Some new surprises in chaos

Bunimovich, Leonid; Vela-Arevalo, Luz V.

doi:10.1063/1.4916330

Cited by 13 publications

(20 citation statements)

References 20 publications

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“…They also indicate how such predictions can be practically made. Numerical simulations [7] suggest that some finite time predictions for nonuniformly hyperbolic systems are also possible. However finite time predictions in this case will be not as simple as for FDL-systems which are the uniformly hyperbolic systems with the maximal possible uniformity.…”

Section: Discussionmentioning

confidence: 99%

“…It seems natural to expect that for more general classes of chaotic dynamical systems there will be more than two time intervals with different hierarchies of the first hitting probabilities. Some (although too vague for formulating exact conjectures) indications of that can be extracted from computer simulations of dispersing billiards [7] Although the theory of finite time dynamics of chaotic systems is in infancy, it is rather clear what to do next and to which classes of dynamical systems these results should be generalized. For FDL-systems a remaining problem is to prove better estimates of the length of the short times interval.…”

Section: Discussionmentioning

confidence: 99%

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Where and when orbits of chaotic systems prefer to go

Bolding

Bunimovich

2019

Nonlinearity

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We prove that transport in the phase space of the "most strongly chaotic" dynamical systems has three different stages. Consider a finite Markov partition (coarse graining) ξ of the phase space of such a system. In the first short times interval there is a hierarchy with respect to the values of the first passage probabilities for the elements of ξ and therefore finite time predictions can be made about which element of the Markov partition trajectories will be most likely to hit first at a given moment. In the third long times interval, which goes to infinity, there is an opposite hierarchy of the first passage probabilities for the elements of ξ and therefore again finite time predictions can be made. In the second intermediate times interval there is no hierarchy in the set of all elements of the Markov partition. We also obtain estimates on the length of the short times interval and show that its length is growing with refinement of the Markov partition which shows that practically only this interval should be taken into account in many cases. These results demonstrate that finite time predictions for the evolution of strongly chaotic dynamical systems are possible. In particular, one can predict that an orbit is more likely to first enter one subset of phase space than another at a given moment in time. Moreover, these results suggest an algorithm which accelerates the process of escape through "holes" in the phase space of dynamical systems with strongly chaotic behavior.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Where and when orbits of chaotic systems prefer to go

Bolding

Bunimovich

2019

Nonlinearity

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“…Bunimovich and Vela-Arevalo (two pioneers in the field) summarized new insights in the field of dynamical billiards (Bunimovich and Vela-Arevalo 2015).…”

Section: Descriptionmentioning

confidence: 99%

DynamicalBilliards.jl: An easy-to-use, modular and extendable Julia package for Dynamical Billiard systems in two dimensions.

Datseris¹

2017

JOSS

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“…Chaos is a fascinating mathematical and physical phenomenon. The study of chaos shows that simple systems can exhibit complex and unpredictable behaviour [19,20]. Generally, chaos can be defined in many different forms according to conditions, observations, or applications of the object.…”

Section: Introductionmentioning

confidence: 99%

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

Sajid

2020

Abstract and Applied Analysis

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The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions fλ,ax=x+1−λxlnax;x>0, depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions fλ,ax are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of fλ,ax are shown. The existence of chaos in the dynamics of fλ,ax is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

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Some new surprises in chaos

Cited by 13 publications

References 20 publications

Where and when orbits of chaotic systems prefer to go

Where and when orbits of chaotic systems prefer to go

DynamicalBilliards.jl: An easy-to-use, modular and extendable Julia package for Dynamical Billiard systems in two dimensions.

Chaotic Behaviour and Bifurcation in Real Dynamics of Two-Parameter Family of Functions including Logarithmic Map

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