scholarly journals The essential coexistence phenomenon in Hamiltonian dynamics

2021 ◽  
pp. 1-22
Author(s):  
JIANYU CHEN ◽  
HUYI HU ◽  
YAKOV PESIN ◽  
KE ZHANG
Keyword(s):  
Lyapunov Exponents ◽  
Smooth Manifold ◽  
Chaotic Dynamics ◽  
Energy Surface ◽  
Hamiltonian Dynamics ◽  
Invariant Subset ◽  
Hamiltonian Flow ◽  
Positive Volume ◽  
Bernoulli Flow

Abstract We construct an example of a Hamiltonian flow $f^t$ on a four-dimensional smooth manifold $\mathcal {M}$ which after being restricted to an energy surface $\mathcal {M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$ -invariant subset $U\subset \mathcal {M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except for the direction of the flow) and is a Bernoulli flow while, on the boundary $\partial U$ , which has positive volume, all Lyapunov exponents of the system are zero.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

scholarly journals A New Test for Chaotic Dynamics Using Lyapunov Exponents

2003 ◽  
Author(s):  
Fernando Fernández Rodríguez ◽  
Simón Javier Sosvilla Rivero ◽  
Julián Andrada Félix
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Dynamics

CHAOTIC DYNAMICS OF A SIMPLE PARAMETRICALLY DRIVEN DISSIPATIVE CIRCUIT

2011 ◽  
Vol 21 (07) ◽  
pp. 1927-1933 ◽  
Author(s):  
P. PHILOMINATHAN ◽  
M. SANTHIAH ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
S. RAJASEKAR
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classic Period ◽  
Control Signal ◽  
Period Doubling Bifurcation ◽  
Chaotic Circuit ◽  
Route To Chaos ◽  
Parameter Values

We introduce a simple parametrically driven dissipative second-order chaotic circuit. In this circuit, one of the circuit parameters is varied by an external periodic control signal. Thus by tuning the parameter values of this circuit, classic period-doubling bifurcation route to chaos is found to occur. The experimentally observed phenomena is further validated through corresponding numerical simulation of the circuit equations. The periodic and chaotic dynamics of this model is further characterized by computing Lyapunov exponents.

CHAOS AND COMPLEXITY

2001 ◽  
Vol 11 (01) ◽  
pp. 19-26 ◽  
Author(s):  
RAY BROWN ◽  
ROBERT BEREZDIVIN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Topological Conjugacy ◽  
Integer Lattices ◽  
Homoclinic Tangles ◽  
Dimensional Complexity ◽  
Finite Integer ◽  
The Relationship

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

DYNAMICAL CHARACTERISTICS OF DISCRETIZED CHAOTIC PERMUTATIONS

2002 ◽  
Vol 12 (10) ◽  
pp. 2087-2103 ◽  
Author(s):  
NAOKI MASUDA ◽  
KAZUYUKI AIHARA
Keyword(s):  
Invariant Measure ◽  
Periodic Orbit ◽  
Lyapunov Exponents ◽  
Chaos Theory ◽  
Chaotic Dynamics ◽  
Invariant Measures ◽  
Random Sequences ◽  
Dynamical Characteristics ◽  
Chaotic Sequences ◽  
Continuous State

Chaos theory has been applied to various fields where appropriate random sequences are required. The randomness of chaotic sequences is characteristic of continuous-state systems. Accordingly, the discrepancy between the characteristics of spatially discretized chaotic dynamics and those of original analog dynamics must be bridged to justify applications of digital orbits generated, for example, from digital computers simulating continuous-state chaos. The present paper deals with the chaotic permutations appearing in a chaotic cryptosystem. By analysis of cycle statistics, the convergence of the invariant measure and periodic orbit skeletonization, we show that the orbits in chaotic permutations are ergodic and chaotic enough for applications. In the consequence, the systematic differences in the invariant measures and in the Lyapunov exponents of two infinitesimally L∞-close maps are also investigated.

Three-Dimensional Torus Breakdown and Chaos With Two Zero Lyapunov Exponents in Coupled Radio-Physical Generators

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Nataliya V. Stankevich ◽  
Natalya A. Shchegoleva ◽  
Igor R. Sataev ◽  
Alexander P. Kuznetsov
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Parameter Plane ◽  
Poincare Section ◽  
Dimensional Torus ◽  
Typical Structure ◽  
Periodic Oscillations ◽  
Torus Breakdown ◽  
Periodic Dynamics

Abstract Using an example a system of two coupled generators of quasi-periodic oscillations, we study the occurrence of chaotic dynamics with one positive, two zero, and several negative Lyapunov exponents. It is shown that such dynamic arises as a result of a sequence of bifurcations of two-frequency torus doubling and involves saddle tori occurring at their doublings. This transition is associated with typical structure of parameter plane, like cross-road area and shrimp-shaped structures, based on the two-frequency quasi-periodic dynamics. Using double Poincaré section, we have shown destruction of three-frequency torus.

TESTING CHAOTIC DYNAMICS IN SYSTEMS WITH TWO POSITIVE LYAPUNOV EXPONENTS: A BOOTSTRAP SOLUTION

2007 ◽  
Vol 17 (01) ◽  
pp. 169-182 ◽  
Author(s):  
SIMONE GIANNERINI ◽  
RODOLFO ROSA ◽  
DIEGO LUIS GONZALEZ
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Time Series Data ◽  
Computer Experiments ◽  
Series Data ◽  
Resampling Methods ◽  
Bufo Arenarum ◽  
Toad Bufo

The present paper is devoted to the problem of detecting the presence of two positive Lyapunov exponents in time series data. In order to accomplish this task the accuracy of the estimates is essential, but existing estimation approaches do not provide it. We present a procedure exploiting resampling methods for building a statistical test for the presence of two positive exponents of comparable magnitudes through rigorous assessment of confidence intervals. The problem is studied by means of computer experiments performed in a variety of conditions on coupled Lorenz systems. Then, a case study regarding the time series of the cardiovascular activity of the toad Bufo Arenarum is presented. A comparison with other estimator algorithms is also shown.

Chaotic Dynamics in Duffing System with Two External Forcings

2011 ◽  
Vol 216 ◽  
pp. 777-781
Author(s):  
Shu Guang Zhang ◽  
Zhi Yong Zhu ◽  
Zhi Guo
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Homoclinic Bifurcation ◽  
External Forcing ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
External Forcings ◽  
Maximum Lyapunov Exponents

Duffing system with two external forcing terms is investigated in detail. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov's method. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, maximum lyapunov exponents and Poincare map are given to illustrate the theoretical analysis.

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