Construction and implementation method of hyperbolic tangent fractional order memristor chaotic system

Abstract

The invention discloses a construction and implementation method of a hyperbolic tangent fractional order memristor chaotic system. The method comprises the following steps: firstly, designing a fractional order magnetic control memristor model containing a hyperbolic tangent function, and constructing a fractional order memristor chaotic system based on the model; through high-resolution order parameter scanning and in combination with Lyapunov exponent spectrum, bifurcation diagram, phase diagram and complexity analysis, rich dynamic behaviors and high sensitivity of the system under order change are revealed; a difference equation model of the system is obtained through a frequency domain approximation and discretization method, digital modeling is completed on a DSP Builder platform, and hardware description language codes executable by an FPGA are automatically generated; finally, the real-time iterative operation of the system is realized on the FPGA development board, and the physical realizability of the chaotic attractor is verified through the oscilloscope. The invention provides a high-precision and low-power-consumption hardware implementation scheme for the fractional order memristor chaotic system, and is suitable for the information security fields of secret communication, image encryption and the like.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear circuit and chaotic system design, and involves the modeling, dynamic analysis and FPGA implementation of fractional memristor chaotic systems. Background Technology

[0002] Memristors are the fourth fundamental circuit element after resistors, capacitors, and inductors. They possess nonlinear memory characteristics and hold significant promise for applications in chaotic systems, neural networks, and information security. Chaotic systems are a class of systems exhibiting complex and unpredictable motion patterns that are extremely sensitive to initial conditions. Introducing memristors into chaotic systems allows the new systems to generate chaotic phenomena, with dynamic behaviors that are richer and more unpredictable than traditional chaotic systems. Fractional calculus can accurately describe complex dynamic behaviors with memory and genetic properties. Introducing fractional calculus into memristor chaotic systems allows them to exhibit many complex dynamic behaviors not found in integer-order systems, offering significant advantages in secure communication, image encryption, and signal processing.

[0003] In recent years, research on chaotic systems based on memristors has mainly focused on integer-order models, while research on fractional-order memristor chaotic systems is relatively limited. Traditional memristor models have limitations in describing nonlinear characteristics, while memristor models based on the hyperbolic tangent function, due to their smooth and continuous characteristics, can more accurately simulate the dynamic behavior of actual memristors while reducing the complexity of numerical computation. However, the application of existing hyperbolic tangent memristor models in fractional-order chaotic systems has not been fully explored, and most memristor chaotic systems suffer from high complexity and difficulties in hardware implementation, limiting their application in practical engineering.

[0004] Field-Programmable Gate Arrays (FPGAs) have become an ideal platform for hardware implementation of chaotic systems due to their advantages such as strong parallel computing capabilities, high reconfigurability, and short development cycles. However, most existing FPGA-implemented chaotic systems are based on integer-order differential equations, and the FPGA implementation of fractional-order chaotic systems still suffers from high computational complexity and high resource consumption. Therefore, designing a high-precision, low-power digital discrete model and an efficient FPGA implementation method for fractional-order memristor chaotic systems has significant theoretical and engineering value. Summary of the Invention

[0005] The purpose of this invention is to propose a method for constructing and implementing a hyperbolic tangent fractional-order memristor chaotic system. By constructing a novel four-dimensional fractional-order memristor chaotic system model, and combining high-resolution order parameter scanning (resolution up to 0.001) with multi-dimensional dynamic analysis, the complex dynamic behavior of the system under synchronous and single-order parameter changes is revealed, demonstrating its high sensitivity to order parameters. Finally, based on the DSP Builder platform, the FPGA implementation of the fractional-order memristor chaotic system is completed, verifying the physical realizability of the system and providing a reliable theoretical model and hardware foundation for its application in information technology fields such as secure communication and image encryption.

[0006] This invention is achieved through the following technical solutions.

[0007] The present invention discloses a method for constructing and implementing a hyperbolic tangent fractional memristor chaotic system, comprising the following steps:

[0008] Step S1: Based on the definition of memristor and the definition of Caputo fractional derivative, design a fractional magnetically controlled memristor model containing a hyperbolic tangent function;

[0009] Step S2: Analyze the current-voltage characteristics of the fractional-order magnetically controlled memristor at different frequencies, peak voltages, and orders;

[0010] Step S3: Based on the fractional-order magnetically controlled memristor model, construct a fractional-order memristor chaotic system based on hyperbolic tangent;

[0011] Step S4: In the high-resolution order space of the hyperbolic tangent fractional memristor chaotic system from step S3, numerical simulations (phase diagram trajectories, Lyapunov exponent spectra, bifurcation diagrams, etc.) and dynamic analyses are performed when the system order changes synchronously and unidirectionally. The analysis results show that the system exhibits rich dynamic behavior and order sensitivity.

[0012] Step S5: Perform frequency domain approximation and discretization on the hyperbolic tangent fractional memristor chaotic system from step S3, and design its digital discrete model on the DSP Builder 18.0 platform;

[0013] Step S6: Based on the order change of the system in Step S4, the digital discrete model in Step S5 is automatically converted into hardware description language code optimized for the target FPGA model. The code is compiled, synthesized, and placed and routed using Quartus II to generate the corresponding bitstream file and download it to the FPGA development board.

[0014] Step S7: The S06 FPGA generates a digital sequence by performing real-time iterative calculations on the discrete model in step S3, and converts the digital sequence into an analog signal via a digital-to-analog converter; the consistency between the chaotic attractor phase diagram generated by the analog signal captured on an oscilloscope and the numerical simulation results confirms the physical realizability of the system.

[0015] The mathematical model of the fractional-order magnetically controlled memristor containing the hyperbolic tangent function mentioned in step S1 is as follows:

[0016] (1)

[0017] In the formula, Here, q is the Caputo differential operator, k, l, and m represent the memristor parameters, and i(t) and v(t) represent the memristor current and voltage, respectively. This represents the magnetic flux within the memristor's internal state. Represents the memorized derivative function;

[0018] The mathematical model of the hyperbolic tangent fractional memristor chaotic system described in step S3 is as follows:

[0019] (2)

[0020] In the formula, x, y, z, and u are system state variables. For q i q-th order Caputo differential operator i (i=1,2,3,4) represents the order of the corresponding state variable, and q i ∈(0,1). The system parameters a, b, c, d, e are real numbers, the nonlinear term kutanh(u) is the hyperbolic tangent magnetically controlled memristor introduced into the system, and the last equation is the internal state equation of the fractional-order memristor;

[0021] Step S5 describes the frequency domain approximation and discretization of the hyperbolic tangent fractional memristor chaotic system described in step S3, resulting in the following discrete-time difference equations:

[0022] (3)

[0023] In the formula, The sampling index K is an integer, and the sampling time is Δt = 2 × 10. -4 A i B i C i D j These are the discretization coefficients.

[0024] The key features of this invention are as follows: Based on the characteristics of the hyperbolic tangent function and fractional calculus theory, a hyperbolic tangent fractional-order magnetically controlled memristor model is designed, and its nonlinear dynamic characteristics are studied. A corresponding fractional-order memristor chaotic system is constructed, and its equilibrium point and stability are analyzed. Within the order parameter space, the evolution of the system's dynamic behavior with changes in order parameters under synchronous and single-order changes of the fractional order is investigated. Lyapunov exponent spectrum, bifurcation diagrams, phase diagram trajectories, complexity chaos diagrams, and attraction basins are used to analyze the system's dynamic characteristics. This invention proposes an FPGA implementation scheme based on DSPBuilder, improving the implementation accuracy and computational efficiency of the fractional-order chaotic system. Comparison between MATLAB numerical simulations and hardware measurement results verifies the correctness and engineering practicality of the designed system, providing new methods and ideas for the design and implementation of complex nonlinear systems. Attached Figure Description

[0025] Figure 1 The current-voltage characteristic curves of the hyperbolic tangent fractional memristor designed for this invention are shown. (a), (b), and (c) are the current-voltage characteristic curves as a function of frequency, peak voltage, and order, respectively.

[0026] Figure 2 The dynamic characteristics of the system are shown when the order q ∈ [0.6, 1] changes synchronously. (a) The system is shown when q i (a) The Lyapunov exponent spectrum of the system at q (i=1,2,3,4); (b) The Lyapunov exponent spectrum of the system at q i = q(i=1,2,3,4); (c)-(h) are the phase diagrams when q = 0.671, 0.883, 0.908, 0.935, 0.988 and 0.992.

[0027] Figure 3 The following are the dynamic characteristics of the system with a fixed order (q2, q3, q4) = (0.893, 0.875, 0.884) and q1 ∈ [0.1, 1]. (a) is the Lyapunov exponent spectrum of the system with (q2, q3, q4) = (0.893, 0.875, 0.884); (b) is the bifurcation diagram of the system with (q2, q3, q4) = (0.893, 0.875, 0.884); (c)-(h) are the phase diagrams with q1 = 0.219, 0.785, 0.923, 0.959, 0.983 and 0.997.

[0028] Figure 4The present invention uses a fixed order (q1, q3, q4) = (0.983, 0.875, 0.884) to show the dynamic characteristics of the system when q2 ∈ [0.5, 1]. (a) is the Lyapunov exponent spectrum of the system when (q1, q3, q4) = (0.983, 0.875, 0.884); (b) is the bifurcation diagram of the system when (q1, q3, q4) = (0.983, 0.875, 0.884); (c)-(h) are the phase diagrams when q2 = 0.541, 0.655, 0.755, 0.861, 0.929, and 0.955.

[0029] Figure 5 The present invention uses a fixed order (q1, q2, q4) = (0.983, 0.893, 0.884) to show the dynamic characteristics of the system when q3 ∈ [0.5, 1]. (a) is the Lyapunov exponent spectrum of the system when (q1, q2, q4) = (0.983, 0.893, 0.884); (b) is the bifurcation diagram of the system when (q1, q2, q4) = (0.983, 0.893, 0.884); (c)-(h) are the phase diagrams when q3 = 0.679, 0.759, 0.825, 0.858, 0.951 and 0.995.

[0030] Figure 6 The present invention uses a fixed order (q1, q2, q3) = (0.983, 0.893, 0.875) to show the dynamic characteristics of the system when q4 ∈ [0.5, 1]. (a) is the Lyapunov exponent spectrum of the system when (q1, q2, q3) = (0.983, 0.893, 0.875); (b) is the bifurcation diagram of the system when (q1, q2, q3) = (0.983, 0.893, 0.875); (c)-(e) are the phase diagrams when q4 = 0.541, 0.571, and 0.961.

[0031] Figure 7 Let be the system complexity when the order q∈[0.6,1] changes synchronously. (a) is the system complexity as the order q changes. i = SE complexity when q(i=1,2,3,4) varies; (b) is the system complexity as order q changes. i = C0 complexity when q(i=1,2,3,4) changes.

[0032] Figure 8 When three of the orders (q1, q2, q3, q4) = (0.983, 0.893, 0.875, 0.884) in this invention are fixed, the system continues to function with the remaining order q. iThe complexity of the system when (i=1,2,3,4) varies. (a)-(d) are the SE complexity of the system when the order q1, q2, q3 and q4 respectively; (e)-(h) are the C0 complexity of the system when the order q1, q2, q3 and q4 respectively.

[0033] Figure 9 This invention presents a hyperbolic tangent fractional memristor chaotic system model built on the DSP Builder platform.

[0034] Figure 10 This is the FPGA experimental platform for the hyperbolic tangent fractional-order memristor chaotic system of this invention. (a) Corresponding to Figure 11 (b); (b) corresponds to Figure 12 (c)

[0035] Figure 11 For the present invention and Figure 2 The FPGA implementation result corresponding to the phase diagram in (f) has an order of q1=q2=q3=q4=0.935.

[0036] Figure 12 For the present invention and Figure 5 The FPGA implementation result corresponding to the phase diagram in (g) has an order of (q1, q2, q3, q4) = (0.983, 0.893, 0.951, 0.884). Detailed Implementation

[0037] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0038] Example 1: Design and I-V characteristic analysis of a fractional-order magnetically controlled memristor model containing a hyperbolic tangent function

[0039] The initial conditions defined by Caputo are compatible with the classical integer-order calculus form and are widely used for modeling and optimizing complex systems such as engineering control and neural networks. The Caputo fractional derivative is defined as:

[0040] (4)

[0041] in, For fractional order, and , for The Caputo differential operator of order, the function f(t) in the time domain Continuous above, Г(·) is the Gamma function, and its expression is: .

[0042] Based on the definitions of memristors and Caputo's definition, the hyperbolic tangent fractional-order magnetically controlled memristor model is obtained as described by equation (1). Selecting memristor parameters k=1, l=1, and m=-1, equation (1) is transformed into:

[0043] (5)

[0044] Apply a sinusoidal voltage excitation signal source to the memristor , where V m Let f be the peak voltage and f be the frequency. The current-voltage characteristic curve of this memristor can be obtained as follows: Figure 1 As shown. Figure 1 (a) presents the situation when the initial value is maintained. The order q = 0.95 and the peak voltage V m =2V remains constant, and the memristor characteristic curves are set at frequencies f of 0.8Hz, 1.2Hz, 3.0Hz and 25Hz respectively; Figure 1 (b) presents the situation when the initial value is maintained. The order q = 0.95 and the frequency f = 1Hz remain unchanged, and the peak voltage V is set. m Memristor characteristic curves at 1.0V, 2.0V, 3.0V and 3.5V respectively; Figure 1 (c) presents the situation when the initial value is maintained. Peak voltage V m Memristor characteristic curves with constant voltage f=2V and frequency f=1Hz, and with order q set to 1.0, 0.95, 0.8 and 0.75 respectively.

[0045] Example 2: Constructing a hyperbolic tangent fractional memristor chaotic system model.

[0046] 1) The fractional calculus Grunwald-Letnikov (GL) is defined as:

[0047] (6)

[0048] Where Г(·) is the Gamma function, n-1≤q≤n, q is the fractional order, and n is an integer.

[0049] 2) Based on the hyperbolic tangent fractional-order magnetically controlled memristor model (5) in Example 1, and according to the fractional calculus theory defined by Caputo, the hyperbolic tangent fractional-order memristor chaotic system model is obtained as described by formula (2). Selecting system parameters as a = 1.39, b = 0.56, c = 15, d = 36, e = 0.9, l = 1, m = -1, k = 1, then formula (2) is converted to:

[0050] (7)

[0051] Example 3: Analysis of the dynamic characteristics, complexity, and attractor basin of fractional-order memristor chaotic systems as the order changes.

[0052] This invention performs a fine scan in the order parameter space, with a step size set to 0.001 or 0.002. All order values ​​are accurate to the thousandths place to ensure high resolution in the dynamic analysis.

[0053] 1) Set the system order as q1=q2=q3=q4=q, with a step size of q of 0.002 and a variation range of [0.6,1]. Analyze the global dynamic evolution characteristics of the system when all orders q change synchronously. Figure 2 The Lyapunov exponent spectrum and bifurcation diagram of the system are shown, along with the corresponding phase diagram trajectory in Table 1. Table 1 shows the Lyapunov exponent, fractal dimension, and system state of the system when q = 0.671, 0.883, 0.908, 0.935, 0.988, and 0.992.

[0054] Table 1 and Figure 2 Lyapunov exponents, fractal dimension, and system states corresponding to the phase diagram trajectory

[0055]

[0056] 2) With the three orders fixed at different values, only the single order q i The dynamic characteristics of the system when (i=1,2,3,4) change. Figures 3-6 The Lyapunov exponent spectrum and bifurcation diagram of the system are shown, along with the corresponding phase diagram trajectory in Table 2. Table 2 shows the Lyapunov exponent, fractal dimension, and system state under different conditions.

[0057] With the order set to q2=0.893, q3=0.875, q4=0.884, the range of q1 is [0.1,1], with a step size of 0.001. The dynamic behavior of the system is observed when q1 = 0.219, 0.785, 0.923, 0.959, 0.983 and 0.997 are selected.

[0058] With the order set to q1=0.983, q3=0.875, q4=0.884, the range of q2 is [0.5,1], with a step size of 0.001. The dynamic behavior of the system is observed when q2 = 0.541, 0.655, 0.755, 0.861, 0.929, and 0.955.

[0059] With the order set as q1=0.983, q2=0.893, q4=0.884, the range of q3 is [0.5,1], with a step size of 0.001. The dynamic behavior of the system is observed when q3 = 0.679, 0.759, 0.825, 0.858, 0.951 and 0.995.

[0060] With the order set as q1=0.983, q2=0.893, q3=0.875, the range of q4 is [0.5,1], with a step size of 0.001. The dynamic behavior of the system is observed when q4 = 0.541, 0.571, and 0.961 are selected.

[0061] Table 2 and Figures 3-6 Lyapunov exponents, fractal dimension, and system states corresponding to the phase diagram trajectory

[0062]

[0063] 4) The complexity analysis of the proposed fractional memristor chaotic system will be performed using the Spectral Entropy (SE) algorithm and the C0 algorithm.

[0064] Figure 7 It demonstrates the SE complexity and C0 complexity of the system as the order q changes, when the system order is uniformly set to q1=q2=q3=q4=q.

[0065] Figure 8 This demonstrates that when the system is fixed with three of the following orders unchanged: q1=0.983, q2=0.893, q3=0.875, and q4=0.884, only the i-th order q... i The SE and C0 complexities of the system when (i=1,2,3,4) vary independently.

[0066] Example 4: Frequency domain approximation, discretization, and DSPBuilder design of a hyperbolic tangent fractional memristor chaotic system.

[0067] 1) According to the definition of the Laplace transform, the Laplace transform of the fractional memristor chaotic system (2) is:

[0068] (8)

[0069] Where, q i For ∈(0,1)(i=1,2,3,4), s+σ+jw is the complex frequency, and the expressions for H(s), I(s), J(s), N(s), R(s), and O(s) are as follows:

[0070] (9)

[0071] 2) Select as follows Figure 2 The dynamic states shown in (f) and 5(g) are implemented in hardware, with corresponding system orders q1 = q2 = q3 = q4 = 0.935, (q1, q2, q3, q4) = (0.992, 0.992, 0.976, 0.992), and (q1,q2, q3, q4) = (0.983, 0.893, 0.951, 0.884), respectively. According to the Bode plot frequency domain approximation algorithm, assuming the reciprocal of the system's relaxation time constant is 0.01 and the maximum frequency bandwidth is 100 rad / s, the approximate expressions for the integer-order rational functions of the integral operators corresponding to the fractional orders involved can be obtained in the frequency domain. Table 3 shows the specific parameters and error tolerances.

[0072] Table 3. Approximate expressions for integer-order rational functions in the frequency domain of fractional-order integral operators.

[0073]

[0074] make Substituting the approximate expression corresponding to the fractional order into equation (8) yields the system of differential equations:

[0075] (10)

[0076] Tables 4-5 list the relevant... Figure 2 The correlation coefficient values ​​of the differential equation set (10) corresponding to the phase diagram trajectories in (f) and 5(g)

[0077] Table 4 shows the order q1 = q2 = q3 = q4 = 0.935, and... Figure 2 (f) Correlation coefficient values ​​of the corresponding differential equation system (10)

[0078]

[0079] Table 5 shows the order (q1, q2, q3, q4) = (0.983, 0.893, 0.951, 0.884), and... Figure 5 (g) Correlation coefficient values ​​of the corresponding differential equation system (10)

[0080]

[0081] 3) This invention uses the Euler method to discretize the continuous-time differential equations (10) describing a fractional-order memristor chaotic system, resulting in a discrete-time difference equation set as described by formula (3). The sampling time is Δt = 2 × 10⁻⁶. -4The corresponding discretization coefficient A i B i C i D j The values ​​of (i=1,2,…,6,j=1,2,3,4) are shown in Table 4-5.

[0082] In the MATLAB / Simulink environment, basic modules from the DSP Builder standard library can be used to build discretized system equations. The Signal Compiler module automatically converts the algorithm model into a hardware description language (VHDL / Verilog), and directly interfaces with FPGA development tools such as Quartus II for synthesis and implementation, constructing an automated process from algorithm simulation to hardware deployment, greatly improving development efficiency. Therefore, this invention integrates DSP Builder 18.0 in the MATLAB / Simulink R2024b environment to build an experimental platform. Figure 9 A model of a hyperbolic tangent fractional memristor chaotic system is presented based on the above difference equation set (4).

[0083] Example 5: FPGA implementation of a hyperbolic tangent fractional memristor chaotic system.

[0084] 1) After completing the design of the hyperbolic tangent fractional memristor chaotic system model based on DSP Builder, the simulation output of the system model is compared with the numerical simulation results of MATLAB to confirm that the model can reproduce the expected chaotic dynamic behavior, thereby ensuring the accurate conversion from continuous mathematical model to discrete hardware model.

[0085] 2) Configure the Signal Compiler module in DSP Builder, specify the target FPGA device model, and automatically generate optimized hardware description language code. Next, use Quartus II software to compile, synthesize, and place and route the design, successfully generating a bitstream file (.sof). Finally, download this file to the FPGA development board, convert the digital signal to an analog waveform using an extended DAC module, and capture the actual output time-domain trajectory and phase diagram using an oscilloscope.

[0086] 3) The experimental platform used in this invention mainly includes: Cyclone IV E series FPGA development board (EP4CE115F29C7), 14-bit dual-channel DAC module (AD9767) and GWINSTEK GDS-2202E digital oscilloscope. Figure 10 The hardware connection between the FPGA development board and peripheral devices is demonstrated. Figures 11-12 The chaotic attractor phase diagram captured by an oscilloscope in XY mode is shown.

[0087] The results of the hyperbolic tangent fractional memristor chaotic system of this invention displayed on the oscilloscope are consistent with those of the MATLAB numerical simulation. Figure 2 The phase diagram trajectories in (f) and 5(g) match, verifying the correctness and feasibility of the designed hyperbolic tangent fractional memristor chaotic system implemented on the FPGA platform.

Claims

1. A method for constructing and implementing a hyperbolic tangent fractional-order memristive chaotic system, characterized in that, The method comprises the following steps: S1, designing a fractional-order magnetron memristor model containing a hyperbolic tangent function; S2, analyzing the volt-ampere characteristics of the fractional-order magnetron memristor under different frequencies, peak voltages and orders; S3, constructing a hyperbolic tangent fractional-order memristor chaotic system based on the fractional-order magnetron memristor model; S4, performing high-resolution numerical simulation and dynamic analysis on the chaotic system in the order parameter space, and revealing the dynamic behavior of the chaotic system with the change of the order; S5, performing frequency domain approximate solution and discretization processing on the chaotic system to obtain a discrete time difference equation set; S6, constructing a digital model of the discrete system on a DSP Builder platform, and automatically generating hardware description language code for a target FPGA, which is compiled, synthesized and laid out and then downloaded to an FPGA development board; S7, performing real-time iterative operation of the discrete model by FPGA, outputting an analog signal after digital-to-analog conversion, and capturing a chaotic attractor phase diagram by an oscilloscope to verify the physical realizability of the system.

2. The method of claim 1, wherein, The mathematical expression of the fractional-order magnetron memristor model in step S1 is: where i(t), v(t) are the current and voltage of the memristor, respectively, is the internal state magnetic flux, is the Caputo fractional differential operator, q is the order, k, l, m represent the parameters of the memristor, denotes the memductance function.

3. The method of claim 1, wherein, The mathematical model of the four-dimensional hyperbolic tangent fractional-order memristor chaotic system in step S3 is: where x, y, z, u are system state variables, q i (i = 1, 2, 3, 4) are corresponding orders, a, b, c, d, e, l, m, k are system parameters, and kutanh(u) is the hyperbolic tangent magnetic control memristor introduced into the system.

4. The method of claim 1, wherein, The dynamic analysis in step S4 includes Lyapunov exponent spectrum, bifurcation diagram, phase trajectory, complexity analysis and basin of attraction analysis.

5. The method of claim 1, wherein, The discrete time difference equation set in step S5 is: where Δt is the sampling time, K is the sampling index, A i , B i , C i , D j is the discretization coefficient.

6. The method of claim 1, wherein, The DSP Builder platform in step S6 is used to automatically generate VHDL / Verilog code, and the Signal Compiler module is used to interface with Quartus II to complete FPGA synthesis and implementation.

7. The method of claim 1, wherein, The FPGA development board in step S7 includes a Cyclone IVE series FPGA, an AD9767 DAC module and a digital oscilloscope, and is used to realize real-time generation and display of chaotic signals.

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