A threshold-smoothing multi-scroll attractor modulation method and system

Abstract

The present invention relates to the technical field of multi-scroll attractors, and in particular to a threshold-smoothed multi-scroll attractor modulation method and system, comprising selecting two state variables satisfying bipolarity from a multi-level chaotic system, and inputting the two bipolar state variables into a threshold-smoothed hyperbolic tangent function to obtain two bipolar hyperbolic tangent functions; respectively adding odd-numbered triangle wave functions and even-numbered triangle wave functions to obtain odd-numbered and even-numbered modulation dimension equations, and forming a multi-scroll attractor system with the multi-level chaotic system. The present invention solves the problem that the existing multi-scroll chaotic attractor system is complex to implement and has a low success rate.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-scroll attractors, and in particular to a threshold-smoothing multi-scroll attractor modulation method and system. Background Art

[0002] Since chaotic systems are widely used in nonlinear system control, secure communication and other fields, designing chaotic systems with complex dynamic characteristics has important practical significance and has received great attention in recent years.

[0003] Many researchers have studied its system modeling and dynamic characteristics, and design methods through memristor modulation feedback have been proposed, but they still have some defects.

[0004] A major problem is that there is no universal method to construct multi-vortex generation. For example, Zhang Zhaoxia and others are keen on combining triangular wave functions with the original system equations to generate multi-vortices in their research on modeling, control and application of several types of multi-vortex and multi-wing chaotic systems. This method is difficult to implement and has a low success rate because it requires a large number of attempts to find the position of the wave function in the equation and the relationship between the coefficients. This problem limits the flexibility of the generation of multi-vortex (or multi-wing) chaotic attractors and brings difficulties to the application of multi-vortex chaotic attractors.

[0005] Therefore, a design method that can be directly used to generate multiple scrolls remains a challenging and important task. Summary of the invention

[0006] In view of the shortcomings of the existing system, the present invention solves the problems that the existing multi-scroll chaotic attractor system is complex to implement and has a low success rate.

[0007] The technical solution adopted by the present invention is: a threshold-smoothed multi-scroll attractor modulation method comprises the following steps:

[0008] Two state variables satisfying bipolarity are selected from the multi-level chaotic system, and the two bipolar state variables are input into the threshold-smoothed hyperbolic tangent function to obtain two bipolar hyperbolic tangent functions, which are added to odd-numbered triangle wave functions and even-numbered triangle wave functions respectively to obtain odd-numbered and even-numbered modulation dimension equations respectively, and constitute a multi-vortex attractor system with the multi-level chaotic system.

[0009] As a preferred embodiment of the present invention, the multi-level chaotic system includes a Lorenz system and an HNN neural network system.

[0010] As a preferred embodiment of the present invention, the multi-scroll attractor system equation of the Lorenz system is:

[0011]

[0012] in, is the dimensionless form of the dimensional equation obtained through modulation, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v respectively, u is the state variable of the odd-numbered triangular wave function, and v is the state variable of the even-numbered triangular wave function; x, y, z are the state variables of the system equation; are the dimensionless forms of the 1st, 2nd and 3rd dimensions of the system equations respectively.

[0013] As a preferred embodiment of the present invention, the formula of the odd-numbered triangle wave function is:

[0014]

[0015] Where M is a natural number, p is the slope point parameter, q is the delay parameter, and ξ is the state variable.

[0016] As a preferred embodiment of the present invention, the formula of the even-numbered triangle wave function is:

[0017]

[0018] Where N is a natural number, p is the slope point parameter, q is the delay parameter, and ξ is the state variable.

[0019] As a preferred embodiment of the present invention, the HNN neural network system constitutes a multi-scroll attractor system equation:

[0020]

[0021] in, is the dimensionless form of the dimensional equation obtained through modulation, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v respectively, u is the state variable of the odd-numbered triangular wave function, and v is the state variable of the even-numbered triangular wave function; x, y, z are the state variables of the system equation; are the dimensionless forms of the 1st, 2nd and 3rd dimensions of the system equations respectively, and α1, α2, β1 and β2 are the adjustment parameters.

[0022] As a preferred embodiment of the present invention, α1=α2=1.3, β1=2.1, β2=2.7.

[0023] As a preferred embodiment of the present invention, a threshold-smoothed multi-scroll attractor modulation system includes: a memory for storing instructions executable by a processor; and a processor for executing the instructions to implement a threshold-smoothed multi-scroll attractor modulation method.

[0024] As a preferred embodiment of the present invention, a computer readable medium stores a computer program code, and when the computer program code is executed by a processor, a threshold-smoothing multi-scroll attractor modulation method is implemented.

[0025] Beneficial effects of the present invention:

[0026] 1. The number of multi-scrolls generated can be flexibly controlled according to demand by simply changing the modulation function. The method is simple;

[0027] 2. Ensure the integrity of the initial system to the greatest extent possible. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 It is a threshold-smoothing multi-scroll attractor modulation method and system flow chart of the present invention;

[0029] Figure 2 It is the function image of the triangular wave function described in the present invention.

[0030] Figure 3 This is a numerical simulation result diagram of the present invention based on the Lorenz system modulating multi-scroll attractor;

[0031] Figure 4 This is a simulation diagram of multiple scroll attractors when the parameters are optimally adjusted based on the HNN neural network system of the present invention.

[0032] Figure 5 This is a simulation diagram of a multi-scroll attractor when other parameters are adjusted based on the HNN neural network system of the present invention. DETAILED DESCRIPTION

[0033] The present invention is further described below in conjunction with the accompanying drawings and embodiments. This figure is a simplified schematic diagram, which only illustrates the basic structure of the present invention in a schematic manner, and therefore it only shows the components related to the present invention.

[0034] like Figure 1 As shown, a threshold-smoothed multi-scroll attractor modulation method includes:

[0035] Select two bipolar state variables from the multi-level chaotic system. It is easy to know that the state variables x and y in the Lorenz system are bipolar, that is, symmetrical about their own coordinate axis, so they are selected.

[0036] The two bipolar state variables are input into the threshold-smoothed hyperbolic tangent function; the hyperbolic sine function can be tanh(.);

[0037] Get two bipolar hyperbolic tangent functions: tanh(x) and tanh(y);

[0038] Adding the odd-numbered triangle wave functions h1(u) and the even-numbered triangle wave functions h2(v) respectively, we can get the odd-numbered and even-numbered modulation dimension equations respectively. and

[0039] And it forms a multi-vortex attractor system with the multi-level chaotic system.

[0040] Figure 2 The triangular wave function h(ξ) is used as a candidate function for the modulation method. In order to generate odd and even scrolls, the triangular wave function can be written as:

[0041]

[0042] and

[0043]

[0044] Among them, parameters M, N = 1, 2, 3, ..., are used to control the number of scrolls, and the specific parameter selection is shown in Table 1; p is the slope point parameter, q is the time delay parameter, and p, q always satisfy the condition q = 0.02p.

[0045] Taking M=2 or N=2 as an example, Figure 2 The internal state function h(ξ) of the modulator in equations (1) and (2) is plotted as Figure 2 (a) and Figure 2 As shown in (b), it can be seen that: when M = 2, h(ξ) has 5 negative slope intersection points, and when N = 2, h(ξ) has 6 negative slope intersection points; therefore, using this modulated piecewise linear function may generate (2M+1) (odd number) or (2N+2) (even number) multi-vortex attractors. This characteristic can be modulated in an ordinary system with a double-vortex attractor. The selected modulation function and the corresponding number of multi-vortex attractors generated are shown in Table 1.

[0046] Table 1 The form of the complex piecewise nonlinear function h(ξ) and the number of scrolls determined by the value of the control parameter M / N

[0047]

[0048] Figure 3 The present invention uses the Lorenz system as a test system and uses MATLAB numerical simulation software to observe its multi-scroll attractor under different control parameters. The mathematical model formula is:

[0049]

[0050] in, are the dimensionless forms of the 1st, 2nd and 3rd dimensions of the system equation respectively, and x, y and z are the state variables of the system equation.

[0051] The first-dimensional and second-dimensional state sequences are selected and input into the triangular wave function h(ξ) for modulation, and then the bipolar sequence in the original system is filtered and output through the threshold-smoothed hyperbolic tangent function, and the equation can be obtained as follows:

[0052]

[0053] in, is the dimensionless form of the dimensional equation obtained after modulation, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v of equation (1) and equation (2) respectively; the initial value (x0, y0, z0, u0, v0) is set to (0.1, 0.1, 0.1, 0, 0), and the number of multi-vortices is controlled by adjusting the control parameters M and N. With the change of M and N, any number of multi-vortex attractors can be generated; on the other hand, the number of modulated vortices can be achieved by changing the control parameter M / N of the piecewise linear function module, that is, the nonlinear function is made more multi-segmented.

[0054] Figure 4 The present invention uses the HNN neural network system as a test system and uses MATLAB numerical simulation software to obtain a multi-scroll attractor under different control parameters. The mathematical model of the HNN neural network system is:

[0055]

[0056] It is easy to know that the state variables x and z of this system are bipolar, so these two state sequences are selected to input into the modulation function and add appropriate coefficients for modulation. Then, the bipolar sequence in the original system is filtered and output through the hyperbolic tangent function with threshold smoothing, and the equation can be obtained:

[0057]

[0058] Among them, α1, α2, β1, β2 are adjustment parameters, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v in equation (1) and (2), respectively.

[0059] It should be noted that the adjustment parameters of the functions in the fourth and fifth dimensions in formula (6) can be used to find the neatest phase trajectory diagram through round-robin training. The initial value (x0, y0, z0, u0, v0) is set to (0.1, 0.1, 0, 0, 0), and the number of multi-vortices is controlled by changing M and N. As M and N change, any number of multi-vortex attractors can be generated.

[0060] Figure 4 The vortex attractor phase trajectory diagram produced when α1=α2=1.3, β1=2.1, β2=2.7 is the neatest.

[0061] Figure 5 The simulation diagram for other adjustment parameters is shown in Figure 2. Taking multi-scroll 5×5 and 6×6 as examples, Figure 5 As shown in the figure, the results of numerical simulation will be different from the actual situation; when the adjustment parameters are slightly different from the optimal condition parameters, such as selecting α1=α2=1.2, β1=2, β2=2.71, the results of numerical simulation will cause the multi-vortex attractor to be incompletely displayed, such as Figure 5 (a)(c) As shown; when the adjustment parameters are significantly different from the appropriate parameters, such as selecting α1 = α2 = 1.9, β1 = 1.2, β2 = 1.7, the numerical simulation results will cause the shape of the multi-scroll attractor to change, such as Figure 5 (b)(d) As shown; when α1=α2=1, β1=1, β2=1 is selected, the numerical simulation results will cause the multi-vortex attractors to overlap in the longitudinal direction and thus fail to produce a grid attractor visually; this shows that for more complex original system equations, it is very important to adjust the selection of parameters in order to obtain appropriate results.

[0062] It should be noted that the modulation method of the present invention is applicable to all continuous systems with saddle-focus chaotic attractors with an index of 2. Unlike other multi-scroll attractor modulators, the system of the present invention is independent of the modulation system and depends on the state sequence in the modulation system, but will not interact with the modulation system. The present invention takes the HNN neural network system and the Lorenz system as examples, and other continuous systems that produce saddle-focus chaotic attractors with an index of 2 are no longer enumerated.

[0063] Based on the above ideal embodiments of the present invention, the relevant staff can make various changes and modifications without departing from the technical concept of the present invention through the above description. The technical scope of the present invention is not limited to the contents of the specification, and its technical scope must be determined according to the scope of the claims.

Claims

1. A threshold-smoothed multi-scroll attractor modulation method, characterized in that: The following steps are involved: Select two state variables that satisfy bipolarity from the multi-level chaotic system, and input the two bipolar state variables into the threshold-smoothed hyperbolic tangent function to obtain two bipolar hyperbolic tangent functions; Two bipolar hyperbolic tangent functions tanh(x) and tanh(y) are added to the odd-numbered triangle wave functions h1(u) and the even-numbered triangle wave functions h2(v) to obtain the odd-numbered and even-numbered modulation dimension equations. And it forms a multi-vortex attractor system with the multi-level chaotic system; The multi-level chaotic system includes a Lorenz system or an HNN neural network system; wherein, the first dimension x and the second dimension y sequence of the state variables of the Lorenz system are selected and input into the triangular wave function h(ξ) for modulation, and then the bipolar sequence in the original system is filtered and output through the hyperbolic tangent function with threshold smoothing, and the result is: in, is the dimensionless form of the dimensional equation obtained through modulation, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v respectively, u is the state variable of the odd-numbered triangular wave function, and v is the state variable of the even-numbered triangular wave function; x, y, z are the state variables of the system equation; are the dimensionless forms of the 1st, 2nd and 3rd dimensions of the system equations respectively.

2. The threshold-smoothed multi-scroll attractor modulation method according to claim 1, characterized in that: The state variables x and z sequences of the HNN neural network system are selected for modulation, and then the bipolar sequence in the original system is filtered and output through the threshold-smoothed hyperbolic tangent function to obtain: in, is the dimensionless form of the dimensional equation obtained through modulation, h1(u) and h2(v) are the modulated triangular wave functions of the state variables u and v respectively, u is the state variable of the odd-numbered triangular wave function, and v is the state variable of the even-numbered triangular wave function; x, y, z are the state variables of the system equation; are the dimensionless forms of the 1st, 2nd and 3rd dimensions of the system equations respectively, and α1, α2, β1 and β2 are the adjustment parameters.

3. The threshold-smoothed multi-scroll attractor modulation method according to claim 2, characterized in that: The formula for an odd number of triangle wave functions is: Where M is a natural number, p is the slope point parameter, q is the delay parameter, and ξ is the state variable.

4. The threshold-smoothed multi-scroll attractor modulation method according to claim 2, characterized in that: The formula for an even number of triangle wave functions is: Where N is a natural number, p is the slope point parameter, q is the delay parameter, and ξ is the state variable.

5. The threshold-smoothed multi-scroll attractor modulation method according to claim 2, characterized in that: α1＝α2＝1.3，β1＝2.1，β2＝2.7。 6. A threshold-smoothed multi-scroll attractor modulation system, characterized in that: include: a memory for storing instructions executable by a processor; A processor, configured to execute instructions to implement the threshold-smoothed multi-scroll attractor modulation method as described in any one of claims 1 to 5.

7. A computer readable medium storing a computer program code, characterized in that: When the computer program code is executed by a processor, the computer program code implements the threshold-smoothed multi-scroll attractor modulation method according to any one of claims 1 to 5.

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