Construction method of non-autonomous fractional order chaotic system

Abstract

The invention provides a construction method of a non-autonomous fractional order chaotic system, which comprises the steps of system structure design, system parameter design and fractional order setting, and specifically comprises the following steps: adding an external driving signal and two parameters i and theta based on a fractional order generalized Lorenz chaotic system model to obtain the non-autonomous fractional order chaotic system; a parameter constraint condition and a fractional order constraint condition set by the non-autonomous fractional order chaotic system are provided, and it is ensured that the non-autonomous fractional order chaotic system is in a non-chaotic state without external drive; by setting an external driving signal, the system can generate different chaotic oscillation modes; and the chaotic oscillation of the non-autonomous fractional order chaotic system is suppressed by setting parameters i and theta. According to the method, the control of the system balance point and the system dynamic behavior can be realized, the stability and controllability of the chaotic system are improved, the method has a potential new application field, and a new direction is opened up for the development of scientific research and engineering technology.

Description

Technical Field

[0001] The invention belongs to the technical field of automatic control, and in particular relates to a construction method of a non-autonomous fractional-order chaotic system. Background Art

[0002] Chaotic systems have been widely used in secure communications, image encryption and signal detection. Existing chaotic systems still have certain shortcomings in signal detection, such as complex detection methods and few detectable signal types. Therefore, it is necessary to conduct new research on chaotic dynamics behavior to optimize the performance and application scope of chaotic systems.

[0003] Traditional chaotic systems have problems of instability and difficulty in control, and often require the design of complex control algorithms to achieve chaos control and anti-control, which is greatly limited in practical engineering and applications such as signal detection. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a method for constructing a non-autonomous fractional-order chaotic system. The method involves structural design, parameter design and fractional-order design of the fractional-order chaotic system, and specifically includes the following steps:

[0005] Step S1: Based on the original fractional-order generalized Lorenz chaotic system model, an external driving signal f(t) and two parameters Ξ and Θ are added to design a new system structure to obtain a non-autonomous fractional-order chaotic system;

[0006] Step S2: Based on the structure of the non-autonomous fractional-order chaotic system and the stability of the fractional-order system, the parameter constraints and the fractional-order constraints are derived, and the parameter design and fractional-order setting are performed to determine the stability of the non-autonomous fractional-order chaotic system.

[0007] Step S3: based on the parameter constraints and the fractional order constraints, the system parameters and fractional order of the non-autonomous fractional order chaotic system are set to make it in a non-chaotic state, and the non-autonomous fractional order chaotic system generates different dynamic behaviors, i.e., different oscillation modes, by adjusting the external driving signal;

[0008] Step S4: By adjusting the amplitude of the external driving signal f(t), the non-autonomous fractional-order chaotic system can generate chaotic oscillations. At the same time, by setting the values ​​of parameters Ξ and Θ, the chaotic oscillations of the non-autonomous fractional-order chaotic system can be suppressed.

[0009] Wherein, in step S1, an external driving signal f(t) and two parameters Ξ and Θ are added to the original fractional-order generalized Lorenz chaotic system model; by adding the external driving signal, it is turned into a non-autonomous fractional-order chaotic system; by adding the two parameters Ξ and Θ, the dynamic behavior of the non-autonomous system is enriched.

[0010] Furthermore, the non-autonomous fractional-order chaotic system expression is:

[0011]

[0012] In formula (1), x(t), y(t) and z(t) are system states, a 1 , a 2 , a 23 , b 1 , b 2 , b 13 , c 3 and c 12 is the system parameter, is a fractional calculus operator, a and t represent the upper and lower limits respectively; q>0 is the fractional order. Ξ and Θ are two added parameters, and f(t) represents the external driving signal.

[0013] Wherein, in step S2, the following steps are specifically included:

[0014] Step S201: Calculate the equilibrium point of the non-autonomous fractional-order chaotic system, derive parameter constraints according to the stability of the fractional-order system and the properties of the equilibrium point, and perform parameter design on the non-autonomous fractional-order chaotic system based on the parameter constraints, so that the set parameter constraints can ensure the stability of the non-autonomous fractional-order chaotic system;

[0015] Step S202: by deriving the constraints of the fractional order, based on the stability of the fractional order system and the constraints of the fractional order, the fractional order design is performed on the non-autonomous fractional order chaotic system, so that the non-autonomous fractional order chaotic system is in a non-chaotic state, and then the stability of the non-autonomous fractional order chaotic system is determined.

[0016] Specifically, in step S201, it also includes:

[0017] Step S2011: Set the left sides of the three equations of the non-autonomous fractional-order chaotic system (1) to 0;

[0018] Step S2012: Solve the equations to obtain three equilibrium points: E 1 (0,0,0), and

[0019] Step S2013: based on the properties of the equilibrium point, solving the eigenvalues ​​of the Jacobian matrix in the non-autonomous fractional-order chaotic system, and determining the stability of the non-autonomous fractional-order chaotic system through the eigenvalues;

[0020] Step S2014: After the stability of the non-autonomous fractional-order chaotic system is determined, the E 1 The equilibrium point is designed as a saddle point with an index of 1. 2 and E 3 Both equilibrium points are designed as saddle points with an index of 2, so that when the non-autonomous fractional-order chaotic system produces chaotic oscillations, there is a double-scroll attractor;

[0021] Step S2015: setting the parameters of the non-autonomous fractional-order chaotic system, on the one hand, making the set parameters satisfy the parameter constraints, and on the other hand, ensuring the dissipation of the non-autonomous fractional-order chaotic system, thereby realizing the dynamic behavior design of the non-autonomous fractional-order chaotic system.

[0022] Further, in step S2013:

[0023] The Jacobian matrix expression is:

[0024] The three equilibrium points E 1 , E 2 and E 3 At , the eigenvalues ​​of the system Jacobian matrix are as follows:

[0025] At the equilibrium point E 1 At the point where the eigenvalue of the Jacobian matrix of the system is 1,1 , 1,2 and λ 1,3 , whose expression is:

[0026]

[0027] λ 1,3 =c 3 ;

[0028] At the equilibrium point E 2 At the point where the eigenvalue of the Jacobian matrix of the system is 2,1 , 2,2 and λ 2,3 , which is expressed as:

[0029] λ 2,1 =λ 2,2 =α 1 ±jβ 1 ;

[0030] λ 2,3 =γ1 ;

[0031] At the equilibrium point E 3 At the point where the eigenvalue of the Jacobian matrix of the system is 3,1 , 3,2 and λ 3,3 , which is expressed as:

[0032] λ 3,1 =λ 3,2 =α 2 ±jβ 2 ,

[0033] λ 3,3 =γ 2 .

[0034] Among them, considering the complexity of its eigenvalue analytical form, the eigenvalue is expressed as a complex number, α i and β i represent its real and imaginary parts respectively.

[0035] Furthermore, the fractional-order system is divided into a stable domain and an unstable domain in the frequency domain. The stability of the non-autonomous fractional-order chaotic system is determined by the eigenvalue. Specifically, the eigenvalue has an argument (the argument of a complex number). When the eigenvalue falls in the stable domain, the equilibrium point corresponding to the eigenvalue is stable, thereby determining that the system is in a stable state, otherwise the system is in an unstable state.

[0036] Further, in step S2014:

[0037] The E 1 The equilibrium point is designed as a saddle point with an index of 1, which plays the role of connecting the scrolls; the E 2 and E 3 Both equilibrium points are designed as saddle points with an index of 2, which play a role in generating vortices; the two saddle points with an index of 2 that form double vortices can ensure the stability of the attractor. By designing the saddle points, the parameter constraints of the non-autonomous fractional-order chaotic system can be obtained. By designing the parameter constraints, the system has a vortex attractor; the expression of the parameter constraints of the non-autonomous fractional-order chaotic system is:

[0038]

[0039] The saddle point has two directions in the system space, one of which is a maximum and the other is a minimum. When designing the equilibrium point, the equilibrium point needs to be designed as a saddle point with different exponents. By designing saddle points with different exponents, the constraint conditions of the parameters are obtained. Through the constraint conditions, it can be realized that there is a vortex attractor in the non-autonomous fractional-order chaotic system.

[0040] Furthermore, in step S2015, the dissipative property is a necessary condition for spontaneous symmetry breaking and the formation of complex oscillation patterns. Without dissipative property, chaotic oscillations cannot occur in the non-autonomous fractional-order chaotic system.

[0041] To ensure the dissipativeness of the non-autonomous fractional-order chaotic system, the following parameter constraints should be met:

[0042]

[0043] a 1 +b 2 +c 3 <0 (5).

[0044] Furthermore, when the parameters of the non-autonomous fractional-order chaotic system simultaneously satisfy the constraints (2), (3), (4) and (5), the non-autonomous fractional-order chaotic system can generate chaotic oscillations and there is a double vortex attractor, thereby ensuring the dynamic behavior of the non-autonomous fractional-order chaotic system.

[0045] Specifically, in step S202, the following steps are also included:

[0046] Step S2021: Determine a stable region and an unstable region according to the stability theory of a non-autonomous fractional-order system;

[0047] Step S2022: Assume that the unstable characteristic value of the system equilibrium point is: 2,1 =λ 2,2 =λ 3,2 =λ 3,2 =α 1,2 ±jβ 1,2 , then the necessary condition for the existence of a double-scroll strange attractor in a non-autonomous fractional-order system is that the eigenvalue is in the unstable domain, so the fractional order of the system to be chaotic should satisfy: From formula (9), we can get the minimum order of chaos generated by non-autonomous fractional-order chaotic system:

[0048] Step S2023: Based on equation (10), determine that the fractional order of the non-autonomous fractional-order system satisfies the stability constraint condition: q min (11).

[0049] Wherein, in step S3, the system parameters and fractional order of the non-autonomous fractional-order chaotic system are set based on the parameter constraint condition and the fractional-order constraint condition, so that it is in a non-chaotic state; wherein the state includes a chaotic state and a non-chaotic state, and the state is reflected by the Lyapunov exponent of the non-autonomous fractional-order chaotic system;

[0050] ​Furthermore, the non-autonomous fractional-order chaotic system has three state variables, that is, there are three Lyapunov exponents. When at least one Lyapunov exponent is greater than 0 (positive number), it means that the system is in a chaotic state, otherwise, the system is in a non-chaotic state.

[0051] Specifically, in step S3, the dynamic behavior of the non-autonomous fractional-order chaotic system is obtained by adjusting the external driving signal, and the specific dynamic behavior is determined by setting the threshold value Π of the external driving signal f(t);

[0052] The threshold is a critical amplitude for adjusting the amplitude of the external driving signal f(t) so that the non-autonomous fractional-order chaotic system just generates chaotic oscillation.

[0053] The dynamic behavior includes the following cases:

[0054] If the amplitude of the external driving signal f(t) is 0, that is, when there is no external driving signal, the designed non-autonomous fractional-order chaotic system is in an asymptotically stable state;

[0055] If the designed non-autonomous fractional-order chaotic system is to generate stable chaotic oscillation, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to remain unchanged;

[0056] If the designed non-autonomous fractional-order chaotic system is to produce intermittent chaos, the amplitude of the external driving signal f(t) needs to fluctuate around the threshold Π;

[0057] If the designed non-autonomous fractional-order chaotic system is to produce an oscillation mode with multiple chaotic modes coexisting, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to change in real time.

[0058] Wherein, in step S4, the values ​​of the parameters Ξ and Θ are set to perform chaotic control on the chaotic oscillation of the non-autonomous fractional-order chaotic system. In this process, the values ​​of the specific parameters Ξ and Θ can be scanned by gradually increasing the values ​​of the parameters Ξ and Θ from small to large until the values ​​taken when the chaotic oscillation of the non-autonomous fractional-order chaotic system is controlled are taken as the optimal values.

[0059] Compared with the prior art, the present invention has the following advantages:

[0060] 1. The non-autonomous fractional-order chaotic system proposed in the present invention has an external driving signal and two adjustable parameters. The required chaotic oscillation mode can be obtained by setting an external driving signal, and the chaotic oscillation can be suppressed by using the designed two parameters; different chaotic oscillation modes can be designed according to actual needs to realize the control of the system equilibrium point and dynamic behavior, thereby improving the stability and controllability of the chaotic system.

[0061] 2. In terms of chaos control and anti-control of the system, it can be achieved by adjusting the external driving signal and two parameters without the need for complex control methods.

[0062] Other features and advantages of the present invention will be described in the following description, and partly become apparent from the description, or understood by practicing the present invention. The purpose and other advantages of the present invention can be realized and obtained by the structures pointed out in the description, claims and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0064] Figure 1 A flow chart of a construction method of a non-autonomous fractional-order chaotic system is shown.

[0065] Figure 2 A schematic diagram of the stable and unstable regions of a non-autonomous fractional-order chaotic system is shown.

[0066] Figure 3 Shown is the system bifurcation diagram of a non-autonomous fractional-order chaotic system.

[0067] Figure 4 Shown is a Lyapunov exponent diagram of a non-autonomous fractional-order chaotic system.

[0068] Figure 5 The graph shows how the Lyapunov exponent of a non-autonomous fractional-order chaotic system varies with the parameter Ξ and the amplitude of the signal f(t).

[0069] Figure 6 The graph shows how the Lyapunov exponent of a non-autonomous fractional-order chaotic system changes with the parameter Θ and the amplitude of the signal f(t). DETAILED DESCRIPTION

[0070] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the 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.

[0071] Embodiment 1:

[0072] This embodiment provides a method for constructing a non-autonomous fractional-order chaotic system. Figure 1 The construction method involves the structural design, parameter design and fractional order design of the fractional order chaotic system, forming a non-autonomous fractional order chaotic system. The specific process includes the following steps:

[0073] Step S1: Structural design: based on the original fractional-order generalized Lorenz chaotic system model, an external driving signal f(t) and two parameters Ξ and Θ are added to design a new system structure, thus obtaining a non-autonomous fractional-order chaotic system.

[0074] Furthermore, the original fractional-order generalized Lorenz chaotic system model has no parameters Ξ and Θ, and no external driving signal f(t). It is an autonomous chaotic system. Although the system can generate chaotic oscillations, it cannot be applied in practical engineering.

[0075] Furthermore, based on the original fractional-order generalized Lorenz chaotic system model, an external driving signal and two parameters Ξ and Θ are added; by adding the external driving signal, it is transformed into a non-autonomous fractional-order chaotic system; by adding the two parameters Ξ and Θ, the dynamic behavior of the non-autonomous system is enriched.

[0076] Furthermore, the non-autonomous fractional-order chaotic system expression is:

[0077]

[0078] Among them, x(t), y(t) and z(t) are the system states, a 1 , a 2 , a 23 , b 1 , b 2 , b 13 , c 3 and c 12 is the system parameter, is a fractional calculus operator, a and t represent the upper and lower limits respectively; q>0 is the fractional order. Ξ and Θ are two added parameters, and f(t) represents the external driving signal.

[0079] Step S2: Based on the structure of the non-autonomous fractional-order chaotic system and the stability of the fractional-order system, the parameter constraints and the fractional-order constraints are derived, and the parameter design and fractional-order setting are performed to determine the stability of the non-autonomous fractional-order chaotic system.

[0080] Furthermore, the stability of the non-autonomous fractional-order chaotic system is related to the parameter design and fractional order setting of the non-autonomous fractional-order chaotic system. In order to design the dynamic behavior of the non-autonomous fractional-order chaotic system, it is necessary to design the system parameters and fractional order.

[0081] Furthermore, in step S2, the following steps are also specifically included:

[0082] Step S201: Calculate the equilibrium point of the non-autonomous fractional-order chaotic system, derive parameter constraints according to the stability of the fractional-order system and the properties of the equilibrium point, and perform parameter design on the non-autonomous fractional-order chaotic system based on the parameter constraints, so that the set parameter constraints can ensure the stability of the non-autonomous fractional-order chaotic system;

[0083] Wherein, step S201 also includes the following steps:

[0084] Step S2011: Set the left sides of the three equations of the non-autonomous fractional-order chaotic system (1) to 0;

[0085] Step S2012: Solve the equations to obtain three equilibrium points: E 1 (0,0,0), and

[0086] Step S2013: based on the properties of the equilibrium point, solving the eigenvalues ​​of the Jacobian matrix in the non-autonomous fractional-order chaotic system, and determining the stability of the non-autonomous fractional-order chaotic system through the eigenvalues;

[0087] Furthermore, the Jacobian matrix expression is: The three equilibrium points E 1 , E 2 and E 3 At , the eigenvalues ​​of the system Jacobian matrix are as follows:

[0088] At the equilibrium point E 1 At the point where the eigenvalue of the Jacobian matrix of the system is 1,1 , 1,2 and λ 1,3 , whose expression is:

[0089]

[0090] λ 1,3 =c 3 ;

[0091] At the equilibrium point E 2 At the point where the eigenvalue of the Jacobian matrix of the system is 2,1 ,2,2 and λ 2,3 , which is expressed as:

[0092] λ 2,1 =λ 2,2 =α 1 ±jβ 1 ;

[0093] λ 2,3 =γ 1 ;

[0094] At the equilibrium point E 3 At the point where the eigenvalue of the Jacobian matrix of the system is 3,1 , 3,2 and λ 3,3 , which is expressed as:

[0095] λ 3,1 =λ 3,2 =α 2 ±jβ 2 ,

[0096] λ 3,3 =γ 2 ;

[0097] Among them, considering the complexity of its eigenvalue analytical form, the eigenvalue is expressed as a complex number, α i and β i denote its real and imaginary parts respectively;

[0098] For further information, see Figure 2 The fractional-order system is divided into a stable domain and an unstable domain in the frequency domain. The stability of the non-autonomous fractional-order chaotic system is determined by the eigenvalue. Specifically, the eigenvalue has an argument (the argument of a complex number). When the eigenvalue falls in the stable domain, the equilibrium point corresponding to the eigenvalue is stable, and then it is determined that the system is in a stable state, otherwise the system is in an unstable state.

[0099] Step S2014: After the stability of the non-autonomous fractional-order chaotic system is determined, the E 1 The equilibrium point is designed as a saddle point with an index of 1. 2 and E 3 Both equilibrium points are designed as saddle points with an index of 2, so that a double vortex attractor exists when the non-autonomous fractional-order chaotic system produces chaotic oscillations;

[0100] Furthermore, the E 1 The equilibrium point is designed as a saddle point with an index of 1, which plays the role of connecting the scrolls; the E 2 and E 3The two equilibrium points are designed as saddle points with an index of 2, which play a role in generating vortices. The two saddle points with an index of 2 that form double vortices can ensure the stability of the attractor. By designing the saddle points, the parameter constraints of the non-autonomous fractional-order chaotic system can be obtained. By designing the parameter constraints, the non-autonomous fractional-order chaotic system can be realized to have a vortex attractor. The expression of the parameter constraints of the non-autonomous fractional-order chaotic system is:

[0101]

[0102] Furthermore, the saddle point has two directions in the system space, one of which is a maximum and the other is a minimum. When designing the equilibrium point, the equilibrium point needs to be designed as a saddle point with different exponents. By designing saddle points with different exponents, the constraint conditions of the parameters are obtained. Through the constraint conditions, it can be realized that there is a vortex attractor in the non-autonomous fractional-order chaotic system.

[0103] Step S2015: setting the parameters of the non-autonomous fractional-order chaotic system, on the one hand, making the set parameters satisfy the parameter constraints, and on the other hand, ensuring the dissipation of the non-autonomous fractional-order chaotic system, thereby realizing the dynamic behavior design of the non-autonomous fractional-order chaotic system.

[0104] The dissipative property is a necessary condition for the spontaneous occurrence of symmetry breaking and the formation of complex oscillation patterns. Without dissipative property, chaotic oscillations cannot occur in non-autonomous fractional-order chaotic systems.

[0105] Furthermore, to ensure the dissipativeness of the non-autonomous fractional-order chaotic system, the following parameter constraints should be met:

[0106]

[0107] a 1 +b 2 +c 3 <0 (5).

[0108] Furthermore, when the parameters of the non-autonomous fractional-order chaotic system simultaneously satisfy the constraints (2), (3), (4) and (5), the non-autonomous fractional-order chaotic system can generate chaotic oscillations and there is a double vortex attractor, thereby ensuring the dynamic behavior of the non-autonomous fractional-order chaotic system.

[0109] In one embodiment, the system parameters can be set to the following groups:

[0110] a 2 =-a 1 =12, a 23 =0,b 1 =4,b 2 =4,b 13 =-1, c12 =1, c 3 = -1 (6);

[0111] a 2 =-a 1 =9.4,a 23 =0,b 1 =0,b 2 =9 / 4,b 13 =-1,c 3 =-0.1,c 12 =0.5 (7);

[0112] a 2 =-a 1 =7, a 23 =0, b 1 =7, b 2 =2, b 13 =-1, c 12 =1, c 3 =-9 / 4 (8).

[0113] By setting the parameters of equations (6), (7) and (8), the non-autonomous fractional-order chaotic system can generate chaotic oscillations and have a double-scroll attractor under the condition of satisfying parameter constraints, thereby ensuring the dynamic behavior of the non-autonomous fractional-order chaotic system.

[0114] Step S202: by deriving the constraints of the fractional order, based on the stability of the fractional order system and the constraints of the fractional order, the fractional order design is performed on the non-autonomous fractional order chaotic system, so that the non-autonomous fractional order chaotic system is in a non-chaotic state, and then the stability of the non-autonomous fractional order chaotic system is determined;

[0115] Furthermore, when the non-autonomous fractional-order chaotic system is in a chaotic state, among all the eigenvalues ​​of the Jacobian matrix of the system, at least one eigenvalue is in the unstable domain.

[0116] Furthermore, the fractional order design of the non-autonomous fractional order chaotic system comprises the following steps:

[0117] Step S2021: Determine a stable region and an unstable region according to the stability theory of a non-autonomous fractional-order chaotic system;

[0118] Step S2022: Assume that the unstable characteristic value of the system equilibrium point is: 2,1 =λ 2,2 =λ 3,2 =λ 3,2 =α 1,2 ±jβ 1,2, then the necessary condition for the existence of a double-scroll strange attractor in a non-autonomous fractional-order system is that the eigenvalue is in the unstable domain, so the fractional order of the system to be chaotic should satisfy: From formula (9), we can get the minimum order of chaos generated by non-autonomous fractional-order chaotic system:

[0119] Step S2023: Based on equation (10), determine that the fractional order of the non-autonomous fractional-order system satisfies the stability constraint condition: q min (11).

[0120] In one embodiment, under the premise of setting the parameters of equations (6), (7) and (8), the fractional order of the non-autonomous fractional-order system can be set to: q=0.96.

[0121] Step S3: based on the parameter constraints and the fractional order constraints, the system parameters and fractional order of the non-autonomous fractional order chaotic system are set to make it in a non-chaotic state, and the non-autonomous fractional order chaotic system generates different dynamic behaviors, i.e., different oscillation modes, by adjusting the external driving signal;

[0122] Among them, by setting different external driving signals f(t), the non-autonomous fractional-order chaotic system can produce different oscillation modes.

[0123] Further, in step S3, see 3, Figure 3 Where F represents the amplitude of the external driving signal f(t). When setting the amplitude of the external driving signal f(t), as the amplitude of the external driving signal f(t) gradually increases, the chaotic oscillation mode of the non-autonomous fractional-order chaotic system becomes more and more complex, and the chaotic oscillation mode of the non-autonomous fractional-order chaotic system is related to the signal amplitude; if the designed non-autonomous fractional-order chaotic system is to generate chaotic oscillation, it is necessary to make the amplitude of the external driving signal f(t) exceed a certain value, and the value is set as a threshold value, which is represented by the symbol Π; the threshold value is the critical amplitude when the amplitude of the external driving signal f(t) is adjusted so that the non-autonomous fractional-order chaotic system just generates chaotic oscillation.

[0124] Furthermore, the dynamic behavior of the non-autonomous fractional-order chaotic system is obtained by adjusting the external driving signal, and the specific dynamic behavior includes the following steps:

[0125] Step S301: if the amplitude of the external driving signal f(t) is 0, that is, when there is no external driving signal, the designed non-autonomous fractional-order chaotic system is in an asymptotically stable state;

[0126] ​Step S302: If the designed non-autonomous fractional-order chaotic system is to generate stable chaotic oscillation, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to remain unchanged;

[0127] Step S303: If the designed non-autonomous fractional-order chaotic system is to generate intermittent chaos, the amplitude of the external driving signal f(t) needs to fluctuate around the threshold value Π;

[0128] Step S304: If the designed non-autonomous fractional-order chaotic system is to generate an oscillation mode in which multiple chaotic modes coexist, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to change in real time.

[0129] Step S4: By adjusting the amplitude of the external driving signal f(t), the non-autonomous fractional-order chaotic system can generate chaotic oscillations. At the same time, by setting the values ​​of parameters Ξ and Θ, the chaotic oscillations of the non-autonomous fractional-order chaotic system can be suppressed.

[0130] Furthermore, the values ​​of the parameters Ξ and Θ are set to control the chaotic oscillation of the non-autonomous fractional-order chaotic system. In this process, the values ​​of the specific parameters Ξ and Θ can be scanned by gradually increasing the values ​​of the parameters Ξ and Θ from small to large until the values ​​when the chaotic oscillation of the non-autonomous fractional-order chaotic system is controlled are taken as the optimal values.

[0131] Furthermore, in terms of chaos control, that is, suppressing chaotic oscillations of the system, different control algorithms can be used, including state feedback control, adaptive control, neural network control and fuzzy control, but these methods are more complex than the method in this article, have low control efficiency, and are not conducive to practical engineering applications.

[0132] Embodiment 2:

[0133] In this example, see Figure 4 , Figure 5 and Figure 6 , further explanation is given when a non-autonomous fractional-order chaotic system produces chaotic oscillations under the action of an external driving signal f(t).

[0134] When adjusting the external driving signal f(t), the Lyapunov exponent in the non-autonomous fractional-order chaotic system reflects the state of the non-autonomous fractional-order chaotic system, and the state includes a chaotic state and a non-chaotic state. The non-autonomous fractional-order chaotic system has three state variables, that is, there are three Lyapunov exponents. When at least one Lyapunov exponent is greater than 0 (positive number), it means that the system is in a chaotic state, otherwise, the system is in a non-chaotic state.

[0135] In this embodiment, when setting the system parameter constraint conditions, the values ​​of equations (6), (7) and (8) in Embodiment 1 are taken. When setting the system fractional order constraint conditions, the fractional order is taken as: q = 0.96. Then, the change of the Lyapunov exponent of the system under the action of the external signal f(t) is as follows: Figure 4 As shown, Figure 4 Where A represents the amplitude of the external signal f(t). Figure 4 It can be seen that when the amplitude of the external signal f(t) is greater than the threshold Π, the system may become a chaotic state.

[0136] When a non-autonomous fractional-order chaotic system generates chaotic oscillations under the action of an external driving signal f(t), the parameters Ξ and Θ are set to suppress the chaotic oscillations of the system. The values ​​of the parameters Ξ and Θ are gradually increased until the chaotic oscillations of the system are suppressed. At this time, the changes of the system Lyapunov exponent with the parameters Ξ and Θ and the amplitude of the external signal f(t) are as follows: Figure 5 and Figure 6 shown.

[0137] Depend on Figure 5 and Figure 6 It can be seen from the figure that sufficiently large Ξ and Θ can make the Lyapunov exponent become negative, that is, suppress the chaotic oscillation excited by the external signal f(t).

[0138] Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent substitutions for some of the technical features therein; and these modifications or substitutions do not make the essence of the corresponding technical solutions deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for constructing a non-autonomous fractional-order chaotic system, characterized in that: The construction method comprises: Based on the original fractional-order generalized Lorenz chaotic system model, an external driving signal f(t) and two parameters Ξ and Θ are added to design a new system structure, and a non-autonomous fractional-order chaotic system is obtained. Based on the structure of non-autonomous fractional-order chaotic system and the stability of fractional-order system, the parameter constraints and fractional-order constraints are derived, and the parameter design and fractional-order setting are carried out to determine the stability of non-autonomous fractional-order chaotic system. Based on the parameter constraints and fractional order constraints, the system parameters and fractional order of the non-autonomous fractional order chaotic system are set to make it in a non-chaotic state. By adjusting the external driving signal, the non-autonomous fractional order chaotic system produces different dynamic behaviors, that is, different oscillation modes. By adjusting the amplitude of the external driving signal f(t), the non-autonomous fractional-order chaotic system can generate chaotic oscillations. At the same time, by setting the values ​​of parameters Ξ and Θ, the chaotic oscillations of the non-autonomous fractional-order chaotic system can be suppressed.

2. The method for constructing a non-autonomous fractional-order chaotic system according to claim 1, characterized in that: The expression of the non-autonomous fractional-order chaotic system is: In formula (1), x(t), y(t) and z(t) are system states, a1, a2, a 23 , b1, b2, b 13 , c3 and c 12 is the system parameter, is a fractional calculus operator, a and t represent the upper and lower limits respectively; q>0 is the fractional order. Ξ and Θ are two added parameters, and f(t) represents the external driving signal.

3. The method for constructing a non-autonomous fractional-order chaotic system according to claim 2, characterized in that: The method is based on the structure of the non-autonomous fractional-order chaotic system and the stability of the fractional-order system, deriving parameter constraints and fractional-order constraints, and using them to perform parameter design and fractional-order setting to determine its stability, and also includes: Calculate the equilibrium point of the non-autonomous fractional-order chaotic system, derive the parameter constraints according to the stability of the fractional-order system and the properties of the equilibrium point, and design the parameters of the non-autonomous fractional-order chaotic system based on the parameter constraints, so that the set parameter constraints can ensure the stability of the non-autonomous fractional-order chaotic system; By deriving the constraints of fractional order, based on the stability of fractional order system and the constraints of fractional order, the fractional order design of non-autonomous fractional order chaotic system is carried out, so that the non-autonomous fractional order chaotic system is in a non-chaotic state, and then the stability of the non-autonomous fractional order chaotic system is determined.

4. The method for constructing a non-autonomous fractional-order chaotic system according to claim 3, characterized in that: The method further comprises: calculating the equilibrium point of the non-autonomous fractional order chaotic system, deriving parameter constraints according to the stability of the fractional order system and the properties of the equilibrium point, and performing parameter design on the non-autonomous fractional order chaotic system based on the parameter constraints, so that the parameter constraints set can ensure the stability of the non-autonomous fractional order chaotic system. Let the left sides of the three equations of the non-autonomous fractional-order chaotic system (1) be 0; Solving the system of equations, we get three equilibrium points: E1(0,0,0), and Based on the properties of the equilibrium point, the eigenvalues ​​of the Jacobian matrix in the non-autonomous fractional-order chaotic system are solved, and the stability of the non-autonomous fractional-order chaotic system is determined by the eigenvalues. After the stability of the non-autonomous fractional-order chaotic system is determined, the E1 equilibrium point among the three equilibrium points is designed as a saddle point with an index of 1, and the two equilibrium points E2 and E3 are designed as saddle points with an index of 2. By designing the saddle points, the parameter constraints of the non-autonomous fractional-order chaotic system can be obtained. By designing the parameter constraints, it can be realized that the non-autonomous fractional-order chaotic system has a vortex attractor. The parameters of the non-autonomous fractional-order chaotic system are set. On the one hand, the set parameters satisfy the parameter constraints. On the other hand, the dissipation of the non-autonomous fractional-order chaotic system is ensured, thus realizing the dynamic behavior design of the non-autonomous fractional-order chaotic system.

5. The method for constructing a non-autonomous fractional-order chaotic system according to claim 4, characterized in that: The Jacobian matrix expression is: At the three equilibrium points E1, E2 and E3, the eigenvalues ​​of the Jacobian matrix are as follows: At the equilibrium point E1, the eigenvalue of the system Jacobian matrix is ​​λ 1，1 , 1，2 and λ 1，3 , whose expression is: λ 1，3 =c3; At the equilibrium point E2, the eigenvalue of the system Jacobian matrix is ​​λ 2,1 , 2,2 and λ 2,3 , which is expressed as: 2,1 =λ 2,2 =α1±jβ1,λ 2,3 =γ1; At the equilibrium point E3, the eigenvalue of the system Jacobian matrix is ​​λ 3,1 , 3,2 and λ 3,3 , which is expressed as: 3,1 =λ 3,2 =α2±jβ2, λ 3,3 =γ2; where α i and β i represent its real and imaginary parts respectively.

6. The method for constructing a non-autonomous fractional-order chaotic system according to claim 4, characterized in that: The E1 equilibrium point is designed as a saddle point with an index of 1, which plays a role in connecting vortices; the two equilibrium points E2 and E3 are both designed as saddle points with an index of 2, which play a role in generating vortices; the two saddle points with an index of 2 that form a double vortex can ensure the stability of the attractor, and by designing the saddle points, the parameter constraints of the non-autonomous fractional-order chaotic system are obtained; wherein: The saddle point has two directions in the system space, one of which is a maximum and the other is a minimum. The expression of the parameter constraint condition of the non-autonomous fractional-order chaotic system is:

7. The method for constructing a non-autonomous fractional-order chaotic system according to claim 6, characterized in that: To ensure the dissipativeness of the non-autonomous fractional-order chaotic system, the constraints should be met: a1+b2+c3<0(5); When the parameters of the non-autonomous fractional-order chaotic system simultaneously satisfy the constraints (2), (3), (4) and (5), the non-autonomous fractional-order chaotic system can generate chaotic oscillations and there is a double vortex attractor, thereby ensuring the dynamic behavior of the non-autonomous fractional-order chaotic system.

8. The method for constructing a non-autonomous fractional-order chaotic system according to claim 3, characterized in that: The fractional order design of the non-autonomous fractional order chaotic system specifically includes: According to the stability theory of non-autonomous fractional-order systems, determine the stable region and unstable region; Assume that the unstable characteristic value of the system equilibrium point is: 2,1 =λ 2,2 =λ 3,2 =λ 3,2 =α 1,2 ±jβ 1,2 , then the necessary condition for the existence of a double-scroll strange attractor in a non-autonomous fractional-order system is that the eigenvalue is in the unstable domain, so the fractional order of the system to be chaotic should satisfy: From formula (9), we can get the minimum order of chaos generated by non-autonomous fractional-order chaotic system: Based on formula (10), the constraint condition for the stability of the fractional order of the non-autonomous fractional-order system is determined as: min (11).​ 9. The method for constructing a non-autonomous fractional-order chaotic system according to claim 1, characterized in that: Based on the parameter constraints and the fractional order constraints, the system parameters and fractional order of the non-autonomous fractional order chaotic system are set to make it in a non-chaotic state, wherein: The state includes a chaotic state and a non-chaotic state, and the state is reflected by the Lyapunov exponent in the non-autonomous fractional-order chaotic system; The non-autonomous fractional-order chaotic system has three state variables, that is, there are three Lyapunov exponents. When at least one Lyapunov exponent is greater than 0, the system is in a chaotic state, otherwise, the system is in a non-chaotic state.

10. The method for constructing a non-autonomous fractional-order chaotic system according to claim 1, characterized in that: By adjusting the external driving signal, the dynamic behavior of the non-autonomous fractional-order chaotic system is obtained, wherein: The dynamic behavior is determined by setting a threshold value Π of the external driving signal f(t), and the threshold value is a critical amplitude for adjusting the amplitude of the external driving signal f(t) so that the non-autonomous fractional-order chaotic system just generates chaotic oscillations.

11. A method for constructing a non-autonomous fractional-order chaotic system according to any one of claims 1 to 10, characterized in that: The dynamic behaviors include: If the designed non-autonomous fractional-order chaotic system is to generate stable chaotic oscillation, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to remain unchanged; If the designed non-autonomous fractional-order chaotic system is to produce intermittent chaos, the amplitude of the external driving signal f(t) needs to fluctuate around the threshold Π; If the designed non-autonomous fractional-order chaotic system is to produce an oscillation mode with multiple chaotic modes coexisting, the amplitude of the external driving signal f(t) needs to be greater than the threshold Π, and the amplitude of the external driving signal f(t) needs to change in real time.

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