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Fractional order hidden chaotic system without balance point

A fractional-order system, chaotic system technology, applied in digital transmission systems, transmission systems, secure communication through chaotic signals, etc., can solve the optimization without considering time, ignoring time finiteness, only emphasizing system robustness and other problems, Achieving the effect of rich diversity and limited time synchronization

Active Publication Date: 2019-09-24
QILU UNIV OF TECH
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  • Abstract
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Problems solved by technology

Another important issue is that many synchronization methods only emphasize the robustness of the system without considering the optimization of time, ignoring the time finiteness

Method used

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  • Fractional order hidden chaotic system without balance point
  • Fractional order hidden chaotic system without balance point
  • Fractional order hidden chaotic system without balance point

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Embodiment 1

[0044] The fractional-order hidden chaotic system without equilibrium point involved in this embodiment specifically includes the following steps:

[0045] Step 1: First give the definition of the Caputo fractional derivative:

[0046]

[0047] Where q is the order of the differential operator, t and A are limits, w is the smallest positive integer, w-1<q<w. Γ(*) is a gamma function, and f(*) is a continuous function.

[0048] The relevant properties of Caputo fractional differential are as follows:

[0049] Property 1: We consider general fractional differential equations

[0050]

[0051] The general solution of the equation is

[0052] x(t)=x(0)E q (At q ), (3)

[0053] and the Mittag-Leffter function is

[0054]

[0055] Then according to the finite-time stability theory of fractional order systems, the following Lemma 1 and Lemma 2 are introduced.

[0056] Lemma 1: For a general fractional order system, if it satisfies

[0057]

[0058] where x=[x 1 ...

Embodiment 2

[0070] For the fractional-order hidden chaotic system without equilibrium point involved in this embodiment, based on the finite-time stability theory of fractional-order systems, a finite-time synchronous controller and a combined synchronous controller for fractional-order systems with hidden attractors are designed.

[0071] Finite-time synchronization: Assuming the driving system is formula (8), the response system is as follows:

[0072]

[0073] Wherein, m=0, n≠0. let e 1 =x 1 -x,e 2 ==y 1 -y,e 3 =z 1 -z,q=0.99, the error system is

[0074]

[0075] Then we get the following theorem.

[0076] Theorem 1: For the error system (11), we design the finite-time synchronous controller as

[0077]

[0078] Among them, k 1 and B 1 is the scaling parameter, the error system (11) in finite time t 1 converges to zero, and

[0079]

[0080] Proof: From Lemma 1, we get

[0081]

[0082] According to formula (7), we get

[0083]

[0084] Therefore, we co...

Embodiment 3

[0120] In the simulation, in order to observe the chaotic synchronization between different initial values ​​of the fractional order chaotic system, the finite-time synchronization and combined synchronization are studied by using the prediction-correction method. For finite-time synchronization, the total number of iterations is 600, and the order of all fractional-order systems is q = 0.99. The initial value of finite time synchronization is [x(0)y(0)z(0)x 1 (0)y 1 (0)z 1 (0)]=[-0.3 -0.4 -0.6 0.2 0.6 0.4], according to formula (5), get

[0121] Such as Figure 3-4 , limiting the total number of iterations to 600 for clean results, e 1 ,e 2 ,e 3 converges approximately to zero at the 12th iteration, while x-x 1 ,y-y 1 ,z-z 1 Simultaneously achieve synchronization. The results show that the errors between the drive system and the response system converge to zero without considering the calculation error, and the error system gradually stabilizes within a finite tim...

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Abstract

The invention belongs to the technical field of automatic control methods, and particularly relates to a fractional order hidden chaotic system without a balance point. A new fractional order chaotic system without the balance point is provided, a fractional order chaotic system with a hidden chaotic attractor is generated, and a finite time synchronization method and a combination synchronization method of the chaotic system are designed. The system enriches the diversity of a fractional order chaotic system with the hidden attractors, realizes the finite time synchronization of the fractional order chaotic system with the hidden attractors according to a finite time stability theory, provides the reference for the finite time stability of other fractional order chaotic systems and provides a combined synchronization method of the fractional order chaotic system. Due to the natural advantage of the combined synchronization in information transmission application and the complexity of the fractional order system, the method has higher security at the aspect of realizing the secure communication compared with many other types of synchronization and integer order chaotic systems.

Description

technical field [0001] The invention belongs to the technical field of automatic control methods, and in particular relates to a fractional-order hidden chaotic system without an equilibrium point. Background technique [0002] With the development of computer technology, the discovery of new chaotic systems has aroused great interest of many researchers. In 1994, the hidden chaotic system was proposed for the first time. Hidden attractors exist in many natural phenomena and some important fields, such as the famous nasal Hoover oscillator, drilling system, aircraft control, convective fluid motion, etc., security communication, etc. In addition, hidden chaotic attractors also have some disadvantages, which may lead to unexpected and catastrophic results. The hidden chaotic system can be divided into the following categories: the attractor of the chaotic system is a surface equilibrium point, there is no equilibrium point, there is only one stable equilibrium point, a curve...

Claims

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Application Information

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IPC IPC(8): H04L9/00
CPCH04L9/001H04L2209/12
Inventor 刘加勋张芳芳黄明明王培冷森
Owner QILU UNIV OF TECH
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