Labyrinth Chaos

Sprott, J. C.; Chlouverakis, K.E.

doi:10.1142/s0218127407018245

Cited by 45 publications

(32 citation statements)

References 17 publications

(11 reference statements)

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“…We observed that the other expressions which only involved the terms +, -, ×, ÷, without the sine term, were mostly variants of the Lorenz system with almost similar time domain dynamics. After screening with the evolved attractors with LLE > 0.1 in the time delay embedding technique, their true LLE was again computed directly from the structure of the differential equations using symbolic differentiation and are reported in Table 1 -4. There are a few research results on the increase in the number of equilibrium points in the phase space due to the sine terms which is commonly known as the 'Labyrinth chaos' [31], [32]. Similar complex dynamics have also been observed in the phase space due to increase in the number of equilibrium points beside the standard two wing Lorenz like attractor dynamics.…”

Section: Discussionmentioning

confidence: 95%

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: Discussionmentioning

confidence: 95%

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

Pan

Das

2015

Chaos, Solitons & Fractals

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“…A very detailed study of this system was undertaken in [16] and it is presented as a prototypical example of chaos in [17]. As it is noted in [16], "Despite its mathematical simplicity, this system of ordinary differential equations produces a surprisingly rich dynamic behaviour that can serve as a prototype for chaos studies". They also remark that in the case of chaotic walks, the approach of an ensemble of initial conditions to equilibrium is by way of fractional Brownian motion with a Hurst exponent of approximately 0.61 and a slightly leptokurtic distribution.…”

Section: Labyrinth Chaos and Hyperchaosmentioning

confidence: 99%

Hyperchaos & labyrinth chaos: Revisiting Thomas–Rössler systems

Basios

Antonopoulos

2019

Journal of Theoretical Biology

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We consider a multi-dimensional extension of the Thomas-Rössler (TR) systems, that was inspired by Thomas' earlier work on biological feedback circuits, and we report on our first results that shows its ability to sustain a spatio-temporal behaviour, reminiscent of chimera states. The novelty here being that its underlying mechanism is based on "chaotic walks" discovered by René Thomas during the course of his investigations on what he called Labyrinth Chaos. We briefly review the main properties of TR systems and their chaotic and hyperchaotic dynamics and discuss the simplest way of coupling, necessary for this spatio-temporal behaviour that allows the emergence of complex dynamical behaviours. We also recall René Thomas' memorable influence and interaction with the authors as we dedicate this work to his memory.

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“…In fact, the number of subsystems is not limited to two and the subsystems are not necessary to be identical in system parameters and even in dimension. Labyrinth chaos [Thomas, 1999;Sprott & Chlouverakis, 2007] is a good example, including three identical linear subsystems entangled by sine function. Here, we entangle three linear subsystems with different parameters and different dimensions.…”

Section: Chaos Entanglement With Different Subsystemsmentioning

confidence: 99%

Chaos Entanglement: A New Approach to Generate Chaos

Zhang

Liu

Shen

et al. 2013

Int. J. Bifurcation Chaos

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A new approach to generate chaotic phenomenon, called chaos entanglement, is proposed in this paper. The basic principle is to entangle two or multiple stable linear subsystems by entanglement functions to form an artificial chaotic system such that each of them evolves in a chaotic manner. Firstly, a new attractor, entangling a two-dimensional linear subsystem and a one-dimensional one by sine function, is presented as an example. Dynamical analysis shows that both entangled subsystems are bounded and all equilibra are unstable saddle points when chaos entanglement is achieved. Also, numerical computation shows that this system has one positive Lyapunov exponent, which implies chaos. Furthermore, two conditions are given to achieve chaos entanglement. Along this way, by different linear subsystems and different entanglement functions, a variety of novel chaotic attractors have been created and abundant complex dynamics are exhibited. Our discovery indicates that it is not difficult any more to construct new artificial chaotic systems/networks for engineering applications such as chaos-based secure communication. Finally, a possible circuit is given to realize these new chaotic attractors.

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Labyrinth Chaos

Cited by 45 publications

References 17 publications

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

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

Hyperchaos & labyrinth chaos: Revisiting Thomas–Rössler systems

Chaos Entanglement: A New Approach to Generate Chaos

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