Control of chaos in unidimensional maps

Güémez, J.; Matías, Manuel A.

doi:10.1016/0375-9601(93)91119-p

Cited by 127 publications

(51 citation statements)

References 18 publications

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“…Techniques for stabilizing the infinity of unstable periodic orbits appearing in the presence of chaos are based mainly in two categories: the addition of perturbations to the parameters acting over the dynamics (Ott et al, 1990) and to the dynamic variables at play (Güemez and Matías, 1993;Parthasarathy and Sinha, 1995). The first one, more difficult to apply in real systems, needs some previous knowledge of the dynamics of the system and gets less effective as the system gets more complex.…”

Section: Discussionmentioning

confidence: 99%

“…The first one, developed by Güemez and Matías (1993) (GM or proportional feedback method) involves stabilizing one orbit of the many unstable periodic ones of a chaotic system by perturbing the system with periodic pulses, which are proportional to the state the system presents. For some values of intensity and periodicity of the perturbation the orbit is stabilized.…”

Section: State (External) Perturbationsmentioning

confidence: 99%

“…It has been shown that, under some constraints, a chaotic system may be 'controlled': unstable periodic orbits can be stabilized by means of small, periodic perturbations of the system parameters (Ott et al, 1990) or states (Güemez and Matías, 1993;Parthasarathy and Sinha, 1995). In fact, the very nature of chaotic dynamics, with its sensitive dependence on initial conditions and an infinite number of unstable periodic orbits simultaneously embedded in phase space, makes feasible the possibility of chaos control by moving and keeping the system's trajectory close to one of the (unstable) orbits, artificially stabilizing one of them.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Controlling Chaos in Ecology: From Deterministic to Individual-based Models

Solé

Gamarra

Gisbert³

et al. 1999

Bulletin of Mathematical Biology

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The possibility of chaos control in biological systems has been stimulated by recent advances in the study of heart and brain tissue dynamics. More recently, some authors have conjectured that such a method might be applied to population dynamics and even play a nontrivial evolutionary role in ecology. In this paper we explore this idea by means of both mathematical and individual-based simulation models. Because of the intrinsic noise linked to individual behavior, controlling a noisy system becomes more difficult but, as shown here, it is a feasible task allowed to be experimentally tested.

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Section: Discussionmentioning

confidence: 99%

Section: State (External) Perturbationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Controlling Chaos in Ecology: From Deterministic to Individual-based Models

Solé

Gamarra

Gisbert³

et al. 1999

Bulletin of Mathematical Biology

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“…Reference [8] contains some numerical experiments that reveal that this method has the potential to drive a chaotic system (1) into a T -periodic regime after control (2), where T is a multiple of m. Solé et al [15] provided more comments on the PF method, related to its applicability in the control of populations. For m = 1, the controlled system (2) reduces to the difference equation…”

Section: Introductionmentioning

confidence: 98%

Global stabilization of periodic orbits using a proportional feedback control with pulses

Braverman

Liz

2011

Nonlinear Dyn

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We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit.An important feature of our approach is that the required assumptions for which our results hold are met by a general class of maps, improving in this way some previous results. We discuss the applicability of our scheme to some models of population dynamics, and give numerical simulations to illustrate our analytical results.

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“…we getx 0 = 0, and the unique hyperbolic unstable ∆-periodic orbit is given by 5) which is inside the ball Br forr =…”

Section: Affine Equations With Equidistant and Shifted Impulsesmentioning

confidence: 96%