A Fractional-Order Chaotic System with an Infinite Number of Equilibrium Points

Zhou, Ping; Huang, Kun; Yang, Ce

doi:10.1155/2013/910189

Cited by 8 publications

(4 citation statements)

References 25 publications

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“…We use standard nonlinear terms in the Lorenz family like the cross product and square terms in combination with sinusoidal terms, giving rise to multiple equilibrium points and hence very complex dynamical behaviour in the phase space e.g. with infinite number of equilibrium points [12] [13].…”

Section: Introductionmentioning

confidence: 99%

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

Pan

Das

2015

Chaos, Solitons & Fractals

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In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone. Introduction:An important research theme in non-linear dynamics is to identify sets of differential equations along with their parameters which give rise to chaos. Starting from the advent of Lorenz attractors in three dimensional nonlinear differential equations [1][2], its several other family of attractors have been developed like Rossler, Rucklidge, Chen, Lu, Liu, Sprott, Genesio-Tesi, Shimizu-Morioka etc. [3]. Extension of the basic attractors to four or even higher dimensional systems has resulted in a similar family of hyper-chaotic systems [4]. In addition, several other structures like multi-scroll [5] and multi-wing [6] versions of the Lorenz family of attractors have also been developed by increasing the number of equilibrium points. Development of new chaotic attractors has huge application especially in data encryption, secure communication [7] etc. and in understanding the dynamics of many real world systems whose governing equations match with the template of these chaotic systems [8]. There has been several research reports on the application of master-slave chaos synchronization in secure communication [9], where the fresh set of chaotic attractors can play a big role due to their rich phase space dynamics. Here we explore the potential of automatic generation of chaotic attractors which is developed from the basic three dimensional structure of the Lorenz system.We use the Lyapunov exponent -the most popular signature of chaos, to judge whether a computer generated arbitrary nonlinear structure of third order differential equation along with some chosen parameters exhibits chaotic motion in the phase space. Since there could be chaotic behaviour or complex limit cycles or even stable/unstable motions in the phase space, for unknown mathematical expressions of similar third order dynamical system, the computationally tractable way for the 2. Genetic programming to evolve new chaotic attractors 2.

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Section: Introductionmentioning

confidence: 99%

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

Pan

Das

2015

Chaos, Solitons & Fractals

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“…It is noted that the uncontrolled system (13) is a chaotic system with an infinite number of equilibrium points [42].…”

Section: Fshps Between the 3d Fractional Chaotic System And 4d Fractimentioning

confidence: 99%

“…When considering the effects of fractional derivatives on systems with hidden attractors, a few fractional-order forms of systems with hidden attractors have been introduced [39]. Fractional-order forms of systems without equilibrium were reported in [39-41, 43, 44], while fractional-order forms of systems with an infinite number of equilibrium points were presented in [42,45,46]. Sifeu et al investigated the fractional-order form of a threedimensional chaotic autonomous system with only one stable equilibrium [65].…”

Section: Introductionmentioning

confidence: 99%

A fractional-order form of a system with stable equilibria and its synchronization

et al. 2018

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There has been an increasing interest in studying fractional-order chaotic systems and their synchronization. In this paper, the fractional-order form of a system with stable equilibrium is introduced. It is interesting that such a three-dimensional fractional system can exhibit chaotic attractors. Full-state hybrid projective synchronization scheme and inverse full-state hybrid projective synchronization scheme have been designed to synchronize the three-dimensional fractional system with different four-dimensional fractional systems. Numerical examples have verified the proposed synchronization schemes.

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“…In studying dynamical systems, it is important to verify that all the equations are independent and contribute to the dynamics. The presence of extraneous equations and their apparent additional dimensions can lead to false conclusions such as the claim that the initial conditions are bifurcation parameters or that a system has infinitely many equilibria [9]. However, what might be viewed as a flaw in Ref.…”

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confidence: 99%