Stabilization of unstable periodic orbits for a chaotic system

In this paper we present a new method of controlling periodic orbits in chaotic systems. This method can be applied in situations when the chaotic system depends on one system parameter, which can be changed over a continuous interval or over a discrete, two-element set. We compare the new method to other ones, discuss its properties, and illustrate our approach with a numerical example.

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Ubiquitous quantum scarring does not prevent ergodicity

Nature Communications ◽

10.1038/s41467-021-21123-5 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Saúl Pilatowsky-Cameo ◽

David Villaseñor ◽

Miguel A. Bastarrachea-Magnani ◽

Sergio Lerma-Hernández ◽

Lea F. Santos ◽

...

Keyword(s):

Phase Space ◽

Periodic Orbits ◽

Chaotic System ◽

Chaotic Regime ◽

Quantum States ◽

Unstable Periodic Orbits ◽

Quantum Ergodicity ◽

Long Time ◽

Dicke Model ◽

Random States

AbstractIn a classically chaotic system that is ergodic, any trajectory will be arbitrarily close to any point of the available phase space after a long time, filling it uniformly. Using Born’s rules to connect quantum states with probabilities, one might then expect that all quantum states in the chaotic regime should be uniformly distributed in phase space. This simplified picture was shaken by the discovery of quantum scarring, where some eigenstates are concentrated along unstable periodic orbits. Despite that, it is widely accepted that most eigenstates of chaotic models are indeed ergodic. Our results show instead that all eigenstates of the chaotic Dicke model are actually scarred. They also show that even the most random states of this interacting atom-photon system never occupy more than half of the available phase space. Quantum ergodicity is achievable only as an ensemble property, after temporal averages are performed.

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Stabilizing Unknown and Unstable Periodic Orbits in DC-DC Converters by Temporal Perturbations of the Switching Time

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e98.a.331 ◽

2015 ◽

Vol E98.A (1) ◽

pp. 331-339 ◽

Cited By ~ 2

Author(s):

Hanh Thi-My NGUYEN ◽

Tadashi TSUBONE

Keyword(s):

Periodic Orbits ◽

Switching Time ◽

Unstable Periodic Orbits

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Unstable periodic orbits in a four-dimensional Faraday disk dynamo

Geophysical & Astrophysical Fluid Dynamics ◽

10.1080/03091929.2010.511206 ◽

2011 ◽

Vol 105 (2-3) ◽

pp. 273-286 ◽

Cited By ~ 2

Author(s):

Irene M. Moroz

Keyword(s):

Periodic Orbits ◽

Unstable Periodic Orbits ◽

Disk Dynamo

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On the Potential for Entangled States Between Chaotic Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500771 ◽

2014 ◽

Vol 24 (06) ◽

pp. 1450077 ◽

Cited By ~ 3

Author(s):

Matthew A. Morena ◽

Kevin M. Short

Keyword(s):

Dynamical System ◽

Periodic Orbits ◽

Chaotic Systems ◽

Entangled States ◽

Future Research ◽

Unstable Periodic Orbits ◽

Qualitative Information ◽

Chaotic Dynamical System ◽

Naturally Occurring ◽

External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

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