A method for implementing a conservative charge-controlled memcapacitive hyperchaotic circuit of multiple coexisting attractors

Abstract

The application discloses a method for realizing a conservative charge-controlled memcapacitive hyperchaotic circuit with multiple coexisting attractors. First, a six-dimensional conservative memcapacitive circuit model is constructed by a charge-controlled memcapacitive device, and the memcapacitive characteristics are verified. Then, the balance point and dissipation are used to verify the conservativeness. By changing the system parameters and initial conditions, a larger Lyapunov exponent can be obtained, and it is found that the internal parameter e of the circuit can not only cause the circuit to exhibit complex transient chaotic phenomena, but also has large-scale range chaotic characteristics. The attractor basin, coexisting attractors and complexity reflect the extreme multistability and complex nonlinear dynamic behavior of the circuit. In addition, the initial offset boosting behavior is also found. The application combines the FPGA technology to design a digital conservative charge-controlled memcapacitive hyperchaotic circuit model, and the hardware simulation result is consistent with the numerical calculation result, thereby providing a new idea for the digital realization of a nonlinear storage system.

Description

Technical Field

[0001] This invention belongs to the field of digital circuit technology and involves memory containers, chaos theory, and the design and simulation of chaotic circuits. Background Technology

[0002] The concept of memcells was proposed by Chua's research team in 2009, who designed a piecewise linear memcell model. Prior to the memcell concept, there were reports of hysteresis characteristics in nanocapacitors, such as nano-oxide capacitors deposited by ion and electron beams exhibiting hysteresis due to the hysteresis between the voltage and charge across the oxide capacitor. Memcell modeling primarily employs two methods: one is to construct a mathematical model of the memcell to impart hysteresis characteristics; the other is to establish an equivalent circuit of the memcell to realize its memory characteristics and enable its application as a basic component in circuits. The design of the memcell model allows for porting of the device, facilitating its replacement and application in other chaotic circuits to construct chaotic systems with more complex dynamics. Currently, there are some research results in memcell chaos research. Examples include a chaotic system simultaneously incorporating TiO memristors and charge-controlled memcells, a novel charge-controlled memcell chaotic oscillation circuit with multiple stability and coexisting attractors, and a chaotic system constructed using a memcell model analogous to a quadratic curve, etc.

[0003] Compared to dissipative systems, conservative systems represent an important research direction in nonlinear science. Therefore, the design of conservative chaotic circuits has attracted significant attention, with conservative systems incorporating memristors exhibiting substantial advantages and characteristics in nonlinear dynamics. The number of memristor systems with conservative properties is limited, including 4D memristor hyperchaotic conservative systems based on Jerk systems, 3D conservative circuits composed of memristors and capacitors, and conservative memristor circuits exhibiting two different initial offset boosting behaviors. Research on conservative memcapacitive chaotic systems is rarely reported. Traditional methods for analyzing chaotic systems include Lyapunov spectra, bifurcation diagrams, and equilibrium points. For chaotic systems with complex nonlinear dynamic behaviors, methods include complexity analysis and attractor basin analysis. Because conservative chaotic systems are more sensitive to initial conditions, changes in these conditions can reveal different nonlinear dynamic characteristics, such as extreme multistability, initial offset boosting behavior, and transient transition behavior. The multistability of nonlinear dynamic systems implies that the system possesses attractor basins of different shapes; when different initial states are chosen, the system trajectory may be drawn into different attractor basins. With the in-depth research on memcapacitor chaotic circuits, it has become clear that analog memcapacitor circuits are easily affected by the external environment, while FPGA technology offers advantages such as high flexibility, parallel computing, low power consumption, and high reliability. Therefore, designing memcapacitor chaotic circuits using FPGAs is of great significance for practical applications. Currently, various memcapacitor chaotic circuits have been implemented using FPGAs, such as FPGA-based fractional-order memcapacitor chaotic oscillators and FPGA-based non-volatile memcapacitor-assisted hyperchaotic circuits, etc. Summary of the Invention

[0004] The purpose of this invention is to propose a method for realizing a conservative charge-controlled memcapacitor hyperchaotic circuit with multiple coexisting attractors. First, a six-dimensional conservative memcapacitor hyperchaotic circuit is constructed using two capacitors, two inductors, and one charge-controlled memcapacitor. The circuit is verified to be a hyperchaotic system with zero dissipation and two positive Lyapunov exponents. Furthermore, the nonlinear dynamic behavior of the circuit is analyzed. This circuit is highly sensitive to changes in parameter e, exhibiting not only large-scale chaotic characteristics of e∈(-1000, 1000), but also revealing... Two transient transition behaviors were investigated: the system transitions from chaotic state 1 to chaotic state 2 and from chaotic state 3 to chaotic state 4. The chaotic characteristics of the circuit were studied using complexity, coexistence attractors, and the attraction basin method, revealing rich complex dynamic behaviors, such as three types of coexistence attractors (nested coexistence, symmetric nested coexistence, and homomorphic coexistence) and initial offset boosting behavior. Finally, a conservative memcapacitive hyperchaotic circuit model was implemented using FPGA technology. The experimental results are consistent with the numerical simulation results, indicating that this circuit has potential application value in fields such as secure communication and digital circuits.

[0005] The present invention provides a method for implementing a conservative charge-controlled memcapacitive hyperchaotic circuit with multiple coexisting attractors, comprising the following steps:

[0006] (S01): Construct a new load-controlled memory container model and analyze the characteristic curves of the load-controlled memory container;

[0007]

[0008] Where q(t) and u(t) are the charge and voltage of the memory container at time t, respectively; σ is the continuous time integral of charge q(t) passing through the memory container; and φ(σ) is the magnetic flux through the memory container, which is a function of σ. The reciprocal of the capacitance of the memory capacitor, α, β, and e are parameters. The range of values ​​for parameters α, β, and e is determined to ensure that the system circuit characteristics are chaotic.

[0009] (S02): Construct a six-dimensional conservative charge-controlled memcapacitor hyperchaotic circuit composed of the novel charge-controlled memcapacitor, inductor, and capacitor described in (S01), and analyze its nonlinear dynamic behavior. The conservative memcapacitor hyperchaotic oscillating circuit is composed of the charge-controlled memcapacitor C. M It consists of capacitor C1, capacitor C2, inductor L1, and inductor L2. According to Kirchhoff's laws:

[0010]

[0011] Where C1 and C2 are capacitors, V CM V1 is the voltage across capacitor C1, V2 is the voltage across capacitor C2, L1 and L2 are inductors, and i L1 i L2 Let be the inductor current, and α, β, and e be parameters, the values ​​of which are chosen to ensure that the circuit is a hyperchaotic system.

[0012] According to formula (1), V CM =ασcos(βσ)q CM Then we have:

[0013]

[0014] (S021): The conservative charge-controlled memcapacitor hyperchaotic system model is derived from the conservative charge-controlled memcapacitor hyperchaotic circuit in (S02):

[0015]

[0016] Where x = V1, y = V2, z = i L1 u = i L2 v = q CM w = σ CMa, b, c, d, and e are parameters, and the range of values ​​for parameters a, b, c, d, and e is determined to ensure the chaotic characteristics of the system circuit.

[0017] (S022): The Lyapunov exponent increases due to changes in internal parameters and initial conditions in the analysis model; the large-scale chaotic characteristics and transient behavior of the circuit internal parameter e in the conservative charge-controlled memcapacitive hyperchaotic system model in (S021) are analyzed.

[0018] (S023): Select different initial conditions (x0, y0, z0, u0v0, w0) of the conservative load-controlled memcomplex hyperchaotic system model in (S021) and analyze the attraction basin, coexisting attractor, initial offset boosting behavior and complexity distribution.

[0019] (S03): The conservative load-controlled memcapacitive hyperchaotic system model in (S02) is designed using FPGA technology, and compared with the numerical calculation in (S02) to verify the correctness and reliability of the designed system model.

[0020] The invention is characterized by the following: a conservative charge-controlled memcached hyperchaotic circuit is constructed using a charge-controlled memcached capacitor containing trigonometric function terms, along with two capacitors and two inductors. This circuit not only exhibits continuous chaos over a very large range of parameters e∈(-1000, 1000), but also displays characteristics such as switching between hyperchaos and chaos. When parameters e=112.4 and e=315 are selected, two different complex transient transition behaviors are observed. This circuit is extremely sensitive to initial conditions and system parameters. The attractor basin reflects the circuit's multiple stability and chaotic characteristics. With changes in initial values, three different coexisting attractors (nested coexistence, symmetric nested coexistence, and homomorphic coexistence) are discovered, and the attractor can be boosted to any position, i.e., initial offset boosting behavior. Changing the circuit parameters and initial conditions increases the complexity. The complexity distribution and Lyapunov exponent distribution with respect to the two parameters (b, c) are shown. Figure 1 The conservative memcell circuit model containing trigonometric function terms was implemented on a hardware platform using FPGA technology, and the experimental results were consistent with the numerical simulation results. Attached Figure Description

[0021] Figure 1 The characteristic curves of the charge-controlled memory container constructed in this invention are shown, wherein (a) is the characteristic curve of qu as a function of frequency f, and (b) is the characteristic curve of qu as a function of charge Q.

[0022] Figure 2 The diagram shows the six-dimensional conservative charge-controlled memcapacitor hyperchaotic circuit constructed for this invention.

[0023] Figure 3 This is a diagram of the chaotic attractor state trajectories for six different variables (x, y, z, u, v, w) when the initial conditions of this invention are I0 = (0.01, 0.01, 0.01, 0.01, 0.01) and the parameters are (a, b, c, d, e) = (0.8, 4, -4, 1, -4).

[0024] Figure 4 This is the Poincaré section corresponding to the chaotic attractor state trajectory diagrams of the six different variables (x, y, z, u, v, w) of this invention.

[0025] Figure 5 The graph shows the increase of the Lyapunov exponent when the initial conditions of this invention are (x0, y0, z0, u0v0, w0) = (0.1, 0.1, 0.1, 10, 0.1, 0.01) and the parameters are (a, b, c, d, e) = (0.8, 2000, -4, 1, -3000).

[0026] Figure 6 This is the Lyapunov exponent spectrum of the present invention as a function of the two parameters b∈(2.7,6) and c∈(-5,5). Among them, (a) is the largest Lyapunov exponent, and (b) is the second largest Lyapunov exponent spectrum.

[0027] Figure 7 This is the bifurcation diagram of the present invention as the parameter e∈(-1000, 1000) varies.

[0028] Figure 8 The diagrams show the time-domain waveforms and coexisting attractors for parameters e = 112.4 and e = 315 in this invention. Specifically, (a) is the time-domain waveform for parameter e = 112.4 (chaotic state 1 transforming into chaotic state 2), (b) is the attractor trajectory for parameter e = 112.4, (c) is the time-domain waveform for parameter e = 315 (chaotic state 3 transforming into chaotic state 4), and (d) is the attractor trajectory for parameter e = 315.

[0029] Figure 9 This invention presents an attraction basin map and multiple sets of coexisting attractors that vary with initial values ​​u(0) and w(0). Specifically, (a) is an attraction basin map varying with two initial values ​​u(0)∈(-5, 5) and w(0)∈(-1, 3); (b) is a coexisting attractor with initial conditions I2 / I6=(0.01, 0.01, 0.01, -1, 0.01, 0.12 / -0.12); (c) is a coexisting attractor with initial conditions I3 / I5=(0.01, 0.01, 0.01, -1, 0.01, 0 / 0.01); and (d) is a coexisting attractor with initial conditions I1 / I4=(0.01, 0.01, 0.01, -1, 0.01, 2.0 / 2.1).

[0030] Figure 10 The invention varies with the initial value z(0) = (0.10, 0.09, 0.08, 0.07, 0.06), that is, the initial conditions are I7 / I8 / I9 / I 10 / I 11 Nested coexistence attractor graphs. Among them, (a) is the coexistence attractor trajectory in the uv plane, (b) is the coexistence attractor trajectory in the vw plane, (c) is the coexistence attractor trajectory in the xy plane, and (d) is the coexistence attractor trajectory in the xyu space.

[0031] Figure 11 The invention varies with the initial value z(0) = (0.10, 0.09, 0.08, 0.07, 0.06), that is, the initial conditions are I7 / I8 / I9 / I 10 / I 11 The coexistence bifurcation diagram and coexistence time-domain waveform diagram are shown. Among them, (a) is the coexistence bifurcation diagram, (b) is the overall coexistence time-domain waveform diagram, and (c) is the partial coexistence time-domain waveform diagram.

[0032] Figure 12 The invention varies with the initial value z(0) = (0.05, 0.04, 0.03, 0.02, 0.01), i.e., the initial condition is I. 12 / I 13 / I 14 / I 15 Nested symmetric coexistence attractor diagrams at / I0. Among them, (a) is the coexistence attractor trajectory in the uv plane, (b) is the coexistence attractor trajectory in the vw plane, (c) is the coexistence attractor trajectory in the xy plane, and (d) is the coexistence attractor trajectory in the xuv space.

[0033] Figure 13 The invention varies with the initial value z(0) = (0.05, 0.04, 0.03, 0.02, 0.01), i.e., the initial condition is I. 12 / I 13 / I 14 / I 15 The coexistence bifurcation diagram and coexistence time-domain waveform diagram at / I0 are shown. Among them, (a) is the coexistence bifurcation diagram, (b) is the overall coexistence time-domain waveform diagram, and (c) is the partial coexistence time-domain waveform diagram.

[0034] Figure 14 The invention varies with the initial value w(0) = (0.05, 0.04, 0.03, 0.02, 0.01), i.e., the initial condition is I. 16 / I 17 / I 18 / I 19Homomorphic coexistence attractor diagrams at / I0. Among them, (a) is the coexistence attractor trajectory in the uv plane, (b) is the coexistence attractor trajectory in the vw plane, (c) is the coexistence attractor trajectory in the xy plane, and (d) is the coexistence attractor trajectory in the xvw space.

[0035] Figure 15 The invention varies with the initial value w(0) = (0.05, 0.04, 0.03, 0.02, 0.01), i.e., the initial condition is I. 16 / I 17 / I 18 / I 19 The coexistence bifurcation diagram and coexistence time-domain waveform diagram at / I0 are shown. Among them, (a) is the coexistence bifurcation diagram, (b) is the overall coexistence time-domain waveform diagram, and (c) is the partial coexistence time-domain waveform diagram.

[0036] Figure 16 This diagram illustrates the spectral entropy complexity distribution of the present invention under different initial conditions and with varying two parameters. Specifically, (a) shows the complexity distribution with two parameters (b, e) when the initial condition is I0 = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01), and (b) shows the complexity distribution with two parameters (b, c) when the initial condition is I0 = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). 20 The complexity distribution of the given condition (= (0.01, 0.01, 0.01, 0.01, 0.01, 1) as a function of the two parameters (b, e), where (b) is the initial condition I. 20 The complexity distribution of the given value (0.01, 0.01, 0.01, 0.01, 0.01, 1) as a function of the two parameters (b, c).

[0037] Figure 17 This is a connection diagram of the FPGA experimental platform of the present invention.

[0038] Figure 18 The diagram shows the chaotic attractor traject ... Detailed Implementation

[0039] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0040] The present invention provides a method for implementing a conservative charge-controlled memcapacitive hyperchaotic circuit with multiple coexisting attractors, comprising the following steps:

[0041] Step 1: Construct a novel load-controlled memory container model and analyze the characteristic curves of the load-controlled memory container.

[0042] 1) Construct a load-controlled memory container model containing trigonometric function terms. That is:

[0043]

[0044] Where q(t) and u(t) are the charge and voltage of the memory container at time t, respectively; σ is the continuous time integral of charge q(t) passing through the memory container; and φ(σ) is the magnetic flux through the memory container, which is a function of σ. It is the reciprocal of the capacitance of the memory capacitor, and α, β, and e are parameters.

[0045] 2) The values ​​of parameters α, β, and e should be chosen to ensure the chaotic characteristics of the system circuit. To achieve the best chaotic characteristics, parameters α = -4, β = 1, and e = -4 are set. Other value ranges for parameters α, β, and e will be discussed in subsequent experiments. With a sinusoidal excitation q(t) = Qsin(2πft), and a given charge Q = 2C (C is the unit of charge coulomb), the characteristic curves of the charge-controlled memory container at frequencies f of 1Hz, 2Hz, and 3Hz are as follows: Figure 1 As shown in (a); Figure 1 (b) shows the characteristic curves under the influence of charges Q of 1C, 2C, and 3C at a frequency of f = 1Hz.

[0046] Step 2: Construct a six-dimensional conservative charge-controlled memcapacitor hyperchaotic circuit composed of the novel charge-controlled memcapacitor described in Step 1 and capacitor and inductor components, and analyze its nonlinear dynamic behavior.

[0047] 1) such as Figure 2 The conservative memcapacitor-controlled hyperchaotic oscillator circuit shown consists of a charge-controlled memcapacitor C. M It consists of capacitor C1, capacitor C2, inductor L1, and inductor L2. According to Kirchhoff's laws:

[0048]

[0049] Where C1 and C2 are capacitors, q CM Let V1 be the voltage across capacitor C1, V2 be the voltage across capacitor C2, and L1 and L2 be inductors. L1 i L2 Let be the inductor current, and α, β, and e be parameters.

[0050] 2) Based on the conservative memcapacity hyperchaotic circuit model, set x = V1, y = V2, z = i L1 u = i L2 v = q CM w = σ CMChoosing parameters a = 1 / C1, b = 1 / C2, 1 / L1 = 1 / L2 = 1, c = α, d = β, the simple memristor hyperchaotic system model is:

[0051]

[0052] Where a, b, c, d, and c are parameters.

[0053] 3) To ensure the best performance of the chaotic characteristics of the system circuit, the parameters are set as follows: a = 1 / C1 = 0.8, b = 1 / C2 = 4, 1 / L1 = 1 / L2 = 1, c = α = -4, d = β = 1, e = -4, and the initial condition is I0 = (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). The chaotic attractors corresponding to different variables (x, y, z, u, v, w) are as follows: Figure 3 As shown, the corresponding Poincaré section is as follows: Figure 4 As shown, the corresponding Lyapunov exponents are calculated using the classic Jacobi algorithm as LE1 = 0.012, LE2 = 0.002, LE3 = 0, LE4 = 0, LE5 ​​= -0.001, and LE6 = -0.013. Among them, LE1 and LE2 are greater than zero, LE3 and LE4 are equal to zero, and LE5 and LE6 are less than zero. Therefore, the polarity of the Lyapunov exponents is (+, +, 0, 0, -, -). Since there are two positive Lyapunov exponents, the circuit is a hyperchaotic system.

[0054] When the parameters a, b, c, d, and e take other values, such as a = 0.7, b = 10, c = -3, d = 2, and e = -3, the corresponding Lyapunov exponents calculated using the classic Jacobi algorithm are LE1 = 0.026, LE2 = 0.002, LE3 = 0.002, LE4 = 0, LE5 ​​= -0.009, and LE6 = -0.021. When a = 0.9, b = 100, c = -5, d = 8, and e = -5, the corresponding Lyapunov exponents calculated using the classic Jacobi algorithm are LE1 = 0.1, LE2 = 0.003, LE3 = 0.003, LE4 = -0.004, LE5 ​​= -0.014, and LE6 = -0.088. Systems with the above parameter values ​​exhibit hyperchaotic characteristics.

[0055] Step 2.1: Based on the six-dimensional conservative load-controlled memcompressive hyperchaotic system model derived in the previous steps, analyze the rich dynamic behaviors exhibited by the changes in internal parameters and initial conditions in the model.

[0056] 1) By changing the system parameters (a, b, c, d, e) = (0.8, 2000, -4, 1, -3000) and the initial conditions (x0, y0, z0, u0v0, w0) = (0.1, 0.1, 0.1, 10, 0.1, 0.01), a larger Lyapunov exponent can be obtained, such as... Figure 5 As shown;

[0057] 2) Keeping other parameters unchanged, change the range of values ​​for parameters b and c, where b ∈ (2.7, 6) and c ∈ (-5, 5). The initial condition is (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). Simultaneously, the Lyapunove exponent spectrum distribution as a function of the two parameters (b, c) is shown in the figure. Figure 6 As shown.

[0058] Step 2.2: Analyze the large-scale chaotic characteristics and transient behavior of the circuit internal parameter e of the conservative charge-controlled memcapacitive hyperchaotic system model in Step 2.1;

[0059] 1) When only the value of parameter e is changed, while other parameters remain unchanged, and the initial conditions of the circuit are (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). Figure 7 The bifurcation graph is shown as a function of parameter e, where c ∈ (-1000, 1000);

[0060] 2) Setting e=112.4 and e=315 yields two complex transient transition behaviors. Figure 8 (a) and (b) show the time-domain waveform and attractor trajectory diagram of the transition from chaos 1 to chaos 2, respectively. Figure 8 (c) and (d) show the time-domain waveform and attractor trajectory diagrams of the transition from chaos 3 to chaos 4, respectively.

[0061] Step 2.3: Select different initial conditions (x0, y0, z0, u0v0, w0) for the conservative load-controlled memcomplex hyperchaotic system model in Step 2.1, and demonstrate the system's sensitivity to initial conditions and more complex chaotic characteristics through analysis of attraction basins, coexisting attractors, initial offset boosting behavior, and complexity distribution.

[0062] 1) Select the location of the attraction basin as (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, u(0), 0.01, w(0)), where u(0)∈(-5, 5) and w(0)∈(-1, 3). The circuit parameters remain unchanged. Figure 9 (a) shows the attraction basin as the initial values ​​u(0) and w(0) change; Figure 9 (b), (c), and (d) respectively demonstrate the coexistence attractors with initial conditions I2 / I6 = (0.01, 0.01, 0.01, -1, 0.01, 0.12 / -0.12), I3 / I5 = (0.01, 0.01, 0.01, -1, 0.01, 0 / 0.01), and I1 / I4 = (0.01, 0.01, 0.01, -1, 0.01, 2.0 / 2.1).

[0063] 2) According to the definition of an attraction basin, changing the initial values ​​allows the attractor to be shifted to any position, i.e., initial shift boosting behavior. This can be achieved by changing the initial values ​​z(0) or w(0), while keeping other initial values ​​unchanged. Figure 10 The initial value is z(0) = (0.10, 0.09, 0.08, 0.07, 0.06), which means the initial conditions are I7 / I8 / I9 / I 10 / I 11 Coexistence attractor of time Figure 11 (a), (b), and (c) are the corresponding coexistence bifurcation diagram and coexistence time-domain waveform diagram, respectively. Figure 12 The initial value is z(0) = (0.05, 0.04, 0.03, 0.02, 0.01), that is, the initial condition is I. 12 / I 13 / I 14 / I 15 Coexistence attractor at / I0 Figure 13 (a), (b), and (c) are the corresponding coexistence bifurcation diagram and coexistence time-domain waveform diagram, respectively. Figure 14 The initial value is w(0) = (0.05, 0.04, 0.03, 0.02, 0.01), which means the initial condition is I. 16 / I 17 / I 18 / I 19 Coexistence attractor at / I0 Figure 15 (a), (b), and (c) are the corresponding coexistence bifurcation diagram and coexistence time-domain waveform diagram, respectively;

[0064] 3) According to the definition of spectral entropy complexity, Figure 16 (a) and (b) are the complexity distributions of the two parameters (b, c) and (b, e) respectively, when other parameters remain unchanged and the initial conditions are I0 = (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). Figure 16 (c) and (d) respectively show the initial condition I with other parameters remaining unchanged. 20The complexity distribution of (x0, y0, z0, u0v0, w0) as a function of the two parameters (b, c) and (b, e) when (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 1).

[0065] Step 3: Design the conservative load-controlled memcapacitive hyperchaotic system model from Step 3 using FPGA technology, and compare it with the numerical calculation from Step 3 to verify the correctness and reliability of the designed system model.

[0066] 1) Set the initial conditions of model (6) as (x0, y0, z0, u0v0, w0) = (0.01, 0.01, 0.01, 0.01, 0.01, 0.01). Considering the accuracy issue and the simplicity of the circuit, the improved Euler algorithm is used to discretize model (6). The discretized conservative load-controlled memcapacitive hyperchaotic circuit is as follows:

[0067]

[0068] Where, Δt=1×10 -11 It is the time sampling step size.

[0069] 2) The discretized circuit (7) was designed using Verilog programming and fixed-point number method. The fixed-point number method uses a format of 1-bit signal bits, 1-bit integer bits, and 30-bit fractional bits. Utilizing the logical integrity, clear engineering structure, and flexibility of the state machine, a state machine was written in Verilog to realize the parallel computation of the discretized circuit (7) and the effective output results. The main experimental platform used was a Cyclone IVE series FPGA main chip, model EP4CE10F17C8N, a 14-bit dual-channel AD9767 DAC chip, and an oscilloscope. Figure 17 The connection diagram of the FPGA experimental platform is shown;

[0070] 3) Observe the attractor using a digital oscilloscope and compare it with the numerical simulation results one by one. Figure 18 (a), (b), (c), and (d) are the chaotic attractor trajector trajector trajectories of the xy, xu, vw, and uv planes, respectively. Comparing them with Figure (3), it can be found that the state trajectories of the two are consistent, which shows that the numerical design of the conservative charge-controlled mem-capacitor hyperchaotic circuit is correct and feasible.

Claims

1. A method for implementing a conservative charge-controlled memcapacitive hyperchaotic circuit with multiple coexisting attractors, characterized in that: Includes the following steps: Step S01: Construct a novel load-controlled memory container model and analyze the characteristic curves of the novel load-controlled memory container; （1） Wherein, q(t) and u(t) represent the charge and voltage of the novel charge-controlled memory container at time t, respectively. It is electric charge The internal state variables of the novel load-controlled memory container. Internal state variables The first derivative with respect to time, It is the reciprocal of the capacitance of the novel charge-controlled memory capacitor, where α, β, and e are parameters. The range of values ​​for parameters α, β, and e is set to ensure that the system circuit characteristics are chaotic. Step S02: Construct a six-dimensional novel charge-controlled memcached circuit composed of the novel charge-controlled memcached capacitor described in Step S01, an inductor, and a capacitor, and analyze its nonlinear dynamic behavior. The six-dimensional novel charge-controlled memcached circuit is composed of the novel charge-controlled memcached capacitor C. M It consists of capacitor C1, capacitor C2, inductor L1, and inductor L2. According to Kirchhoff's laws: （2） Where C1 and C2 are capacitors, V CM The voltage of the novel load-controlled memory container. It is electric charge The integral, where V1 is the voltage across capacitor C1, V2 is the voltage across capacitor C2, and L1 and L2 are inductors. , It is the inductor current; According to formula (1), V CM = Then we have: （3） Step S021: Derive the conservative charge-controlled memcapacitor hyperchaotic system model from the six-dimensional novel charge-controlled memcapacitor circuit described in step S02: （4） Where x = V1, y = V2, , , , a, b, c, d, e are parameters. The range of values ​​for parameters a, b, c, d, e is set to ensure the conservative hyperchaotic characteristics of the system circuit. Step S022: Analyze the increase in Lyapunov exponent caused by changes in internal parameters and initial conditions in the model; analyze the large-scale chaotic characteristics and transient behavior of the circuit internal parameter e of the conservative charge-controlled memcapacitive hyperchaotic system model in step S021. Step S023: Select different initial conditions (x0, y0, z0, u0 v0, w0) for the conservative load-controlled memcomplex hyperchaotic system model in step S021, and analyze the attraction basin, coexisting attractor, initial offset propulsion behavior and complexity distribution. Step S03: Use FPGA technology to design the conservative load-controlled memcapacitor hyperchaotic system model in step S021, and compare it with the numerical simulation results of the conservative load-controlled memcapacitor hyperchaotic system model in step S021 to verify the correctness and reliability of the designed system model.

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