Self-Reproducing Hidden Attractors in Fractional-Order Chaotic Systems Using Affine Transformations

Fractional order systems have a wider range of applications. Hidden attractors are a peculiar phenomenon in nonlinear systems. In this paper, we construct a fractional-order chaotic system with hidden attractors based on the Sprott C system. According to the Adomain decomposition method, we numerically simulate from several algorithms and study the dynamic characteristics of the system through bifurcation diagram, phase diagram, spectral entropy, and C0 complexity. The results of spectral entropy and C0 complexity simulations show that the system is highly complex. In order to apply such research results to engineering practice, for such fractional-order chaotic systems with hidden attractors, we design a controller to synchronize according to the finite-time stability theory. The simulation results show that the synchronization time is short and the robustness is stable. This paper lays the foundation for the study of fractional order systems with hidden attractors.

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Bursting, Dynamics, and Circuit Implementation of a New Fractional-Order Chaotic System With Coexisting Hidden Attractors

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4043003 ◽

2019 ◽

Vol 14 (7) ◽

Cited By ~ 3

Author(s):

Meng Jiao Wang ◽

Xiao Han Liao ◽

Yong Deng ◽

Zhi Jun Li ◽

Yi Ceng Zeng ◽

...

Keyword(s):

Numerical Simulation ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Systems ◽

Equilibrium Points ◽

Neuroendocrine Cells ◽

Order System ◽

Hidden Attractors ◽

Main Mode ◽

Circuit Realization

Systems with hidden attractors have been the hot research topic of recent years because of their striking features. Fractional-order systems with hidden attractors are newly introduced and barely investigated. In this paper, a new 4D fractional-order chaotic system with hidden attractors is proposed. The abundant and complex hidden dynamical behaviors are studied by nonlinear theory, numerical simulation, and circuit realization. As the main mode of electrical behavior in many neuroendocrine cells, bursting oscillations (BOs) exist in this system. This complicated phenomenon is seldom found in the chaotic systems, especially in the fractional-order chaotic systems without equilibrium points. With the view of practical application, the spectral entropy (SE) algorithm is chosen to estimate the complexity of this fractional-order system for selecting more appropriate parameters. Interestingly, there is a state variable correlated with offset boosting that can adjust the amplitude of the variable conveniently. In addition, the circuit of this fractional-order chaotic system is designed and verified by analog as well as hardware circuit. All the results are very consistent with those of numerical simulation.

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Parameter Identification of Fractional-Order Discrete Chaotic Systems

Entropy ◽

10.3390/e21010027 ◽

2019 ◽

Vol 21 (1) ◽

pp. 27 ◽

Cited By ~ 11

Author(s):

Yuexi Peng ◽

Kehui Sun ◽

Shaobo He ◽

Dong Peng

Keyword(s):

Parameter Identification ◽

Fractional Order ◽

Chaotic System ◽

Chaos Synchronization ◽

Chaotic Systems ◽

Random Noise ◽

Hidden Attractors ◽

Noise Interference ◽

Efficient Performance ◽

Two Samples

Research on fractional-order discrete chaotic systems has grown in recent years, and chaos synchronization of such systems is a new topic. To address the deficiencies of the extant chaos synchronization methods for fractional-order discrete chaotic systems, we proposed an improved particle swarm optimization algorithm for the parameter identification. Numerical simulations are carried out for the Hénon map, the Cat map, and their fractional-order form, as well as the fractional-order standard iterated map with hidden attractors. The problem of choosing the most appropriate sample size is discussed, and the parameter identification with noise interference is also considered. The experimental results demonstrate that the proposed algorithm has the best performance among the six existing algorithms and that it is effective even with random noise interference. In addition, using two samples offers the most efficient performance for the fractional-order discrete chaotic system, while the integer-order discrete chaotic system only needs one sample.

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Fractional-Order Chaotic Systems with Hidden Attractors

10.1007/978-3-030-75821-9_9 ◽

2021 ◽

pp. 199-238

Author(s):

Xiong Wang ◽

Guanrong Chen

Keyword(s):

Fractional Order ◽

Chaotic Systems ◽

Hidden Attractors

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Poincaré maps for detecting chaos in fractional-order systems with hidden attractors for its Kaplan-Yorke dimension optimization

AIMS Mathematics ◽

10.3934/math.2022326 ◽

2022 ◽

Vol 7 (4) ◽

pp. 5871-5894

Author(s):

Daniel Clemente-López ◽

◽

Esteban Tlelo-Cuautle ◽

Luis-Gerardo de la Fraga ◽

José de Jesús Rangel-Magdaleno ◽

...

Keyword(s):

Fractional Order ◽

Chaotic Systems ◽

Chaotic Behavior ◽

Just In Time ◽

Hidden Attractors ◽

Poincaré Maps ◽

Poincare Maps ◽

Accelerated Particle Swarm Optimization ◽

Python Programming ◽

Parameter Values

<abstract><p>The optimization of fractional-order (FO) chaotic systems is challenging when simulating a considerable number of cases for long times, where the primary problem is verifying if the given parameter values will generate chaotic behavior. In this manner, we introduce a methodology for detecting chaotic behavior in FO systems through the analysis of Poincaré maps. The optimization process is performed applying differential evolution (DE) and accelerated particle swarm optimization (APSO) algorithms for maximizing the Kaplan-Yorke dimension ($ D_{KY} $) of two case studies: a 3D and a 4D FO chaotic systems with hidden attractors. These FO chaotic systems are solved applying the Grünwald-Letnikov method, and the Numba just-in-time (jit) compiler is used to improve the optimization process's time execution in Python programming language. The optimization results show that the proposed method efficiently optimizes FO chaotic systems with hidden attractors while saving execution time.</p></abstract>

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