Circuit implementation method of double-scroll chaotic system
By applying the deformed Julia fractal to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed, which solves the problem of the inability to verify characteristics in the circuit in the prior art. It realizes circuit verification and verification of chaotic characteristics of digital circuits, enhances the complexity and flexibility of attractor signals, and expands the application space.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SHIJIAZHUANG UNIVERSITY
- Filing Date
- 2026-05-25
- Publication Date
- 2026-06-19
AI Technical Summary
Existing technologies cannot effectively verify the characteristics of a double-vortex chaotic system through circuitry.
By applying the deformed Julia form to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed. Its dynamic characteristics are analyzed, the attractor shape is determined to be four-winged, and based on the influence of the quadratic coefficient k and parameter a on the maximum Lyapunov exponent, analog and digital circuits are constructed to output electrical signals characterizing the four-winged attractor.
This study demonstrates the ability to verify the characteristics of a double-vortex chaotic system through circuitry, expanding the application scope of chaotic systems, enhancing the complexity and flexibility of attractor signals, and broadening their practical applicability in the era of big data.
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Figure CN122247593A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of dual-vortex chaotic system technology, and in particular to a circuit implementation method for a dual-vortex chaotic system. Background Technology
[0002] Chaos science and fractal theory are important components of nonlinear science. The attractors formed by chaotic motion in phase space are fractals, and the two are closely related. The ideas of chaotic fractals have been widely applied in physics, medicine, computer science, and even textile manufacturing, continuously promoting the development of these disciplines. Julia sets are sets of points that form fractals on the complex plane. Although their complex function forms are simple, they can generate very complex fractal patterns. Applying the mapping relationships generated by Julia fractals to chaotic systems can construct new systems with even more complex forms.
[0003] Therefore, how to verify the characteristics of a dual-vortex chaotic system through circuitry has become a technical problem that urgently needs to be solved by those skilled in the art. Summary of the Invention
[0004] This invention provides a circuit implementation method for a dual-vortex chaotic system, which solves the defect in the prior art that the characteristics of a dual-vortex chaotic system cannot be verified by circuit means.
[0005] In a first aspect, the present invention provides a circuit implementation method for a dual-vortex chaotic system, comprising: By applying the deformed Julia fractal with quadratic coefficient k to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed. Analyzing the dynamic characteristics of the dual-vortex chaotic system, the attractor shape was determined to be four-winged; Determine the influence of the quadratic term coefficient k on the four-wing attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent. Based on the influence of the quadratic coefficient k on the four-winged attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent, an analog circuit and / or a digital circuit are constructed. Both the analog circuit and the digital circuit are used to output an electrical signal characterizing the four-winged attractor. The four-winged attractor shape corresponding to the electrical signal has more wings than the two-winged attractor shape of the deformed Liu system.
[0006] According to a circuit implementation method for a dual-vortex chaotic system provided by the present invention, the analog circuit includes a modular unit circuit and a main circuit. The modular unit circuit includes at least a number of sub-modules for implementing different signal combination functions. Each sub-module is composed of signal processing elements and adjustment elements. The signal combination functions include at least signal exponentiation and subtraction combination, multiplication combination, exponentiation and addition combination, fraction and exponentiation combination, and exponentiation series combination. The main circuit includes multiple channel circuits that generate voltage signals for each state variable, and each channel circuit includes a computing element, an energy storage element, and multiple regulating elements. The state variables of the dual-vortex chaotic system correspond to the voltage signal output by the main circuit. The value of k can be adjusted by adjusting the adjustment element associated with the quadratic term coefficient k. The value of k only changes the size of the output dual-vortex attractor, without changing the shape of the attractor. The analog circuit uses a preset power supply voltage, the regulating element is a resistor with a preset resistance value, and the energy storage element is a capacitor with a preset capacitance value.
[0007] According to a circuit implementation method for a dual-vortex chaotic system provided by the present invention, the sub-module includes: The first module uses two multipliers, three resistors, and an operational amplifier to perform power-subtraction combinations of signals. The second module uses a multiplier, two resistors, and an operational amplifier to perform signal multiplication and combination. The third module uses two multipliers, three resistors, and an operational amplifier to perform power-addition combinations of signals. The fourth module, through a multiplier, two resistors, and an operational amplifier, combines with the third module to realize the combination of fractional and exponential functions of the signal; The fifth module, through a multiplier, two resistors, and an operational amplifier, combines with the third module to realize the combination of fractional and exponential functions of the signal; The sixth module, through three resistors and an operational amplifier, combines with the first and second modules to realize a series of power-law combinations of signals; The seventh module combines a multiplier, three resistors, and an operational amplifier with the first module to realize a series of power-law combinations of signals.
[0008] According to the circuit implementation method of the dual-vortex chaotic system provided by the present invention, the plurality of channel circuits include a first channel circuit, a second channel circuit and a third channel circuit; The first channel circuit includes at least two operational elements, one energy storage element, and multiple regulating elements. The output terminal of the operational element outputs a voltage signal corresponding to the first state variable. The second channel circuit includes at least two operational elements, one energy storage element, and multiple regulating elements. The output terminal of the operational element outputs a voltage signal corresponding to the second state variable. The third channel circuit includes at least two operational elements, one energy storage element, and multiple adjustment elements. The output terminal of the operational element outputs a voltage signal corresponding to the third state variable, and the resistance value of the adjustment element associated with the quadratic coefficient k is adjusted according to the value of the quadratic coefficient k.
[0009] According to the present invention, a circuit implementation method for a dual-vortex chaotic system is provided, wherein the digital circuit includes a hardware part and a software part; The hardware component includes a microcontroller development board, a digital-to-analog converter, and a signal display device. The microcontroller development board is used to convert the processed digital signal into an analog signal through the digital-to-analog converter and then output it to the signal display device. The software component is used to control the hardware component to implement the discretization operation and signal output of the dual-vortex chaotic system. The software component includes at least an initialization function, an iterative operation function, and a scaling function.
[0010] According to the circuit implementation method of the dual-scroll chaotic system provided by the present invention, the initialization function includes at least a peripheral initialization function for initializing peripheral devices, a timer initialization function for initializing timers, a port initialization function for initializing general-purpose input / output ports, a controller initialization function for initializing the direct memory access controller, a digital-to-analog converter initialization function for initializing the digital-to-analog converter, and a clock initialization function for initializing the system clock. The iterative operation function uses a preset discretization method to discretize the state equation of the double-vortex chaotic system; The scaling function is used to scale the input signal value to a preset target range.
[0011] According to a circuit implementation method for a dual-vortex chaotic system provided by the present invention, determining the influence of the quadratic term coefficient k on the four-wing attractor of the dual-vortex chaotic system includes: As the coefficient k of the quadratic term increases, the size of the four-winged attractor of the double-vortex chaotic system gradually decreases and tends to converge, while the shape of the four-winged attractor remains unchanged.
[0012] According to the circuit implementation method of the dual-vortex chaotic system provided by the present invention, the dimensionless state equation of the dual-vortex chaotic system is: ; In the formula, Represents the state variable, where, , , , , Indicates system parameters, This represents the coefficient of the quadratic term in the transformed Julia fractional form.
[0013] According to a circuit implementation method for a dual-vortex chaotic system provided by the present invention, determining the influence of parameter 'a' of the dual-vortex chaotic system on the maximum Lyapunov exponent of the dual-vortex chaotic system includes: When the maximum Lyapunov exponent is greater than 0, the dual-vortex chaotic system is in a chaotic state. When the maximum Lyapunov exponent is less than or equal to 0, the double-vortex chaotic system is in a periodic state. When system parameter a is in the first preset interval, most of the region of the dual-vortex chaotic system is in a chaotic state, and there is a parameter sub-interval corresponding to a preset periodic window in the chaotic state region. When system parameter a is in the second preset interval, the dual-vortex chaotic system is in a periodic state.
[0014] The circuit implementation method for a dual-vortex chaotic system provided by the present invention further includes: When k = 1, the range of x for the attractor of the dual-vortex chaotic system is [-2.1, 2.1], and the range of the voltage signal u1 is ±2V. When k = 10, the range of x for the attractor of the dual-vortex chaotic system is [-0.7, 0.7], and the range of the voltage signal u1 is ±0.7V. When k = 100, the range of x for the attractor of the double-vortex chaotic system is [-0.21, 0.21].
[0015] This invention provides a circuit implementation method for a double-vortex chaotic system. It constructs a fractal-based double-vortex chaotic system by applying a deformed Julia fractal with a quadratic coefficient k to a deformed Liu system. The dynamic characteristics of the double-vortex chaotic system are analyzed, and the attractor shape is determined to be four-winged. The influence of the quadratic coefficient k on the four-winged attractor and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent is determined. Based on the influence of the quadratic coefficient k on the four-winged attractor and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent, analog and / or digital circuits are constructed. The Julia fractal-based double-vortex chaotic system is implemented through analog and digital circuits. The consistency between circuit implementation and numerical analysis better verifies the chaotic characteristics of the system, expanding the application space of chaotic systems. Attached Figure Description
[0016] To more clearly illustrate the technical solutions in this invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of this invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.
[0017] Figure 1 This is a schematic flowchart of the circuit implementation method of the dual-vortex chaotic system provided in this embodiment; Figure 2a This is a schematic diagram of the attractor generated by the modified Liu system provided in this embodiment; Figure 2b This is a schematic diagram of the attractor generated by the dual-vortex chaotic system provided in this embodiment; Figure 3a This is the bifurcation diagram of the variable x as parameter a=[5, 15] changes with parameter a, provided in this embodiment; Figure 3b This is the bifurcation diagram of the variable x as parameter a=[8, 8.5] changes with parameter a, provided in this embodiment; Figure 4a This is the Lyapunov exponent spectrum when the parameter a=[5, 15] varies, as provided in this embodiment; Figure 4b This is the Lyapunov exponent spectrum when the parameter a=[8, 8.5] varies in this embodiment; Figure 5a This is the Poincaré mapping diagram of the deformed Liu system when the variable z=2 provided in this embodiment; Figure 5b This is the Poincaré mapping of the double-vortex chaotic system provided in this embodiment when the variable z=2; Figure 6a This is a schematic diagram of the attractor morphology of the double-vortex chaotic system when the quadratic term coefficient k=10 provided in this embodiment; Figure 6b This is a schematic diagram of the attractor morphology of the double-vortex chaotic system when the quadratic term coefficient k=100 provided in this embodiment; Figure 7 This is a schematic diagram of the maximum Lyapunov exponent spectrum of the dual-parameter control of the dual-vortex chaotic system provided in this embodiment. Figures 8a-8g This is a schematic diagram of the modular unit circuit in the analog circuit provided in this embodiment; Figures 9a-9c This is a schematic diagram of the main circuit in the analog circuit provided in this embodiment; Figure 10a This is a schematic diagram of the simulation results of the analog circuit when the quadratic term coefficient k=1 provided in this embodiment; Figure 10b This is a schematic diagram of the simulation results of the analog circuit when the quadratic coefficient k=10 provided in this embodiment; Figure 11 This is a hardware implementation framework diagram of the digital circuit provided in this embodiment; Figure 12 This is a schematic diagram of the digital circuit hardware implementation experimental platform provided in this embodiment; Figure 13a This is a schematic diagram of the hardware implementation result of the digital circuit when the quadratic term coefficient k=1 provided in this embodiment; Figure 13b This is a schematic diagram of the hardware implementation result of the digital circuit when the quadratic coefficient k=10 provided in this embodiment. Detailed Implementation
[0018] To make the objectives, technical solutions, and advantages of this invention clearer, the technical solutions of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this invention. All other embodiments obtained by those skilled in the art based on the embodiments of this invention without creative effort are within the scope of protection of this invention.
[0019] Figure 1 This is a schematic flowchart of the circuit implementation method of the dual-vortex chaotic system provided in this embodiment.
[0020] like Figure 1 As shown, the circuit implementation method of the dual-vortex chaotic system provided in this embodiment of the invention mainly includes the following steps: 101. By applying the deformed Julia fraction with quadratic coefficient k to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed.
[0021] Specifically, the state equation of the deformed Liu system is as follows (1): (1) In the formula, Represents the state variable, where, , , , , Indicates system parameters.
[0022] The modified Julia fractional form with quadratic coefficients set Applied to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed, and its dimensionless state equation is shown in Equation 2.
[0023] (2) In the formula, Represents state variables, , , , , Indicates system parameters, Julia derivation forms The coefficient of the quadratic term (or simply the quadratic coefficient) ).in, , , , , , It is a variable value. They cannot all be 0.
[0024] The construction process can be as follows: First, the system parameters of the Liu system are slightly adjusted to construct a modified Liu system, the mathematical form of which is: ; Where a=10, b=20, c=2.5, d=10, e=4, compared with the original Liu system, the modified Liu system sets the coefficient of the second term uw to 10. Under this parameter, the system is still in a chaotic state.
[0025] Julia's fractal form is Z. n+1 =Z n 2 +Z0, where Z n =x n +iy n Z n+1 =x n+1 +iy n+1 Z0 = x0 + iy0 are both complex numbers. This paper focuses on circuit design and ignores the influence of complex constants, i.e., Z0 = 0. In addition, a coefficient k is introduced into the quadratic term of the Julia fractional form to study its influence on the shape and size of the chaotic attractor.
[0026] This yields the deformed Julia fraction Z. n+1 =kZ n 2 Then we have: ; Introducing the Julia fraction with coefficients into the three-dimensional coordinate system (u, v, w), then (u, v, w) corresponds to (k(x)). 2 -y 2After differentiating ), 2kxy, z), the first derivative relationship of the corresponding variables becomes: ; Finding the inverse matrix and substituting it into the mathematical formula of the deformed Liu system yields a new system based on the deformed Julia fractal with coefficients: ; This yields the dimensionless equation of state: .
[0027] 102. Analyze the dynamic characteristics of the double-vortex chaotic system and determine that the attractor has a four-winged shape.
[0028] When parameter At that time, the double-vortex chaotic system produced a double-vortex attractor, such as Figure 2b As shown, the attractor generated by the deformed Liu system ( Figure 2a Compared to the previous system, the introduction of Julia fractals increases the number of attractors from two to four, making the system more complex and its attractor morphology more complex and diverse.
[0029] 103. Determine the influence of the quadratic coefficient k on the four-wing attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent.
[0030] Figure 3a A bifurcation diagram is presented for a double-vortex chaotic system with system parameter a=[5, 15] as variable x changes with parameter a. In most of the region a=[5, 12.5], the system is in a chaotic state, subsequently entering a periodic state. However, in some regions of the chaotic state, a clear periodic window also appears. Zooming in on the region a=[8, 8.5], a periodic window can be clearly observed when a=[8.13, 8.23], as shown below. Figure 3b As shown.
[0031] To more effectively analyze the motion characteristics of the system during parameter changes, the Lyapunov exponent spectrum of a fractal-based double-vortex chaotic system was studied, such as... Figure 4a and Figure 4b As shown. Figure 4a The Lyapunov exponent of the system as a function of parameter a is given when a = [5, 15]. Figure 4a As can be seen, when a = [12.5, 15], only LE1 = 0 in the Lyapunov exponent of the system, while all other quantities are less than zero, corresponding to periodic motion; while in most other regions, LE1 > 0, corresponding to chaotic motion [7, 8]. However, in some regions of the chaotic state, there are also cases where LE1 = 0, such as... Figure 4b As shown, magnifying the Lyapunov exponent spectrum in the region a=[8, 8.5] clearly reveals that LE1=0 when a=[8.13, 8.23], indicating that a periodic window has appeared in the system at this point. Comparing the bifurcation plot and the Lyapunov exponent spectrum clearly shows that their behavior with parameter a is completely consistent.
[0032] Figure 5a and Figure 5b Poincaré cross-sections of the deformed Liu system and the fractal-based double-vortex chaotic system at z=2 are presented. The figures show that the deformed Liu system exhibits two clusters of dense points on its Poincaré cross-section, resembling two wings, while the fractal-based double-vortex chaotic system shows four clusters of dense points on its Poincaré cross-section, resembling four wings. This further demonstrates that, compared to the deformed Liu system, the new system incorporating Julia fractals produces a double-vortex phenomenon, and its attractor morphology is indeed more complex and richer.
[0033] When k takes different values, the double-vortex attractors generated by the fractal-based double-vortex chaotic system are as follows: Figure 6a and Figure 6b As shown. Comparison Figure 2b It is evident that the shape of the chaotic attractor remains unchanged regardless of the value of k; only its size changes. When k=1, the attractor is confined to the range x=[-2.1, 2.1]; when k=10, the attractor's size significantly decreases, confining it to the range x=[-0.7, 0.7]. Figure 6a When k=100, the size of the attractor is further reduced and confined to the range of x=[-0.21,0.21]. Figure 6b As can be seen, the size of the phase diagram decreases continuously as the coefficient k increases, and the larger the value of k, the more the attractors gather.
[0034] To more clearly analyze the influence of system parameters on system characteristics, the distribution of the maximum Lyapunov exponent λ1 of the fractal-based double-vortex chaotic system in the two-parameter space was calculated, as follows: Figure 7 As shown, the darker the color, the smaller the value of λ1; the lighter the color, the larger the value of λ1. It can be seen that the overall hue of the graph changes parallel to the vertical axis as the parameter a changes. That is, for the same parameter a, regardless of the value of k, the value of the maximum Lyapunov exponent λ1 remains unchanged. This further illustrates that the quadratic term coefficient k only changes the size of the attractor but has no effect on the chaotic state of the system. When λ1≤0, the system is in periodic motion; when λ1>0, the system is in a chaotic state. Figure 7 In the middle, observing the state change of the system along the direction of k=10, we can see that... Figure 3a The bifurcation diagram of the system shown fits well. In most regions of a=[5, 12.5], the system is in a chaotic state and then enters a periodic state. However, in some regions of the chaotic state, such as the region of a=[8.13, 8.23], there are also obvious periodic windows.
[0035] 104. Based on the influence of the quadratic coefficient k on the four-wing attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent, construct analog circuits and / or digital circuits. Both analog and digital circuits are used to output electrical signals characterizing the four-wing attractor. The four-wing attractor shape corresponding to the electrical signal has more wings than the two-wing attractor shape of the deformed Liu system.
[0036] The process of constructing an analog circuit is as follows: To further verify the characteristics of the fractal-based double-vortex chaotic system and study the influence of the coefficient k in the Julia fractal on the system's attractor, a system circuit was designed and simulated on the Multisim platform. The analog circuit includes modular unit circuits and a main circuit. The modular unit circuits include at least several sub-modules for implementing different signal combination functions. Each sub-module is composed of signal processing elements and adjustment elements. The signal combination functions include at least signal exponentiation-subtraction combination, multiplication combination, exponentiation-addition combination, fraction and exponentiation combination, and exponentiation series combination. The main circuit includes multiple channel circuits corresponding to the voltage signals generated for each state variable. Each channel circuit includes an operational element, an energy storage element, and multiple adjustment elements. The state variables of the double-vortex chaotic system correspond to the voltage signals output by the main circuit. The value of k is adjusted by regulating the adjustment element associated with the quadratic coefficient k. The value of k only changes the size of the output double-vortex attractor, not its shape. The analog circuit uses a preset power supply voltage, the adjustment element is a resistor with a preset resistance value, and the energy storage element is a capacitor with a preset capacitance value.
[0037] In the modular unit circuit, voltages u1, u2, and u3 correspond to state variables x, y, and z, respectively. An operational amplifier with a power supply voltage of E = ±24V and analog multipliers and dividers with a gain of 1 are used. Combining formula (2), let... , , Then the system equation becomes (3): (3) To make the circuit clearer and more understandable, a modular circuit design concept is adopted, making: (4) Then system equation (2) can be rewritten as: (5) Combining formulas (3) and (4), modular circuits such as power combination and fraction were designed, such as... Figures 8a-8g As shown in the figure. The resistors Ra, R1, R2, R5, and R10 are each 0.5 ohms. 1 2 5 and 10 Then we have: (6); (7); (8); (9); (10); (11); (12); like Figures 8a-8g As shown, these are modules one, two, three, four, five, six, and seven, respectively. Module one implements the power-subtraction combination of signals using two multipliers, three resistors, and operational amplifier U7A, i.e., (x... 2 -y 2 The second module implements the multiplication combination of signals, i.e., 2xy, using a multiplier, two resistors, and operational amplifier U11A. The third module implements the power-addition combination of signals, i.e., 2(xyy), using two multipliers, three resistors, and operational amplifier U15A. 2 +y 2 The fourth module, through a multiplier, two resistors, and an operational amplifier U6A, combines with module 3 to realize the combination of fractional and exponential functions of the signal, i.e., x / (2x). 2 +2y 2 The fifth module, through a multiplier, two resistors, and operational amplifier U9A, combines with module 3 to realize the fractional and exponential combination of the signal, i.e., y / (2x 2 +2y 2 The sixth module, through three resistors and operational amplifier U1, combines with modules 1 and 2 to realize a series of power-law combinations of signals, namely 20xy-10(x... 2 -y 2 The seventh module, combined with module 1 using a multiplier, three resistors, and operational amplifier U3, implements a series of power-law combinations of signals, namely 20(x). 2 -y 2 )-10z(x 2 -y 2 ).
[0038] Combining formula (5), the main circuit of the fractal-based double-vortex chaotic system is designed. In order to observe the output waveform more clearly, a time scale transformation is performed on t, setting t=10. -3 If τ, then the main circuit equation becomes: (13) The main circuit was designed based on formula (13), as follows: Figures 9a-9c As shown. The values of resistors R1, R4, and R10 are 1... 4 and 10 In the integrating circuit, capacitors C1, C2, and C3 are all 10nF. When the coefficient k is 1, the resistance Rb is 2.5Ω. When the coefficient k is 10, no other circuit changes are needed and the resistance value remains the same; only the resistance of resistor Rb needs to be adjusted to 0.025. That's it. Main circuit and modular unit circuit ( Figures 8a-8g By combining these elements, the overall analog circuit of the system can be constructed, making the circuit design clearer and more understandable.
[0039] The main circuit comprises three channel circuits. The first channel circuit includes at least two operational elements, one energy storage element, and multiple adjustment elements. The output terminal of the operational element outputs the voltage signal corresponding to the first state variable, which is generated as state variable signal x (u1) through operational amplifiers U2 and U16A, capacitor C1, resistors R1 and R10 (3 units), and two multipliers. The second channel circuit includes at least two operational elements, one energy storage element, and multiple adjustment elements. The output terminal of the operational element outputs the voltage signal corresponding to the second state variable, which is generated as state variable signal y (u2) through operational amplifiers U5 and U17A, capacitor C2, resistors R1 and R10 (3 units), and two multipliers. The third channel circuit includes at least two operational elements, one energy storage element, and multiple adjustment elements. The output terminal of the operational element outputs the voltage signal corresponding to the third state variable, and the resistance value of the adjustment element associated with the quadratic coefficient k is adjusted according to the value of the quadratic coefficient k. That is, the state variable signal z, i.e. u3, is generated through operational amplifiers U12A and U18A, capacitor C3, resistors R1, R4, R10, Rb and a multiplier.
[0040] Specifically, in the first channel circuit, the non-inverting input of operational amplifier U16A is grounded, and the inverting input is connected to two 10-bit circuits. The parallel terminals of resistors (R10) are connected, and the other ends of the two parallel resistors are connected to the output terminals of two multipliers (A8 and A11), respectively. The first and second non-inverting input terminals of multiplier A8 are connected to the output terminals of modules 5 and 7, respectively, and the first and second non-inverting input terminals of multiplier A11 are connected to the output terminals of modules 4 and 6, respectively. A 1 A resistor (R1) is connected across the inverting input and output terminals; the non-inverting input of operational amplifier U2 is grounded, and the inverting input is connected through a 10Ω resistor. Resistor (R10) is connected to the output terminal of operational amplifier U16A, and capacitor C1 is connected across the inverting input terminal and the output terminal.
[0041] In the second channel circuit, the non-inverting input of operational amplifier U17A is grounded, and the inverting input is connected to two 10-bit circuits. The parallel terminals of resistors (R10) are connected, and the other ends of the two parallel resistors are connected to the output terminals of two multipliers (A12 and A13), respectively. The first and second non-inverting input terminals of multiplier A12 are connected to the output terminals of modules 4 and 7, respectively. The first and second non-inverting input terminals of multiplier A13 are connected to the inverted value of the output terminal of module 5 and the output terminal of module 6, respectively. A resistor (R1) is connected across the inverting input and output terminals; the non-inverting input of operational amplifier U5 is grounded, and the inverting input is connected through a 10Ω resistor. Resistor (R10) is connected to the output terminal of operational amplifier U17A, and capacitor C2 is connected across the inverting input terminal and the output terminal.
[0042] In the third channel circuit, the non-inverting input of operational amplifier U18A is grounded, and the inverting input is connected to a resistor with a resistance of 2.5Ω. (when k=1) and 4 The parallel terminals of the two resistors (Rb and R4) are connected. The other end of Rb is connected to the output of multiplier A15. The first and second non-inverting inputs of multiplier A15 are both connected to the output of module 1. The other end of R4 is connected to the inverting value of the output of operational amplifier U12A in this channel circuit. A resistor (R1) is connected across the inverting input and output terminals; the non-inverting input of operational amplifier U12A is grounded, and the inverting input is connected through a 10Ω resistor. Resistor (R10) is connected to the output of operational amplifier U18A, and capacitor C3 is connected across the inverting input and output. When k=10, the resistance of resistor Rb is adjusted to 0.025Ω. That's all.
[0043] Figure 10a and Figure 10bThe circuit simulation results of the double-vortex Liu system based on the deformed Julia fractal with coefficients are presented for different values of coefficient k. It can be seen that the circuit simulation produces a four-winged chaotic attractor with the same shape for different values of k. When k=1, the attractor is confined to a range of about ±2V. Figure 10a When k=10, the size of the attractor decreases and is limited to a range of approximately ±0.7V. Figure 10b The circuit simulation results further verified the influence of the coefficient k in the Julia fractional form on the system attractor. Different values of k do not affect the shape of the chaotic attractor, but only affect the size of the attractor. The larger the value of k, the more the attractor gathers.
[0044] Digital circuits consist of hardware and software components. The hardware component includes a microcontroller development board, a digital-to-analog converter (DAC), and a signal display device. The microcontroller development board converts the processed digital signal into an analog signal via the DAC and outputs it to the signal display device. The software component controls the hardware to implement the discretization calculations and signal output of the dual-vortex chaotic system. The software component includes at least initialization functions, iterative calculation functions, and scaling functions. The process of constructing a digital circuit is as follows: In the system's digital circuit design, the STM32 F103 series development board is used for hardware implementation. Figure 11 This is a hardware implementation framework diagram of the system. Figure 12 This section describes the hardware implementation of the system as an experimental platform. Software programming, based on the Keil uVision5 development environment, mainly includes a series of initialization functions, iterative operation functions, and scaling functions. These functions handle data processing, conversion, and signal output to a dual-trace oscilloscope. The initialization functions include HAL_Init(), MX_TIM2_Init(), MX_GPIO_Init(), MX_DMA_Init(), MX_DAC_Init(), and SystemClock_Config(), which initialize all peripheral devices, the STM32 microcontroller's timer TIM2, GPIO (general purpose input / output) ports, the DMA controller, the DAC (digital-to-analog converter), and the system clock. The iterative operation function iterate(&x,&y,&z) uses Runge-Kutta discretization to discretize the system. The scaling function uint8_t scaleValue(int inputValue) scales the input value (e.g., -2100 to 2100) to a target range (0 to 255).
[0045] In the experiment, the same parameters and initial values as the new system were set for iterative calculations. After debugging, the data processing results were output to a dual-trace oscilloscope after passing through a D / A converter, and the xy-plane attractor of the system was obtained as follows: Figure 13a and Figure 13b As shown, the results are compared with those obtained from Matlab numerical calculations. Figure 2a , Figure 2b , Figure 6a , Figure 6b Multisim circuit simulation results Figure 10a and Figure 10b As can be seen, the hardware implementation results agree well with the numerical calculation and circuit simulation results. Coefficient k=1 ( Figure 13a ) and k=10 ( Figure 13b The attractor's shape remains largely unchanged, but when k=1, the attractor range is [-150, 150] mV, while when k=10, the attractor range is [-50, 50] mV. This indicates that the size of the chaotic attractor changes but its shape remains the same when the coefficient k takes different values, and different values of k do not affect the chaotic state of the system. Hardware implementation further verifies the chaotic characteristics of the new system.
[0046] The method of this embodiment enhances the complexity and flexibility of attractor signals and further expands the practical application of chaotic systems in the era of big data.
[0047] Complexity refers to the fact that fractal-based double-vortex chaotic systems possess both mathematical reconfigurability and physical controllability. By applying Julia fractals to the deformed Liu system, a new system with more complex forms was constructed, resulting in double-vortex attractors with richer morphologies. Simultaneously, the phase diagram, Lyapunov exponent spectrum, bifurcation diagram, and other key characteristics of the new system were analyzed on the Matlab platform, verifying its chaotic properties.
[0048] The flexibility refers to the application of the Julia fractal in the fractal-based double-vortex chaotic system, which sets the quadratic coefficient k to achieve flexible changes in the size of the double-vortex attractor. At the same time, the two-parameter exponential spectrum of the system in the two-dimensional parameter space is analyzed, proving that the quadratic coefficient k can guarantee the function of controlling the size of the attractor without affecting its shape.
[0049] Practical applicability refers to the circuit design, verification, and application of the fractal-based dual-vortex chaotic system. The analog circuit employs a modular design concept, utilizing modular unit circuits such as exponentiation and fractions on the Multisim platform to simplify the main circuit design. The digital circuit implementation, specifically the generation of actual digital signals, was achieved using an STM32 F103 series development board, verifying the system's chaotic characteristics and practicality. Both circuit simulation and digital circuit implementation results are consistent with Matlab numerical calculations, fully validating the feasibility of combining fractals and chaos. Therefore, information from the fractal-based dual-vortex chaotic system can be widely applied to cutting-edge applications such as secure communication, signal encryption, and noise simulation.
[0050] Taking the encryption of large data products such as images and videos with large data volumes and strong correlation between adjacent data as an example, the application of fractal-based double-vortex chaotic system signals is more secure—even if the encryption method and the shape of the encrypted signal are known, the correct key cannot be determined because the quadratic term coefficient k controls the size of the encrypted signal.
[0051] The device embodiments described above are merely illustrative. The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this embodiment according to actual needs. Those skilled in the art can understand and implement this without any creative effort.
[0052] Through the above description of the embodiments, those skilled in the art can clearly understand that each embodiment can be implemented by means of software plus necessary general-purpose hardware platforms, and of course, it can also be implemented by hardware. Based on this understanding, the above technical solutions, in essence or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as ROM / RAM, magnetic disk, optical disk, etc., and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments or some parts of the embodiments.
[0053] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A circuit implementation method for a dual-vortex chaotic system, characterized in that, include: By applying the deformed Julia fractal with quadratic coefficient k to the deformed Liu system, a fractal-based double-vortex chaotic system is constructed. Analyzing the dynamic characteristics of the dual-vortex chaotic system, the attractor shape was determined to be four-winged; Determine the influence of the quadratic term coefficient k on the four-wing attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent. Based on the influence of the quadratic coefficient k on the four-winged attractor of the double-vortex chaotic system and the parameter a of the double-vortex chaotic system on the maximum Lyapunov exponent, an analog circuit and / or a digital circuit are constructed. Both the analog circuit and the digital circuit are used to output an electrical signal characterizing the four-winged attractor. The four-winged attractor shape corresponding to the electrical signal has more wings than the two-winged attractor shape of the deformed Liu system.
2. The circuit implementation method of the dual-vortex chaotic system according to claim 1, characterized in that, The analog circuit includes modular unit circuits and a main circuit; The modular unit circuit includes at least a number of sub-modules for implementing different signal combination functions. Each sub-module is composed of signal processing elements and adjustment elements. The signal combination functions include at least signal exponentiation and subtraction combination, multiplication combination, exponentiation and addition combination, fraction and exponentiation combination, and exponentiation series combination. The main circuit includes multiple channel circuits that generate voltage signals for each state variable, and each channel circuit includes a computing element, an energy storage element, and multiple regulating elements. The state variables of the dual-vortex chaotic system correspond to the voltage signal output by the main circuit. The value of k can be adjusted by adjusting the adjustment element associated with the quadratic term coefficient k. The value of k only changes the size of the output dual-vortex attractor, without changing the shape of the attractor. The analog circuit uses a preset power supply voltage, the regulating element is a resistor with a preset resistance value, and the energy storage element is a capacitor with a preset capacitance value.
3. The circuit implementation method of the dual-vortex chaotic system according to claim 2, characterized in that, The submodule includes: The first module uses two multipliers, three resistors, and an operational amplifier to perform power-subtraction combinations of signals. The second module uses a multiplier, two resistors, and an operational amplifier to perform signal multiplication and combination. The third module uses two multipliers, three resistors, and an operational amplifier to perform power-addition combinations of signals. The fourth module, through a multiplier, two resistors, and an operational amplifier, combines with the third module to realize the combination of fractional and exponential functions of the signal; The fifth module, through a multiplier, two resistors, and an operational amplifier, combines with the third module to realize the combination of fractional and exponential functions of the signal; The sixth module, through three resistors and an operational amplifier, combines with the first and second modules to realize a series of power-law combinations of signals; The seventh module combines a multiplier, three resistors, and an operational amplifier with the first module to realize a series of power-law combinations of signals.
4. The circuit implementation method of the dual-vortex chaotic system according to claim 2, characterized in that, The plurality of channel circuits includes a first channel circuit, a second channel circuit, and a third channel circuit; The first channel circuit includes at least two operational elements, one energy storage element, and multiple regulating elements. The output terminal of the operational element outputs a voltage signal corresponding to the first state variable. The second channel circuit includes at least two operational elements, one energy storage element, and multiple regulating elements. The output terminal of the operational element outputs a voltage signal corresponding to the second state variable. The third channel circuit includes at least two operational elements, one energy storage element, and multiple adjustment elements. The output terminal of the operational element outputs a voltage signal corresponding to the third state variable, and the resistance value of the adjustment element associated with the quadratic coefficient k is adjusted according to the value of the quadratic coefficient k.
5. The circuit implementation method of the dual-vortex chaotic system according to claim 1, characterized in that, The digital circuit includes hardware and software components; The hardware component includes a microcontroller development board, a digital-to-analog converter, and a signal display device. The microcontroller development board is used to convert the processed digital signal into an analog signal through the digital-to-analog converter and then output it to the signal display device. The software component is used to control the hardware component to implement the discretization operation and signal output of the dual-vortex chaotic system. The software component includes at least an initialization function, an iterative operation function, and a scaling function.
6. The circuit implementation method of the dual-vortex chaotic system according to claim 5, characterized in that, The initialization functions include at least a peripheral initialization function for initializing peripheral devices, a timer initialization function for initializing timers, a port initialization function for initializing general-purpose input / output ports, a controller initialization function for initializing the direct memory access controller, a digital-to-analog converter initialization function for initializing the digital-to-analog converter, and a clock initialization function for initializing the system clock. The iterative operation function uses a preset discretization method to discretize the state equation of the double-vortex chaotic system; The scaling function is used to scale the input signal value to a preset target range.
7. The circuit implementation method of the dual-vortex chaotic system according to claim 1, characterized in that, Determining the influence of the quadratic term coefficient k on the four-winged attractor of the double-vortex chaotic system includes: As the coefficient k of the quadratic term increases, the size of the four-winged attractor of the double-vortex chaotic system gradually decreases and tends to converge, while the shape of the four-winged attractor remains unchanged.
8. The circuit implementation method of the dual-vortex chaotic system according to claim 1, characterized in that, The dimensionless state equation of the dual-vortex chaotic system is: ； In the formula, Represents the state variable, where, , , , , Indicates system parameters, This represents the coefficient of the quadratic term in the transformed Julia fractional form.
9. The circuit implementation method of the dual-vortex chaotic system according to claim 8, characterized in that, The determination of the influence of parameter 'a' on the maximum Lyapunov exponent of the dual-vortex chaotic system includes: When the maximum Lyapunov exponent is greater than 0, the dual-vortex chaotic system is in a chaotic state. When the maximum Lyapunov exponent is less than or equal to 0, the double-vortex chaotic system is in a periodic state. When system parameter a is in the first preset interval, most of the region of the dual-vortex chaotic system is in a chaotic state, and there is a parameter sub-interval corresponding to a preset periodic window in the chaotic state region. When system parameter a is in the second preset interval, the dual-vortex chaotic system is in a periodic state.
10. The circuit implementation method of the dual-vortex chaotic system according to claim 8, characterized in that, Also includes: When k = 1, the range of x for the attractor of the dual-vortex chaotic system is [-2.1, 2.1], and the range of the voltage signal u1 is ±2V. When k = 10, the range of x for the attractor of the dual-vortex chaotic system is [-0.7, 0.7], and the range of the voltage signal u1 is ±0.7V. When k = 100, the range of x for the attractor of the double-vortex chaotic system is [-0.21, 0.21].