Hidden attractors in dynamical systems

Dudkowski, Dawid; Jafari, Sajad; Kapitaniak, Tomasz; Kuznetsov, Nikolay V.; Леонов, Г. А.; Prasad, Awadhesh

doi:10.1016/j.physrep.2016.05.002

Cited by 569 publications

(344 citation statements)

References 161 publications

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“…The perpetual points are defined by the system of equationsR = 0,Ṙ = 0. These points can be used as the initial conditions to achieve different attractors of the system [5,6]. Often, all attractors of a dynamical system can be obtained this way.…”

Section: Main Equation and Resultsmentioning

confidence: 99%

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Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

2018

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Abstract.A model describing the dynamics of a spherical gas bubble in a compressible viscous liquid is studied. The bubble is oscillating close to an elastic wall of finite thickness under the influence of an external pressure field which simulates a contrast agent oscillating close to a blood vessel wall. Here we investigate numerically the coexistence of chaotic and periodic attractors in this model. One of the tools applied for seeking coexisting attractors is the perpetual points method. This method can be helpful for localizing coexisting attractors, occurring in various physically realistic ranges of variation of the control parameters. We provide some examples of coexisting attractors to demonstrate the importance of the multistability problem for the applications.

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Section: Main Equation and Resultsmentioning

confidence: 99%

“…The choice of the initial data becomes a non-trivial problem when attractors can coexist. We apply the perpetual points method to overcome this problem [5,6]. It does not require calculations of the attractors basins and, thus, is much less CPU consuming.…”

Section: Introductionmentioning

confidence: 99%

Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

2018

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“…We underline that an implementation of our approach reveals, with a high probability, if there is another unexpected attractor in the range of the considered perturbations. Indeed, as our method implies testing the system for a large number of initial conditions even hidden attractors (see, e.g., [1]) can automatically be found (when producing Figs. 5 and 8 more than 50,000 initial conditions where tested in each system).…”

Section: Discussionmentioning

confidence: 99%

How to find simple nonlocal stability and resilience measures

Lundström

2018

Nonlinear Dyn

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Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow-Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a highdimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and meth-N. L. P. Lundström (B) Department of Mathematics and Mathematical Statistics, Umeå University, 90187 Umeå, Sweden e-mail: niklas.lundstrom@umu.se ods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.

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“…An attractor is called rare if it has relatively small basin of attraction volume, while the basin of the hidden attractor does not intersect with the neighborhood of any fixed point [4]. Recently, many new examples of hidden attractors have been discovered [5][6][7][8][9] and this topic is gaining increasing interest.…”

Section: Introductionmentioning

confidence: 99%

Experimental investigation of perpetual points in mechanical systems

Brzeski

Virgin

2017

Nonlinear Dyn

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In dissipative dynamical systems, equilibrium (stationary) points have a dominant organizing effect on transient motion in phase space, especially in nonlinear systems. These time-independent solutions are readily defined in the context of ordinary differential equations, that is, they occur when all the time derivatives are simultaneously zero. However, there has been some recent interest in perpetual points: points at which the higher time derivatives are zero, but not necessarily the first. Previous work has focused on analytic work (including simulation) and some experimental studies of electric circuits. This paper focuses attention on the occurrence of these points in a simple mechanical system, including experimental verification. Thus, points of zero acceleration can be found in which the corresponding velocity is a maximum or minimum, but not zero. Specifically, the rigid-arm pendulum is used to generate data for which acceleration (and its derivative) can be evaluated. In this paper an experimental (mechanical) setup is described, specifically designed to investigate perpetual points, including a description of the data analysis approaches developed to identify their location.

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Hidden attractors in dynamical systems

Cited by 569 publications

References 161 publications

Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

Hidden Attractors in a Model of a Bubble Contrast Agent Oscillating Near an Elastic Wall

How to find simple nonlocal stability and resilience measures

Experimental investigation of perpetual points in mechanical systems

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