Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

Parekh, Nita; Kumar, V. Ravi; Kulkarni, Bhaskar D.

doi:10.1007/bf02845637

Cited by 16 publications

(16 citation statements)

References 51 publications

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“…In addition to the familiar approach of obtaining the stability of the synchronized state, we also show that the method could be used to estimate the onset of synchronization. Basically, a given system is stable, if there exist a continuous, positive definite, differentiable, and radially unbounded function (the Lyapunov function, V ) defined along the system's trajectory, such that its time derivativeV ≤ 0, as t → ∞ [27,28,30]. To begin with, we shall apply the following lemma to prove the main theorem of this paper.…”

Section: Stability and Synchronization Criteriamentioning

confidence: 99%

“…Proof Letting k 11 = k, k 22 = k, and k 12 = k 21 = 0, the inequalities (28)-(29) can be obtained according to the partial synchronization criteria (25)- (27).…”

Section: Stability and Synchronization Criteriamentioning

confidence: 99%

“…The Lyapunov stability theory employs the classical method of Lyapunov functionals which could be used for examining the synchronization behaviour in coupled and driven oscillators (see, for example [27,28]). In addition to the familiar approach of obtaining the stability of the synchronized state, we also show that the method could be used to estimate the onset of synchronization.…”

Section: Stability and Synchronization Criteriamentioning

confidence: 99%

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Multi-stability and basin crisis in synchronized parametrically driven oscillators

2010

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This paper studies the synchronization dynamics of two linearly coupled parametrically excited oscillators. The Lyapunov stability theory is employed to obtain some sufficient algebraic criteria for global asymptotic stability of the synchronization of the systems, and an estimated critical coupling, k cr , for which synchronization could be observed is determined. The synchronization transition is found to be associated with the boundary crisis of the chaotic attractor. In the bistable states, where two asymmetric T-periodic attractors co-exist, we show that the coupled oscillators can attain multi-stability via a new dynamical transition-the basin crisis wherein two co-existing attractors are destroyed while new co-existing attractors are created. The stability of the steady states is examined and the possible bifurcation routes identified.

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Section: Stability and Synchronization Criteriamentioning

confidence: 99%

“…Proof Letting k 11 = k, k 22 = k, and k 12 = k 21 = 0, the inequalities (28)-(29) can be obtained according to the partial synchronization criteria (25)- (27).…”

Section: Stability and Synchronization Criteriamentioning

confidence: 99%

Section: Stability and Synchronization Criteriamentioning

confidence: 99%

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Multi-stability and basin crisis in synchronized parametrically driven oscillators

2010

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“…Here, we also show that the method could be used to estimate the onset of synchronization. The basic idea is that a given system is stable, if there exist a continuous positive definite differentiable function (the Lyapunov function, V ) defined along the system's trajectory, such that its time derivativeV ≤ 0, as t → ∞ [34][35][36]. To begin with, we shall apply the following lemma to prove the main theorem of this paper.…”

Section: Sufficient Criteria For Global Synchronizationmentioning

confidence: 99%

“…The Lyapunov stability theory employs Lyapunov functionals which could be used for the analysis and synthesis of synchronization dynamics; and has been employed by earlier authors (see for example refs [34,35]). However, attention has been paid mostly on the stability of the synchronization.…”

Section: Sufficient Criteria For Global Synchronizationmentioning

confidence: 99%

Global chaos synchronization of coupled parametrically excited pendula

2009

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In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.

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Global synchronization of fractional‐order and integer‐order N component reaction diffusion systems: Application to biochemical models

Mesdoui

Shawagfeh

Ouannas

2020

Math Methods in App Sciences

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This study considers the problem of control and synchronization between fractional-order and integer-order, N-components reaction-diffusion systems with nonidentical coefficients and different nonlinear parts. The control scheme is designed using the Lyapunov direct method. The results are exemplified by two significant biochemical models, namely, the fractional-order Lengyel-Epstein model and the Gray-Scott model. To illustrate the effectiveness of the proposed scheme, numerical simulations are performed in one and two space dimensions using Homotopy Analysis Method (HAM).

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Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

Cited by 16 publications

References 51 publications

Multi-stability and basin crisis in synchronized parametrically driven oscillators

Multi-stability and basin crisis in synchronized parametrically driven oscillators

Global chaos synchronization of coupled parametrically excited pendula

Global synchronization of fractional‐order and integer‐order N component reaction diffusion systems: Application to biochemical models

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