Predetermined time adaptive synchronization method and system for fractional order chaotic system

Abstract

The application discloses a predetermined time adaptive synchronization method and system of a fractional order chaotic system, wherein the predetermined time adaptive synchronization method of the fractional order chaotic system comprises the following steps: S1, obtaining two fractional order chaotic systems with time delay, i.e., a driving system and a response system, according to the dynamic characteristics of the fractional order chaotic system; S2, establishing a predetermined time adaptive synchronization error system according to the time delay error of the driving system and the response system in the step S1; and S3, obtaining a predetermined time synchronization controller according to the driving system and the response system in the step S1 and the predetermined time adaptive synchronization error system in the step S2, so as to realize adaptive control of the predetermined time adaptive synchronization error system. The predetermined time adaptive synchronization method of the fractional order chaotic system has a wider application scenario for the synchronization of the fractional order chaotic system, improves the stability of the system, avoids severe shaking caused by the use of a sliding surface, and increases the stability of the synchronization.

Description

Technical Field

[0001] This invention belongs to the field of automatic control technology, specifically relating to a predetermined time adaptive synchronization method and system for a fractional-order chaotic system. Background Technology

[0002] Finite-time stability refers to the stability of a system's state trajectory within a relatively definite time interval after being subjected to an initial disturbance. Finite-time stability is closely related to the initial state of the system. Fixed-time stability refers to the stability of the driving and response systems reaching equilibrium within a finite time, with an estimable upper bound on the settling time. Fixed-time stability is closely related to the gain parameter.

[0003] However, in some practical engineering applications, the initial state and gain parameters are difficult to predict or constantly changing, which makes it impossible to accurately estimate the system's settling time. Therefore, in 2014, Sanchez-Torres proposed a predetermined time stability method, which can enable the system to reach a stable state before a pre-set time even when the initial state and gain parameters are unknown. Meanwhile, in the paper "Projection Synchronization of Multi-Vortex Chaotic Systems Based on Sliding Mode Control Technology, Complexity 2020 (2020)," the authors proposed a synchronization controller based on a correction function and studied the predetermined time stability problem of hyperchaotic systems on a sliding surface. Numerical simulation results show the effectiveness of this theory; however, this method causes severe jitter when using a sliding surface, reducing the stability of the system and the stability of the synchronization. Summary of the Invention

[0004] To address the aforementioned issues, this invention provides a predetermined time adaptive synchronization method and system for fractional-order chaotic systems. This expands the application scenarios for synchronization of fractional-order chaotic systems, improves system stability, avoids severe jitter caused by the use of a sliding surface, and increases synchronization stability.

[0005] Another object of the present invention is to provide a predetermined time adaptive synchronization system for fractional-order chaotic systems.

[0006] The technical solution adopted in this invention is:

[0007] A predetermined time adaptive synchronization method for a fractional-order chaotic system includes the following steps:

[0008] S1. Based on the dynamic characteristics of fractional-order chaotic systems, two fractional-order chaotic systems with time delay are obtained: the driving system and the response system.

[0009] S2. Based on the delay errors of the driving system and the response system in S1, establish a predetermined time adaptive synchronization error system;

[0010] S3. Obtain a predetermined time synchronization controller based on the driving system and response system in S1 and the predetermined time adaptive synchronization error system in S2, for adaptive synchronization of the predetermined time adaptive synchronization error system.

[0011] Preferably, in step S1, the state equation of the drive system is:

[0012]

[0013] The state equation of the response system is:

[0014]

[0015] In the formula, i = 1, 2, ..., n, 0 < α < 1 represents the order of the system, t > t0, t0 represents the initial time, and x i (t) and y i (t) represent the state variables of the driving system and the response system, respectively, and σ i >0 represents the self-suppression of the system, τ>0 represents the time delay constant, X(0)=(x1(0),x2(0),...,x n (0)) T Y(0) = (y1(0), y2(0), ..., y n (0)) T f represents the state vectors of the driving system and the response system, respectively. i (·) represents a continuous nonlinear function, F i (·) is a continuously increasing delayed activation function, u i (t) represents the input of the synchronous controller.

[0016] Preferably, step S2 specifically includes the following steps:

[0017] S21. Subtract the state variables of the driving system and the response system in S1 to obtain the predetermined time projection synchronization error:

[0018] e i (t)=y i (t)-x i (t);

[0019] S22. Calculate the α-th derivative of the predetermined time projection synchronization error in S21 to obtain the predetermined time adaptive synchronization error system:

[0020]

[0021] set up If is the state vector of the predetermined time adaptive synchronization error system, then the initial value of the predetermined time adaptive synchronization error system is E(0) = Y(0) - X(0).

[0022] Preferably, step S3 specifically includes the following steps:

[0023] S31: Substituting the state equations of the driving system and the response system in S1 into the predetermined time adaptive synchronization error system in S22, we can obtain:

[0024] D α e i (t)=-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t)

[0025] Among them, h i (e i (t))=f i (Y,t)-f i (X,t), H i (e i (t-τ))=F i (Y,t-τ)-F i (X,t-τ);

[0026] S32. Based on the predetermined time adaptive synchronization error system in S31, the predetermined time synchronization controller can be obtained as follows:

[0027]

[0028] In the formula

[0029]

[0030] In the formula, β2, λ2, k, and p are all constants, and L = sup{|F i (Y,t-τ)|+|F i (X,t-τ)|},T c C is an adjustable parameter. v These are system parameters.

[0031] Preferably, step S3 further includes the following steps:

[0032] S33. Obtain the steady-state time calculation method of the fractional-order chaotic system according to the predetermined time synchronization controller in S32, so as to confirm that the steady-state time is determined by the adjustable parameters.

[0033] Preferably, in step S33, the steady-state time of the fractional-order chaotic system is calculated as follows:

[0034] Let the function The function is a continuous and strictly monotonically increasing function in [0, +∞) and satisfies:

[0035] (1) in It is a non-empty set representing a globally stable chaotic attractor at a fixed time.

[0036] (2)T c ∈{r1,…,r b} is an adjustable parameter;

[0037] (3) For any V(x) > 0, there exist constants α,k,b,p > 0, 0 < p ≤ 1, and qk ≥ 1. Therefore

[0038]

[0039] in,

[0040] For any V(x) > 0, we have

[0041]

[0042] The formula for calculating the settling time T(x0) is:

[0043]

[0044] According to Jensen's inequality, from 0 < p ≤ 1, we know...

[0045]

[0046] get

[0047]

[0048] when

[0049] As V(x0) → +∞, we obtain

[0050]

[0051] Preferably, in step S33, the steady-state time of the fractional-order chaotic system is calculated as follows:

[0052] Let the function The function is a continuous and strictly monotonically increasing function in [0, +∞) and satisfies:

[0053] (1) in It is a non-empty set representing a globally stable chaotic attractor at a fixed time.

[0054] (2)T c ∈{r1,…,r b} is an adjustable parameter;

[0055] (3) For any V(x) > 0, there exist constants α, b > 0 and p > 1 such that:

[0056]

[0057] in,

[0058]

[0059] According to Jensen's inequality, since p > 1, we know

[0060]

[0061] The formula for calculating the settling time T(x0) is:

[0062]

[0063] when

[0064] As V(x0) → +∞, we obtain

[0065]

[0066] Preferably, step S3 further includes the following step:

[0067] S34. Confirm the adaptive stability of the predetermined time adaptive synchronization error system in S31 based on the Lyapunov function.

[0068] Preferably, S33 specifically comprises:

[0069] Let the Lyapunov function be:

[0070]

[0071] Substituting the predetermined time adaptive synchronization error system in S31 into the Lyapunov function and taking the derivative, we can obtain:

[0072]

[0073] Where β2 and λ2 are both positive parameters, if we assume Then it can be formed in the following form:

[0074]

[0075] Consistent with the assumption of a monotonically increasing function, the adaptive stability of the predetermined time adaptive synchronization error system is confirmed.

[0076] Another technical solution of the present invention is implemented as follows:

[0077] A system applying the predetermined time adaptive synchronization method for fractional-order chaotic systems, comprising:

[0078] System delay partitioning module: Based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are obtained: the driving system and the response system;

[0079] Adaptive synchronization error module: Based on the delay error of the driving system and the response system, establish a predetermined time adaptive synchronization error system;

[0080] Synchronization controller module: Based on the drive system, response system and the predetermined time adaptive synchronization error system, a predetermined time synchronization controller is obtained for adaptive control of the predetermined time adaptive synchronization error system.

[0081] Compared with existing technologies, the predetermined time adaptive synchronization method and system for fractional-order chaotic systems of the present invention first obtains two fractional-order chaotic systems with delays based on the dynamic characteristics of the fractional-order chaotic system: a driving system and a response system; based on the delay errors of the driving system and the response system, a predetermined time adaptive synchronization error system is established; a predetermined time synchronization controller is obtained based on the driving system, the response system, and the predetermined time adaptive synchronization error system for adaptive synchronization of the predetermined time adaptive synchronization error system. This allows for a wider range of applications for the synchronization of fractional-order chaotic systems, improves system stability, avoids severe jitter caused by the use of a sliding surface, and increases synchronization stability. Attached Figure Description

[0082] Figure 1 This is a flowchart of a predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention;

[0083] Figure 2 This is a chaotic phase diagram of the driving system of a predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention.

[0084] Figure 3 This is a response system chaotic behavior trajectory diagram without control input for a predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention;

[0085] Figure 4The predetermined time synchronization controller of the predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention is located at T. c Synchronization error response curve with inputs of 1 and p=0.5;

[0086] Figure 5 The predetermined time synchronization controller of the predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention is located at T. c Synchronization error response curve with input p=1.5 and p=0.5;

[0087] Figure 6 The predetermined time synchronization controller of the predetermined time adaptive synchronization method for a fractional-order chaotic system provided in Embodiment 1 of the present invention is located at T. c Synchronization error response curve with inputs of p=1 and p=2. Detailed Implementation

[0088] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0089] Example 1

[0090] Embodiment 1 of the present invention provides a predetermined time adaptive synchronization method and system for fractional-order chaotic systems, such as... Figure 1-6 As shown, it includes the following steps:

[0091] S1. Based on the dynamic characteristics of fractional-order chaotic systems, two fractional-order chaotic systems with time delay are obtained: the driving system and the response system.

[0092] S2. Based on the delay errors of the driving system and the response system in S1, establish a predetermined time adaptive synchronization error system;

[0093] S3. Obtain a predetermined time synchronization controller based on the driving system and response system in S1 and the predetermined time adaptive synchronization error system in S2, for adaptive synchronization of the predetermined time adaptive synchronization error system.

[0094] Thus, based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are first obtained: a driving system and a response system. Based on the delay errors of the driving and response systems, a predetermined-time adaptive synchronization error system is established. Based on the driving and response systems and the predetermined-time adaptive synchronization error system, a predetermined-time synchronization controller is obtained for adaptive synchronization of the predetermined-time adaptive synchronization error system. This allows for a wider range of applications for the synchronization of the fractional-order chaotic system, improves system stability, avoids severe jitter caused by the use of a sliding surface, and increases synchronization stability.

[0095] In S1, the state equation of the driving system is:

[0096]

[0097] The state equation of the response system is:

[0098]

[0099] In the formula, i = 1, 2, ..., n, 0 < α < 1 represents the order of the system, t > t0, t0 represents the initial time, and x i (t) and y i (t) represent the state variables of the driving system and the response system, respectively, and σ i >0 represents the self-suppression of the system, τ>0 represents the time delay constant, X(0)=(x1(0),x2(0),...,x n (0)) T Y(0) = (y1(0), y2(0), ..., y n (0)) T f represents the state vectors of the driving system and the response system, respectively. i (·) represents a continuous nonlinear function, F i (·) is a continuously increasing delayed activation function, u i (t) represents the input of the synchronous controller.

[0100] In this way, we can obtain the state equations of two fractional-order chaotic systems with time delays: the driving system and the response system, for use in subsequent steps.

[0101] S2 specifically includes the following steps:

[0102] S21. Subtract the state variables of the driving system and the response system in S1 to obtain the predetermined time projection synchronization error:

[0103] e i (t)=y i (t)-x i (t); (3)

[0104] S22. Calculate the α-th derivative of the predetermined time projection synchronization error in S21 to obtain the predetermined time adaptive synchronization error system:

[0105]

[0106] set up If is the state vector of the predetermined time adaptive synchronization error system, then the initial value of the predetermined time adaptive synchronization error system is E(0) = Y(0) - X(0).

[0107] In this way, a predetermined time adaptive synchronization error system can be derived based on the driving system and the response system.

[0108] S3 specifically includes the following steps:

[0109] S31: Substituting the state equations of the driving system and the response system in S1 into the predetermined time adaptive synchronization error system in S22, we can obtain:

[0110] D α e i (t)=-σ i e i (t)+h i (e i (t))+H i (e i (t-τ))+u i (t); (5)

[0111] Among them, h i (e i (t))=f i (Y,t)-f i (X,t), H i (e i (t-τ))=F i (Y,t-τ)-F i (X,t-τ);

[0112] S32. Based on the predetermined time adaptive synchronization error system in S31, the predetermined time synchronization controller can be obtained as follows:

[0113]

[0114] In the formula

[0115]

[0116] In the formula, β2, λ2, k, and p are all constants, and L = sup{|F i (Y,t-τ)|+|F i(X,t-τ)|},T c C is an adjustable parameter. v These are system parameters.

[0117] Thus, by substituting the state equations of the drive system and the response system into the predetermined time adaptive synchronization error system, the predetermined time synchronization controller can be obtained.

[0118] S3 further includes the following steps:

[0119] S33. Obtain the steady-state time calculation method of the fractional-order chaotic system according to the predetermined time synchronization controller in S32, so as to confirm that the steady-state time is determined by the adjustable parameters.

[0120] In S33, the steady-state time of the fractional-order chaotic system is calculated as follows:

[0121] Let the function The function is a continuous and strictly monotonically increasing function in [0, +∞) and satisfies:

[0122] (1) in It is a non-empty set representing a globally stable chaotic attractor at a fixed time.

[0123] (2)T c ∈{r1,…,r b} is an adjustable parameter;

[0124] (3) For any V(x) > 0, there exist constants α,k,b,p > 0, 0 < p ≤ 1, and qk ≥ 1. Therefore

[0125]

[0126] in,

[0127] For any V(x) > 0, we have

[0128]

[0129] The formula for calculating the settling time T(x0) is:

[0130]

[0131] According to Jensen's inequality, from 0 < p ≤ 1, we know...

[0132]

[0133] get

[0134]

[0135]

[0136] when

[0137] As V(x0) → +∞, we obtain

[0138]

[0139] Thus, according to C when 0 < p ≤ 1 v Thus, the formula for calculating the steady-state time of a fractional-order chaotic system is obtained.

[0140] In S33, the steady-state time of the fractional-order chaotic system is calculated as follows:

[0141] Let the function The function is a continuous and strictly monotonically increasing function in [0, +∞) and satisfies:

[0142] (1) in It is a non-empty set representing a globally stable chaotic attractor at a fixed time.

[0143] (2)T c ∈{r1,…,r b} is an adjustable parameter;

[0144] (3) For any V(x) > 0, there exist constants α, b > 0 and p > 1 such that:

[0145]

[0146] in,

[0147]

[0148] According to Jensen's inequality, since p > 1, we know

[0149]

[0150] The formula for calculating the settling time T(x0) is:

[0151]

[0152] when

[0153] As V(x0) → +∞, we obtain

[0154]

[0155] Thus, according to C when p>1 v Thus, the formula for calculating the steady-state time of a fractional-order chaotic system is obtained.

[0156] S3 further includes the following steps:

[0157] S34. Confirm the adaptive stability of the predetermined time adaptive synchronization error system in S31 based on the Lyapunov function.

[0158] Specifically, S33 is:

[0159] Let the Lyapunov function be:

[0160]

[0161] Substituting the predetermined time adaptive synchronization error system in S31 into the Lyapunov function and taking the derivative, we can obtain:

[0162]

[0163] Where β2 and λ2 are both positive parameters, if we assume Then it can be formed in the following form:

[0164]

[0165] Consistent with the assumption of a monotonically increasing function, the adaptive stability of the predetermined time adaptive synchronization error system is confirmed.

[0166] This aligns with the assumption of a monotonically increasing function in this embodiment, thus proving that the predetermined time stability theory for fractional-order chaotic systems is effective and feasible. Similarly, if the control system parameters satisfy certain conditions, such that when e(t) = 0, This enables a pre-time adaptive synchronization method for fractional-order chaotic systems with delay elements.

[0167] In particular, to more intuitively demonstrate the effectiveness and feasibility of the pre-time adaptive synchronization method for fractional-order chaotic systems proposed in this invention, a computer simulation experiment was conducted using MATLAB software. The specific process is as follows:

[0168] The state equation of the fractional-order drive system is:

[0169]

[0170] The driving system is a four-dimensional chaotic system with initial values ​​X(0) = [0.6, 0.7, 0.3, 0.4]. T , Figure 2 This is the chaotic phase diagram of the driving system.

[0171] The state equation of a fractional-order response system is:

[0172]

[0173] The response system is a four-dimensional chaotic system of the same order, with an initial value of Y(0) = [3, -4, 2, 2]. T In the formula, σ i =0.1(i=1,2,3,4), a=10, b=40, c=2.5, k=10, h=4, r=2.5, a1=10, b1=28, c1=8 / 3, r1=1, Figure 3 The three-dimensional trajectory of the response system when there is no control input.

[0174] The error setting for the drive-response system is:

[0175] e i (t)=y i (t)-x i (t); (25)

[0176] The scheduled time synchronization controller is:

[0177]

[0178] In the formula, h i (e i (t))=f i (Y,t)-f i (X,t), L=sup{|F i (Y,t-τ)|+|F i (X,t-τ)|}.

[0179] Substituting formulas (23) and (24) into (26), the controller is specifically...

[0180]

[0181] (1-1) Assume 0 < p ≤ 1, α = 0.82, i = 1, 2, 3, 4, σ i =0.1, μ=12.9, ω=11, k=5.2; take p=0.5, then When T c When = 1, under the input of the synchronization controller (27), the synchronization error trajectory of the system is as follows: Figure 4 As shown;

[0182] (1-2) Assume 0 < p ≤ 1, α = 0.82, i = 1, 2, 3, 4, σ i =0.1, μ=12.9, ω=11, k=5.2; take p=0.5, then When T c When = 1.5, under the input of the synchronization controller (27), the synchronization error trajectory of the system is as follows: Figure 5 As shown;

[0183] (1-3) Assume p > 1, α = 0.82, i = 1, 2, 3, 4, σ i =0.1, μ=12.9, ω=11, k=5.2; take p=2, then When T c When = 1, under the input of the synchronization controller (27), the synchronization error trajectory of the system is as follows: Figure 6 As shown;

[0184] (1-4) In practical applications, the system is inevitably affected by delay. Therefore, this invention designs an active controller to compensate for the impact of delay on the system synchronization error. The control parameters are as shown above. Simulation experiments are conducted, and the simulation curves show that the method provided by this invention can realize the projection synchronization control of the drive-response system under delay and arbitrary initial state.

[0185] The present invention discloses a predetermined time adaptive synchronization method for fractional-order chaotic systems. First, based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are obtained: a driving system and a response system. Based on the delay errors of the driving and response systems, a predetermined time adaptive synchronization error system is established. A predetermined time synchronization controller is obtained based on the driving and response systems and the predetermined time adaptive synchronization error system, and is used for adaptive synchronization of the predetermined time adaptive synchronization error system. This expands the application scenarios of fractional-order chaotic system synchronization, improves system stability, avoids severe jitter caused by the use of a sliding surface, and increases synchronization stability. Simultaneously, the synchronization time of the driving and response systems can be preset as needed, thereby enabling better and more accurate prediction of the adaptive synchronization time.

[0186] Example 2

[0187] Embodiment 2 of the present invention provides a system applying the predetermined time adaptive synchronization method for fractional-order chaotic systems, comprising:

[0188] System delay partitioning module: Based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are obtained: the driving system and the response system;

[0189] Adaptive synchronization error module: Based on the delay error of the driving system and the response system, establish a predetermined time adaptive synchronization error system;

[0190] Synchronization controller module: Based on the drive system, response system and the predetermined time adaptive synchronization error system, a predetermined time synchronization controller is obtained for adaptive control of the predetermined time adaptive synchronization error system.

[0191] The present invention discloses a predetermined time adaptive synchronization system for fractional-order chaotic systems. First, based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are obtained: a driving system and a response system. Based on the delay errors of the driving and response systems, a predetermined time adaptive synchronization error system is established. A predetermined time synchronization controller is obtained based on the driving and response systems and the predetermined time adaptive synchronization error system for adaptive synchronization of the predetermined time adaptive synchronization error system. This expands the application scenarios of fractional-order chaotic system synchronization, improves system stability, avoids severe jitter caused by the use of a sliding surface, and increases synchronization stability. Simultaneously, the synchronization time of the driving and response systems can be preset as needed, thereby enabling better and more accurate prediction of the adaptive synchronization time.

[0192] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A predetermined time adaptive synchronization method for a fractional-order chaotic system, characterized in that, Includes the following steps: S1. Based on the dynamic characteristics of fractional-order chaotic systems, two fractional-order chaotic systems with time delay are obtained: the driving system and the response system. S2. Based on the delay errors of the driving system and the response system in S1, establish a predetermined time adaptive synchronization error system; S3. Obtain a predetermined time synchronization controller based on the driving system and response system in S1 and the predetermined time adaptive synchronization error system in S2, for adaptive synchronization of the predetermined time adaptive synchronization error system. In S1, the state equation of the driving system is: ； The state equation of the response system is: ； In the formula, , Represents the order of the system. , Represents the initial time. and These represent the state variables of the driving system and the response system, respectively. Represents the self-inhibition of the system. Represents the delay constant. , These represent the state vectors of the driving system and the response system, respectively. Represents a continuous nonlinear function. It is a continuously increasing delayed activation function. This represents the input to the synchronous controller; Specifically, S2 includes the following steps: S21. Subtract the state variables of the driving system and the response system in S1 to obtain the predetermined time projection synchronization error: ； S22, calculate the predetermined time projection synchronization error in S21. The first derivative yields the predetermined time adaptive synchronization error system: set up If is the state vector of the predetermined time adaptive synchronization error system, then the initial value of the predetermined time adaptive synchronization error system is... ; Specifically, S3 includes the following steps: S31: Substituting the state equations of the driving system and the response system in S1 into the predetermined time adaptive synchronization error system in S22, we can obtain: in, , ; S32. Based on the predetermined time adaptive synchronization error system in S31, the predetermined time synchronization controller can be obtained as follows: ； In the formula In the formula, , , , All are constants. , It is an adjustable parameter. These are system parameters.

2. The predetermined time adaptive synchronization method for a fractional-order chaotic system according to claim 1, characterized in that, S3 further includes the following steps: S33. Obtain the steady-state time calculation method of the fractional-order chaotic system according to the predetermined time synchronization controller in S32, so as to confirm that the steady-state time is determined by the adjustable parameters.

3. The predetermined time adaptive synchronization method for a fractional-order chaotic system according to claim 2, characterized in that, In S33, the steady-state time of the fractional-order chaotic system is calculated as follows: Let the function V(·): exist It is a continuous and strictly monotonically increasing function, and satisfies: (1) ,in It is a non-empty set representing a globally stable chaotic attractor in a fixed time. (2) It is an adjustable parameter; (3) For any Both have constants. , , ,thus ， in, , For any They all ； Stabilization time The calculation formula is: According to Jensen's inequality, by It can be known get when , when ,get 。 4. The predetermined time adaptive synchronization method for a fractional-order chaotic system according to claim 2, characterized in that, In S33, the steady-state time of the fractional-order chaotic system is calculated as follows: Let the function V(·): exist It is a continuous and strictly monotonically increasing function, and satisfies: (1) ,in It is a non-empty set representing a globally stable chaotic attractor in a fixed time. (2) It is an adjustable parameter; (3) For any Both have constants. and , so that: in, According to Jensen's inequality, by It can be known Stabilization time The calculation formula is: ， when , when ,get 。 5. The predetermined time adaptive synchronization method for a fractional-order chaotic system according to claim 3 or 4, characterized in that, S3 further includes the following steps: S34. Confirm the adaptive stability of the predetermined time adaptive synchronization error system in S31 based on the Lyapunov function.

6. The predetermined time adaptive synchronization method for a fractional-order chaotic system according to claim 5, characterized in that, Specifically, S33 is: Let the Lyapunov function be: ； Substituting the predetermined time adaptive synchronization error system in S31 into the Lyapunov function and taking the derivative, we can obtain: ， in, , All are positive parameters; if we set , Then it can be transformed into the following form: Consistent with the assumption of a monotonically increasing function, the adaptive stability of the predetermined time adaptive synchronization error system is confirmed.

7. A system applying the predetermined time adaptive synchronization method for a fractional-order chaotic system according to any one of claims 1-6, characterized in that, include System delay partitioning module: Based on the dynamic characteristics of the fractional-order chaotic system, two fractional-order chaotic systems with delays are obtained: the driving system and the response system; Adaptive synchronization error module: Based on the delay error of the driving system and the response system, establish a predetermined time adaptive synchronization error system; Synchronization controller module: Based on the drive system, response system and the predetermined time adaptive synchronization error system, a predetermined time synchronization controller is obtained for adaptive control of the predetermined time adaptive synchronization error system.

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