Controlling chaos in discontinuous dynamical systems

Danca, Marius‐F.

doi:10.1016/j.chaos.2004.02.032

Cited by 36 publications

(19 citation statements)

References 19 publications

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“…When a = 0, the model (1) is a discontinuous neural network. But for the activation function f 2 C 1 ; by utilizing the properties of generalized derivative and based on the detailed discussion in Danca (2002Danca ( , 2004, the chaotic behavior could remain chaotic when the trajectories cross the discontinuous surface (due to the discontinuous points fq i k g : xðtÞ ¼ yðtÞ ¼ 0Þ as shown in Fig. 2 with a = -0.025.…”

Section: Illustrative Examplesmentioning

confidence: 98%

“…Since the pioneer work of Pecora and Carrol (1990), chaotic synchronization has become a hot topic in nonlinear dynamics owing to their theoretical significance and potential applications. For a dynamical system with discontinuous vector field, there have been a few works about the synchronization issue of discontinuous dynamical systems (Danca 2002(Danca , 2004. To the best of our knowledge, however, there are few works about the synchronization of discontinuous delayed neural networks.…”

Section: Introductionmentioning

confidence: 95%

“…Sometimes physical laws are expressed by discontinuous functions, examples include dry friction, impacting machines, power circuits, switching in electronic circuits and many others (Danca 2004).…”

Section: Introductionmentioning

confidence: 99%

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Local synchronization of one-to-one coupled neural networks with discontinuous activations

Liu

Cao

2010

Cogn Neurodyn

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In this paper, local synchronization is considered for coupled delayed neural networks with discontinuous activation functions. Under the framework of Filippov solution and in the sense of generalized derivative, a novel sufficient condition is obtained to ensure the synchronization based on the Lyapunov exponent and the detailed analysis in Danca (Int J Bifurcat Chaos 12(8):1813-1826, 2002; Chaos Solitons Fractals 22:605-612, 2004). Simulation results are given to illustrate the theoretical results.

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Section: Illustrative Examplesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 95%

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Local synchronization of one-to-one coupled neural networks with discontinuous activations

Liu

Cao

2010

Cogn Neurodyn

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“…The analysis and control of discontinuous chaotic systems is not new, and has been solved in different ways [16], [17], [18], [19]. The novel contribution of this paper is the reinterpretation of these systems as hybrid automata, which leads to an alternative modelling and simulation framework.…”

Section: Motivationmentioning

confidence: 99%

“…The controlled Lorenz hybrid automaton, by following the computer science divide-and-conquer principle, is obtained by means of the composition of three Lorenz hybrid automata. Although the discontinuous Lorenz system can be effectively stabilised with smooth or even linear controllers, we propose a discontinuous-type controller in order to include within the system a third discontinuity surface, and show the potential of the framework to model and analyse discontinuous systems with multiple switching elements.The analysis and control of discontinuous chaotic systems is not new, and has been solved in different ways [16], [17], [18], [19]. The novel contribution of this paper is the reinterpretation of these systems as hybrid automata, which leads to an alternative modelling and simulation framework.…”

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confidence: 99%

Bringing order to chaos: Hybrid modelling of a discontinuous chaotic system

Navarro-López

Barajas-Ramírez

2010

2010 11th International Workshop on Variable Structure Systems (VSS)

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Abstract-We propose a computational-oriented perspective within the study of discontinuous chaotic systems, and provide insights into the modelling, control and simulation of chaotic systems with switching dynamics. In particular, the Lorenz system in its piecewise-linear version is studied. This system is reinterpreted within the hybrid-automaton framework, and what is referred to as the Lorenz hybrid automaton is established. Furthermore, a discontinuous control which eliminates the chaotic behaviour and steers the trajectories to a desired equilibrium is proposed. An integral characteristic of the modelling framework is that the controlled system, exhibiting three discontinuity surfaces, is reduced to the composition of several Lorenz hybrid automata. The approach proposed here is especially useful in order to specify the transitions between the different system operation modes, which becomes a crucial problem due to the existence of multiple switching surfaces. I. MOTIVATIONWhat is the best model to describe variable structure systems with multiple switching elements and exhibiting several types of complex behaviours? How to specify all their transitions in a deterministic way to facilitate the simulation? Every mathematical model is an approximation of the real world, and is full of limitations. However, some models are better than others at describing the evolution of certain physical and engineering systems. The hybrid systems framework, in particular, the computational hybridautomaton one is very adequate for modelling and controlling complex systemsThis paper is devoted to the specification and the computational abstraction of a particular discontinuous system with chaotic behaviour: the piecewise-linear (PWL) Lorenz system. The computational framework used is that of hybrid automata [2] faced when the trajectories either cross or slide on the discontinuity surfaces. Furthermore, due to the coexistence of several discontinuity surfaces, the problem of specification of the transitions between the system operation modes is critical, and special care has to be taken in the numerical integration [13], [14], [15]. The use of a hybrid automaton to describe the system is presented here as a solution to this problem. This paper is inspired by Navarro's previous results related to the modelling and discrete abstraction of discontinuous dynamical systems (DDSs) under the hybrid-automaton framework [4], [5], [6].The Lorenz hybrid automaton with 6 discrete locations is given. To prove the efficacy of the hybrid-automaton framework, a second hybrid automaton with 18 locations is proposed. This second automaton corresponds to the controlled PWL Lorenz system with three discontinuity surfaces, which is obtained by considering a sliding-modebased control. The control strategy is based on inserting a discontinuity surface on which the system presents the desired behaviour and forces the system trajectories to slide on this surface. The controlled Lorenz hybrid automaton, by following the computer science divide-and-conq...

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