Full-grid multi-scroll chaotic attractor generation method based on position-adjustable robust chaos

Abstract

The invention provides a full-grid multi-scroll chaotic attractor generation method based on position-adjustable robust chaos, and the method comprises the steps: analyzing the influence of system parameters on a balance point and a characteristic equation of a system based on a power system local stability theory, and designing a position-adjustable robust chaotic system; designing a multi-valued weighting function, controlling position parameters of the position-adjustable robust chaotic system, and expanding the number of scrolls in the direction of the position-adjustable variable; designing a time-varying shift control function, realizing shift control of a non-position adjustable variable, and expanding the number of scrolls in the direction of the non-position adjustable variable; and analyzing and verifying the internal relation between the full-grid multi-scroll chaotic features such as the scroll number, the scroll spacing and the scroll geometric distribution and the multi-valued weighting function and the time-varying shift control function.

Description

Technical Field

[0001] The invention relates to the field of chaotic systems, in particular to a method for generating full-grid multi-scroll chaotic attractors based on position-adjustable robust chaos. Background Art

[0002] The modeling, analysis and application of chaotic systems with complex dynamic characteristics involve classical field theories and quantum field theories such as electronics, communications, biology, computers, control, mathematics, chemistry, and physics, and is a multidisciplinary field. In-depth research on chaos can not only help us reveal the dynamic behavior and internal structure of natural systems, but also promote the design of new chaotic systems and the development of chaos theory, and broaden the application scope of chaos, such as artificial intelligence, bioinformatics, brain structure, brain diseases, information security and new generation information technology. In particular, the complex output signals generated by chaotic systems, with their unpredictable and highly random characteristics, have become an ideal choice for security keys and carrier signals in information security communications. For this reason, multimedia data encryption and secure communication based on synchronization technology have become an important application field of chaos theory. Existing studies have shown that compared with single-scroll or double-scroll chaotic systems, multi-scroll (or multi-wing) chaotic systems will exhibit more complex dynamic characteristics, thus better meeting the design requirements of encrypted communication systems.

[0003] Since the famous double-scroll Cai circuit was proposed by Chinese-American scholar Professor Shaotang Cai, researchers have made great efforts to construct and realize multi-scroll chaotic attractors. At present, most studies increase the number of saddle-focus equilibrium points of nonlinear systems by introducing nonlinear elements such as piecewise quadratic functions, sawtooth functions, polynomial functions, hysteresis functions, sine functions, transformation functions, step functions, and trigonometric functions, thereby expanding the number of scrolls. This method has wide applicability in most nonlinear systems. In addition, there are many methods for generating multi-scroll chaotic attractors, such as Julia fractals, heteroclinic rings, symmetric mappings, polynomial transformations, and switching control. It is particularly worth mentioning that in recent years, the method of constructing multi-scroll attractors using memristors and neurons has attracted much attention. Although various design methods have emerged in an endless stream, exploring new multi-scroll chaotic attractor modeling methods is still a highly anticipated and challenging frontier topic. Summary of the invention

[0004] The present application provides a method for generating a full-grid multi-scroll chaotic attractor based on position-adjustable robust chaos, which specifically comprises the following steps:

[0005] Step 1: Based on the theory of local stability of dynamical systems, analyze the influence of system parameters on the system equilibrium point and the characteristic equation at the equilibrium point, and design a position-adjustable robust chaotic system;

[0006] Step 2: Write a Matlab program to draw parameter bifurcation diagrams, Lie index diagrams and phase diagrams to verify the position adjustability and chaos robustness of the position-adjustable robust chaotic system;

[0007] Step 3: Design a multi-valued weighted function to control the position parameters of the position-adjustable robust chaotic system and expand the number of vortices in the direction of the position-adjustable variable;

[0008] Step 4: Design a time-varying shift control function to realize the shift control of the non-position adjustable variable and expand the number of scrolls in the direction of the non-position adjustable variable;

[0009] Step 5: Analyze the intrinsic relationship between the vortex number, vortex distance, vortex geometric distribution, multi-valued weighted function and time-varying shift control function;

[0010] Step 6: Write a Matlab program to draw the parameter bifurcation diagram and attractor phase diagram under the action of multi-valued weighted function and time-varying shift control function to verify the feasibility and effectiveness of the full-grid multi-vortex chaotic attractor generation method.

[0011] As a further solution of the present invention: the model of the N-dimensional position adjustable robust chaotic system is constructed as follows:

[0012]

[0013] In the formula, x1,…x i ,…x N are the state variables of the system; f1,…f i ,…f N ,F i is the state equation function of the system; b is the position parameter, which can adjust the non-zero equilibrium point of the system at x according to a specific functional relationship. j1 ,…x jk The position of the component in the direction, and the adjustment of parameter b will not have any effect on the characteristic equation of the system. At this time, the position parameter b will affect the state variable x according to this functional relationship. j1 ,…x jk The phase space position of the system is modulated without affecting the chaotic dynamic characteristics of the system, that is, the system has a constant Lie index and robust chaotic characteristics with respect to the parameter b.

[0014] As a further solution of the present invention: the construction of the full-grid multi-scroll chaotic system includes the following steps:

[0015] Step 1: Design pulse width and adjustable variable x at each position j1 ,…x jk The multi-valued weighted function H that matches the maximum amplitude of the attractor in the direction j (t), weighted control of the position parameter b, the following dynamic equation is obtained

[0016]

[0017] Thus, in the location parameter b and the multi-valued weighting function H j (t), the position-adjustable robust chaotic system is realized at each position with adjustable variables x j1 ,…x jk The expansion of the number of scrolls in the direction.

[0018] Step 2: Design the time-varying shift control function G i (t), for x j1 ,…x jk Non-positionally adjustable variables x i (i=1,…j1-1,…j k +1,…N) for displacement control, and then the following dynamic equation is obtained

[0019]

[0020] Thus, under the action of the time-varying shift control function Gi(t), the position-adjustable robust chaotic system is realized in the case of non-position-adjustable variables x i (i=1,…j1-1,…j k +1,…N) direction.

[0021] As a further solution of the present invention: taking a three-dimensional position adjustable robust chaotic system as an example, an embodiment of a method for generating a full-grid multi-scroll chaotic attractor is provided:

[0022] Step 1: Construct a three-dimensional position-adjustable robust chaotic system

[0023]

[0024] In the formula, x1, x2, x3 are the state variables of the system; a, b, c are the system parameters, and parameter b is called the position parameter, which can adjust the position of the component of the non-zero equilibrium point of the system in the direction of x3 according to the functional relationship x3 = -b, and parameter b will not have any effect on the characteristic equation of the system. At this time, the position parameter b will modulate the phase space position of the state variable x3 according to the functional relationship x3 = -b, and will have no effect on the chaotic dynamic characteristics of the system, that is, the system has a constant Lie index and robust chaotic characteristics with respect to parameter b.

[0025] Step 2: Design a multi-valued weighting function By weighting the parameter b, the following dynamic equation is obtained:

[0026]

[0027] In the formula, M, A, ω, τ are multi-valued weighted function parameters; in particular, M is a positive integer corresponding to the number of multi-valued weighted function values. Therefore, under the joint action of the position parameter b and the multi-valued weighted function H3(t), the expansion of the number of vortices in the direction of the position adjustable variable x3 of the three-dimensional position adjustable robust chaotic system is realized.

[0028] Step 3: Based on step 2, design time-varying shift control functions G1(t)=0.2t and G2(t)=0.8t, respectively perform shift control on non-position adjustable variables x1 and x2, and obtain the following dynamic equations:

[0029]

[0030] Therefore, under the action of the time-varying shift control functions G1(t) and G2(t), the number of scrolls of the position-adjustable robust chaotic system in the directions of non-position-adjustable variables x1 and x2 is expanded respectively. In this way, the functions G1(t), G2(t) and H3(t) can respectively expand the number of full-grid multi-scrolls of the three-dimensional position-adjustable robust chaotic system in the directions of x1, x2 and x3.

[0031] Compared with the prior art, the beneficial effects of the present invention are as follows: by designing a position-adjustable robust chaotic system and applying a multi-valued weighted function and a time-varying shift control function to it, the expansion of the full-dimensional grid multi-vortex attractor of the position-adjustable robust chaotic system can be achieved, and the number of vortices, the vortex spacing and the vortex geometric distribution of the multi-vortex attractor can be modulated through the relevant parameters of these functions.

[0032] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0034] Figure 1 Phase diagram of the position-adjustable robust chaotic system.

[0035] Figure 2 It is the expansion of the number of vortices in the x3 direction for the position-adjustable robust chaotic system.

[0036] Figure 3 It is the expansion of the number of vortices in the x1 direction for the position-adjustable robust chaotic system.

[0037] Figure 4 It is the expansion of the number of vortices in the x2 direction for the position-adjustable robust chaotic system.

[0038] Figure 5 It is the geometric distribution control of multi-scroll chaotic attractors in the x1x3 direction.

[0039] Figure 6 It is the geometric distribution control of multi-scroll chaotic attractors in the x2x3 direction.

[0040] Figure 7 Schematic diagram of the process of generating full-grid multi-scroll chaotic attractors. DETAILED DESCRIPTION

[0041] The technical solutions in the embodiments of the present invention are described clearly and completely below. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0042] See also Figures 1 to 6 In an embodiment of the present invention, a method for generating a full-grid multi-scroll chaotic attractor based on position-adjustable robust chaos comprises the following steps:

[0043] Step 1: Based on the local stability theory of dynamic systems, analyze the influence of system parameters on the system equilibrium point and the characteristic equation at the equilibrium point, and design a position-adjustable robust chaotic system;

[0044] Step 2: Write a Matlab program to draw parameter bifurcation diagrams, Lie index diagrams and phase diagrams to verify the position adjustability and chaos robustness of the position-adjustable robust chaotic system;

[0045] Step 3: Design a multi-valued weighted function to control the position parameters of the position-adjustable robust chaotic system and expand the number of vortices in the direction of the position-adjustable variable;

[0046] Step 4: Design a time-varying shift control function to realize the shift control of the non-position adjustable variable and expand the number of scrolls in the direction of the non-position adjustable variable;

[0047] Step 5: Analyze the intrinsic relationship between the vortex number, vortex distance, vortex geometric distribution, multi-valued weighted function and time-varying shift control function;

[0048] Step 6: Write a Matlab program to draw the parameter bifurcation diagram and attractor phase diagram under the action of multi-valued weighted function and time-varying shift control function to verify the feasibility and effectiveness of the full-grid multi-vortex chaotic attractor generation method.

[0049] As a further solution of the present invention: the model of the N-dimensional position adjustable robust chaotic system is constructed as follows:

[0050]

[0051] In the formula, x1,…x i ,…x N are the state variables of the system; f1,…f i ,…f N ,F i is the state equation function of the system; b is the position parameter, which can adjust the non-zero equilibrium point of the system at x according to a specific functional relationship. j1 ,…x jk The position of the component in the direction, and the adjustment of parameter b will not have any effect on the characteristic equation of the system. At this time, the position parameter b will affect the state variable x according to this functional relationship. j1 ,…x jk The phase space position of the system can be modulated without any effect on the chaotic dynamic characteristics of the system, that is, the system has a constant Lie index and robust chaotic characteristics with respect to the parameter b.

[0052] As a further solution of the present invention: the construction of the full-grid multi-scroll chaotic system includes the following steps:

[0053] Step 1: Design pulse width and x j1 ,…x jk The multi-valued weighted function H that matches the maximum amplitude of attractors in each direction j (t), weighted control of the position parameter b, the following dynamic equation is obtained

[0054]

[0055] Thus, in the location parameter b and the multi-valued weighting function H j (t), the position-adjustable robust chaotic system is realized at x j1 ,…x jk The expansion of the number of scrolls in the direction.

[0056] Step 2: Design the time-varying shift control function G i (t), for x j1 ,…x jk The remaining variables x i (i=1,…j1-1,…j k +1,…N) for displacement control, and then the following dynamic equation is obtained

[0057]

[0058] Thus, in the time-varying shift control function G i (t), the position-adjustable robust chaotic system is realized at x i (i=1,…j1-1,…j k +1,…N) directions.

[0059] As a further solution of the present invention: taking a three-dimensional position adjustable robust chaotic system as an example, an embodiment of a method for generating a full-grid multi-scroll chaotic attractor is provided:

[0060] Step 1: Construct a three-dimensional position-adjustable robust chaotic system

[0061]

[0062] In the formula, x1, x2, and x3 are the state variables of the system; a, b, and c are system parameters, among which parameter b is called the position parameter, which can adjust the position of the component of the non-zero equilibrium point of the system in the x3 direction according to the functional relationship x3=-b. At the same time, parameter b will not have any effect on the characteristic equation of the system. At this time, the position parameter b will modulate the phase space position of the state variable x3 according to the functional relationship x3=-b, and will have no effect on the chaotic dynamic characteristics of the system, that is, the system has a constant Lie index and robust chaotic characteristics with respect to parameter b. When parameter b takes different values, the position of the position-adjustable robust chaotic system in the x3 direction is effectively regulated, such as Figure 1 shown.

[0063] Step 2: Design a multi-valued weighting function By weighting the parameter b, the following dynamic equation is obtained:

[0064]

[0065] Where M, A, ω, τ are multi-valued weighting function parameters; in particular, M is a positive integer corresponding to the number of multi-valued weighting function values.

[0066] Step 3: Based on step 2, design time-varying shift control functions G1(t)=0.2t and G2(t)=0.8t, and perform shift control on variables x1 and x2 respectively, and obtain the following dynamic equations:

[0067]

[0068] In this way, the functions G1(t), G2(t) and H3(t) can respectively realize the expansion of the number of vortices in the x1, x2 and x3 directions of the three-dimensional position-adjustable robust chaotic system. When the parameters a=2, c=30, A=1, ω=0.02, τ=0.628, M=4, the expansion effect of the number of vortices in the position-adjustable variable x3 direction is as follows: Figure 2 When G1(t) = 0.2t, G2(t) = 0.8t, the expansion effect of the number of scrolls in the direction of the non-position adjustable variables x1 and x2 is as follows: Figure 3 , 4In addition, when G1(t) and G2(t) are selected as time-varying multi-valued control functions, the control results of the multi-vortex chaotic attractor in the x1, x3 direction and the x2, x3 direction are as follows: Figure 5 and Figure 6 As shown. Figure 3 and Figure 4 After comparison, it is found that by designing different forms of time-varying shift control functions G1(t) and G2(t), the geometric distribution and number of vortex attractors in the phase space can be effectively controlled.

[0069] like Figure 7 As shown, the method of the present application includes the following steps: Step 1: Based on the local stability theory of dynamic systems, analyze the influence of system parameters on the system equilibrium point and the characteristic equation at the equilibrium point, and design a position-adjustable robust chaotic system; Step 2: Write a Matlab program to draw a parameter bifurcation diagram, a Lie index diagram and a phase diagram, and verify the position adjustability and chaos robustness of the position-adjustable robust chaotic system; Step 3: Design a multi-valued weighted function to control the position parameters of the position-adjustable robust chaotic system, and expand the number of vortices in the direction of the position-adjustable variable; Step 4: Design a time-varying shift control function to realize the shift control of non-position-adjustable variables, and expand the number of vortices in the direction of non-position-adjustable variables; Step 5: Analyze the internal connection and law of the full-grid multi-vortex chaotic characteristics such as the number of vortices, vortex distance, and vortex geometric distribution and the multi-valued weighted function and the time-varying shift control function; Step 6: Write a Matlab program to draw a parameter bifurcation diagram and an attractor phase diagram under the action of the multi-valued weighted function and the time-varying shift control function, and verify the feasibility and effectiveness of the full-grid multi-vortex chaotic attractor generation method.

[0070] It will be apparent to those skilled in the art that the invention is not limited to the details of the exemplary embodiments described above and that the invention can be implemented in other specific forms without departing from the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A method for generating full-grid multi-scroll chaotic attractors based on position-adjustable robust chaos, characterized in that: Based on the local stability characteristics of the equilibrium point, a position-adjustable robust chaotic system is designed, and a multi-valued weighting function and a time-varying shift control function are introduced to construct a full-grid multi-vortex chaotic attractor generation method with controllable vortex number, vortex spacing, and vortex geometric distribution. The program includes the following steps: Step 1: Based on the local stability theory of dynamic systems, analyze the influence of system parameters on the system equilibrium point and the characteristic equation at the equilibrium point, and design a position-adjustable robust chaotic system; Step 2: Write a Matlab program to draw parameter bifurcation diagrams, Lie index diagrams and phase diagrams to verify the position adjustability and chaos robustness of the position-adjustable robust chaotic system; Step 3: Design a multi-valued weighted function to control the position parameters of the position-adjustable robust chaotic system and expand the number of vortices in the direction of the position-adjustable variable; Step 4: Design a time-varying shift control function to realize the shift control of the non-position adjustable variable and expand the number of scrolls in the direction of the non-position adjustable variable; Step 5: Analyze the intrinsic relationship between the vortex number, vortex distance, vortex geometric distribution, multi-valued weighted function and time-varying shift control function; Step 6: Write a Matlab program to draw the parameter bifurcation diagram and attractor phase diagram under the action of multi-valued weighted function and time-varying shift control function to verify the feasibility and effectiveness of the full-grid multi-vortex chaotic attractor generation method.

2. The method according to claim 1, characterized in that The phase space position of the variable of the position-adjustable robust chaotic system can be changed by adjusting the position parameter; in the step 1, the model of the constructed N-dimensional position-adjustable robust chaotic system is as follows: In the formula, x1,…x i ,…x N are the state variables of the system; f1,…f i ,…f N ,F i is the state equation function of the system; b is the position parameter, which can adjust the non-zero equilibrium point of the system at x according to a specific functional relationship. j1 ,…x jk The position of the component in the direction, and the adjustment of parameter b will not have any effect on the characteristic equation of the system. At this time, the position parameter b will affect the state variable x according to this functional relationship. j1 ,…x jk The phase space position of the system is modulated without affecting the chaotic dynamic characteristics of the system, that is, the system has a constant Lie index and robust chaotic characteristics with respect to the parameter b.

3. The method according to claim 1, characterized in that The generation of multi-scroll attractors in the position-adjustable variable direction of the position-adjustable robust chaotic system can be achieved by designing a multi-valued weighted function of the position parameters.

4. The method according to claim 3, characterized in that: The number of vortices in the direction of the position adjustable variable is expanded according to the number of values ​​of the multi-valued weighted function, and the distance between the vortices is adjusted according to the pulse width of the multi-valued weighted function; in the step three, the model of the full-grid multi-vortex chaotic system is constructed as follows: Step 3.1: Design pulse width and x j1 ,…x jk The multi-valued weighted function H that matches the maximum amplitude of attractors in each direction j (t), weighted control of the position parameter b, the following dynamic equation is obtained Thus, in the location parameter b and the multi-valued weighting function H j (t), the position-adjustable robust chaotic system is realized at x j1 ,…x jk The expansion of the number of scrolls in the direction.

5. The method according to claim 1, characterized in that The generation of multi-scroll attractors in the direction of non-position-adjustable variables of position-adjustable robust chaotic systems can be achieved by designing a time-varying shift control function of the non-position-adjustable variables.

6. The method according to claim 5, characterized in that By designing different forms of time-varying shift control functions, the geometric distribution and number of vortices in the phase space are regulated; in step 4, the model of the full-grid multi-vortex chaotic system is constructed as follows: Step 4.1: Design the time-varying shift control function G i (t), for x j1 ,…x jk The remaining variables x i (i=1,…j1-1,…j k +1,…N) for displacement control, and then the following dynamic equation is obtained Thus, in the time-varying shift control function G i (t), the position-adjustable robust chaotic system is realized at x i (i=1,…j1-1,…j k +1,…N) direction.

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