Unstable periodic orbits and templates of the Rössler system: Toward a systematic topological characterization

1995 ◽  
Vol 5 (1) ◽  
pp. 271-282 ◽  
Author(s):  
C. Letellier ◽  
P. Dutertre ◽  
B. Maheu
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Rossler System ◽  
Rössler System

Topological characterization of deterministic chaos: enforcing orientation preservation

2007 ◽  
Vol 366 (1865) ◽  
pp. 559-567 ◽  
Author(s):  
Marc Lefranc ◽  
Pierre-Emmanuel Morant ◽  
Michel Nizette
Keyword(s):  
Periodic Orbits ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Three Dimensions ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Triangulated Surfaces ◽  
Theoretic Characterization

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

Localization of periodic orbits of the Rössler system under variation of its parameters

2007 ◽  
Vol 33 (5) ◽  
pp. 1445-1449 ◽  
Author(s):  
Konstantin E. Starkov ◽  
Konstantin K. Starkov
Keyword(s):  
Periodic Orbits ◽  
Rossler System ◽  
Rössler System

scholarly journals Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

2021 ◽  
Vol 11 (15) ◽  
pp. 6955
Author(s):  
Andrzej Rysak ◽  
Magdalena Gregorczyk
Keyword(s):  
Optimal Parameter ◽  
Differential Transform Method ◽  
Bifurcation Diagrams ◽  
Transform Method ◽  
Rossler System ◽  
Differential Transform ◽  
Rössler System ◽  
Fractional System ◽  
Parameter Values

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

A SIMPLE OBSERVER OF THE HYPERCHAOTIC RÖSSLER SYSTEM

2010 ◽  
Vol 24 (22) ◽  
pp. 4325-4331
Author(s):  
XING-YUAN WANG ◽  
JUN-MEI SONG
Keyword(s):  
Dynamical System ◽  
Time Domain ◽  
Global Exponential Stability ◽  
Observer Design ◽  
The Other ◽  
State Observation ◽  
Rossler System ◽  
Error System ◽  
Rössler System ◽  
The Time Domain

This paper studies the hyperchaotic Rössler system and the state observation problem of such a system being investigated. Based on the time-domain approach, a simple observer for the hyperchaotic Rössler system is proposed to guarantee the global exponential stability of the resulting error system. The scheme is easy to implement and different from the other observer design that it does not need to transmit all signals of the dynamical system. It is proved theoretically, and numerical simulations show the effectiveness of the scheme finally.

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
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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