along the local attractor basin diagram... or The hierarchical attraction domains in the direction and the extreme multistability behavior in each LBA are rendered with different colors; comparing the two local attraction basins, it can be concluded that the SSDM map has richer extreme multistability than the SCDM map, because the initial value domain of various coexistence behaviors in the SSDM map is much larger than that in the SCDM map; the two maps produce a variety of offset-controlled line coexistence attractors, and there is an offset of width 2 between adjacent attractors. The planar coexistence behavior of initial value regulation for determining the three-dimensional dual discrete memristor hyperchaotic mapping model specifically includes: According to the critical stability condition, SSDM and SCDM are mapped onto... The stability distribution on the plane is visualized, where the initial conditions are... , and The yellow unstable region induced by the memristor initial value simultaneously produces a shift of width 2 in both the horizontal and vertical directions, proving that the planar stability distribution of the two mappings obeys the initial value regulation mechanism; due to the symmetry of the sine and cosine functions, two functions satisfying the condition can be generated within one period. The unstable region indicates that simultaneously in Total number generated on the plane A coexisting unstable domain; The extreme multistability of the coexistence of two mappings can be characterized using local attractor basins. Eighteen mutually separate coexisting local attractor basins were generated, proving the correctness of the stability distribution. Each local attractor basin contains multiple coexisting dynamic behaviors, and each local attractor basin exhibits extreme multistability. Each local attractor basin in... and The axis infinitely replicates itself with an offset of width 2, resulting in the coexistence of planar phenomena that can be generated by adjusting the initial value of the memristor for any kind of dynamic behavior; the extreme multistability phenomenon of the SSDM mapping is richer and the planar coexistence behavior of the two mappings is fundamentally different; the normalized spectral entropy based on Fourier transform is estimated, and the results show that the more complex the dynamic behavior, the larger the value of SE and the darker the corresponding color; the distribution of SE complexity also exhibits periodic cyclic characteristics, and the performance of the dynamic behavior does not change during the adjustment process, possessing lossless control characteristics; when and At that time, an attractor with coexistence of SSDM and SCDM mapping planes was constructed to demonstrate the periodic and cyclic characteristics of various attractors induced by the initial value of the memristor.

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