TRACKING SUSTAINED CHAOS

2000 ◽  
Vol 10 (03) ◽  
pp. 571-578 ◽  
Author(s):  
IRA B. SCHWARTZ ◽  
IOANA TRIANDAF
Keyword(s):  
Dynamical Systems ◽  
Simple Method ◽  
Periodically Driven ◽  
Unstable Solutions ◽  
Chaotic Transients ◽  
Parameter Values

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO 2 laser.

scholarly journals Travelling-Wave Solutions for Wave Equations with Two Exponential Nonlinearities

2018 ◽  
Vol 73 (10) ◽  
pp. 883-892
Author(s):  
Stefan C. Mancas ◽  
Haret C. Rosu ◽  
Maximino Pérez-Maldonado
Keyword(s):  
Dynamical Systems ◽  
Elliptic Equations ◽  
Hypergeometric Functions ◽  
Wave Equations ◽  
Constant Term ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Simple Method ◽  
Solutions Of Wave Equations ◽  
Wave Solutions

AbstractWe use a simple method that leads to the integrals involved in obtaining the travelling-wave solutions of wave equations with one and two exponential nonlinearities. When the constant term in the integrand is zero, implicit solutions in terms of hypergeometric functions are obtained, while when that term is nonzero, all the basic travelling-wave solutions of Liouville, Tzitzéica, and their variants, as as well sine/sinh-Gordon equations with important applications in the phenomenology of nonlinear physics and dynamical systems are found through a detailed study of the corresponding elliptic equations.

The Synchronization of a Periodically Driven Pendulum With Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Fuhong Min
Keyword(s):  
Dynamical Systems ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Invariant Domain ◽  
Periodically Driven

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in Duffing oscillator is developed through the theory of discontinuous dynamical systems. The conditions for the synchronization invariant domain are obtained, and the partial and full synchronizations are illustrated for the analytical conditions.

THREE SCHEMES TO SYNCHRONIZE CHAOTIC FRACTIONAL-ORDER RUCKLIDGE SYSTEMS

2007 ◽  
Vol 21 (12) ◽  
pp. 2033-2044 ◽  
Author(s):  
YANBIN ZHANG ◽  
TIANSHOU ZHOU
Keyword(s):  
Dynamical Systems ◽  
Fractional Order ◽  
System Parameter ◽  
Chaotic Synchronization ◽  
Linear Feedback ◽  
Linear Feedback Control ◽  
Synchronization Problem ◽  
Alternate Choice ◽  
Parameter Values ◽  
Fractional Orders

The synchronization problem of chaotic fractional-order Rucklidge systems is studied both theoretically and numerically. Three different synchronization schemes based on the Pecora–Carroll principle, the linear feedback control and the bidirectional coupling are proposed to realize chaotic synchronization. It is shown that such schemes can achieve the same aim for the same set of system parameter values (including fractional orders). This provides an alternate choice for applications of fractional-order dynamical systems in engineering fields.

Experimental Method for Determining the Damping Parameter of Spectral Lines Emitted by a Microwave Plasma at Atmospheric Pressure

2005 ◽  
Vol 59 (12) ◽  
pp. 1457-1464 ◽  
Author(s):  
I. Santiago ◽  
M. C. García ◽  
M. D. Calzada
Keyword(s):  
Line Profile ◽  
Atmospheric Pressure ◽  
Microwave Plasma ◽  
Argon Plasma ◽  
Spectral Lines ◽  
Experimental Conditions ◽  
Simple Method ◽  
Damping Parameter ◽  
Absorption Phenomenon ◽  
Parameter Values

In this work, a simple method for experimentally obtaining the value of the damping parameter or a-parameter of the spectral lines emitted by an argon plasma generated at atmospheric pressure is presented. The value of this coefficient indicates the proportion existing between the Lorentzian and Doppler components of the total line profile, which can be approximated to a Voigt function for our experimental conditions. The a-parameter values found were within the value interval recorded in the literature. The results obtained showed that the damping coefficient of the lines next to the fundamental level remains practically constant along the plasma column, whereas for the spectral lines involving high-lying levels, the a-parameter is sensitive to the changes in the electron density in the plasma. In this work it is also proved that the self-absorption phenomenon induces errors in the calculation of a, due to an increase in the broadening of the line profile produced by this phenomenon.

A Rigorous Dynamical-Systems-Based Analysis of the Self-Stabilizing Influence of Muscles

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
Melih Eriten ◽  
Harry Dankowicz
Keyword(s):  
Dynamical Systems ◽  
Feedback Mechanism ◽  
Rigid Body Dynamics ◽  
Local Dynamic ◽  
Dynamical Systems Analysis ◽  
Local Dynamic Stability ◽  
Reference Motion ◽  
Self Stabilization ◽  
Parameter Values ◽  
Muscle Models

In this paper, dynamical systems analysis and optimization tools are used to investigate the local dynamic stability of periodic task-related motions of simple models of the lower-body musculoskeletal apparatus and to seek parameter values guaranteeing their stability. Several muscle models incorporating various active and passive elements are included and the notion of self-stabilization of the rigid-body dynamics through the imposition of musclelike actuation is investigated. It is found that self-stabilization depends both on muscle architecture and configuration as well as the properties of the reference motion. Additionally, antagonistic muscles (flexor-extensor muscle couples) are shown to enable stable motions over larger ranges in parameter space and that even the simplest neuronal feedback mechanism can stabilize the repetitive motions. The work provides a review of the necessary concepts of stability and a commentary on existing incorrect results that have appeared in literature on muscle self-stabilization.

scholarly journals Topological chiral modes in random scattering networks

2020 ◽  
Vol 8 (5) ◽  
Author(s):  
Pierre Delplace
Keyword(s):  
Graph Theory ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Tight Binding ◽  
Edge States ◽  
Periodically Driven ◽  
Binding Models ◽  
Regular Networks ◽  
Discrete Time Dynamical Systems ◽  
Elementary Graph

Using elementary graph theory, we show the existence of interface chiral modes in random oriented scattering networks and discuss their topological nature. For particular regular networks (e.g. L-lattice, Kagome and triangular networks), an explicit mapping with time-periodically driven (Floquet) tight-binding models is found. In that case, the interface chiral modes are identified as the celebrated anomalous edge states of Floquet topological insulators and their existence is enforced by a symmetry imposed by the associated network. This work thus generalizes these anomalous chiral states beyond Floquet systems, to a class of discrete-time dynamical systems where a periodic driving in time is not required.

Chaotic Dynamical Systems as Automata

1987 ◽  
Vol 42 (6) ◽  
pp. 547-555 ◽  
Author(s):  
Joseph L. McCauley
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Finite Length ◽  
Discrete Maps ◽  
Chaotic Dynamical Systems ◽  
Periodically Driven ◽  
Driven System

We discuss the replacement of discrete maps by automata, algorithms for the transformation of finite length digit strings into other finite length digit strings, and then discuss what it required in order to replace chaotic phase flows that are generated by ordinary differential equations by automata without introducing unknown and uncontrollable errors. That question arises naturally in the discretization of chaotic differential equations for the purpose of computation. We discuss as examples an autonomous and a periodically driven system, and a possible connection with cellular automata is also discussed. Qualitatively, our considerations are equivalent to asking when can the solution of a chaotic set of equations be regarded as a machine, or a model of a machine.

THE SUBHARMONIC MELNIKOV THEORY FOR DAMPED AND DRIVEN OSCILLATORS REVISITED

2002 ◽  
Vol 12 (08) ◽  
pp. 1915-1923 ◽  
Author(s):  
SERGE BRUNO YAMGOUÉ ◽  
TIMOLÉON CRÉPIN KOFANÉ
Keyword(s):  
Duffing Oscillator ◽  
Poor Agreement ◽  
Weakly Nonlinear ◽  
Periodically Driven ◽  
Damped Oscillations ◽  
Forced Oscillators ◽  
Suggested Technique ◽  
Existence And Stability ◽  
Melnikov Theory ◽  
Parameter Values

The Melnikov theory for detecting subharmonic responses in periodically forced oscillators, the so-called subharmonic Melnikov theory, is a well-known technique within the dynamical system fields. The method succeeds in establishing the existence and stability of subharmonics in perturbed Hamiltonian systems as well as in discussing their bifurcations. But the parameter values estimated for the bifurcations are often found to be in poor agreement with those determined numerically. For damped and periodically driven oscillators, an approach which may substantially improve the agreement between analytical predictions and numerical investigations is proposed. The suggested technique consists in analyzing the forced regime of the oscillations by variational procedure, starting from the (approximate) solution for the free but damped oscillations. Applying this approach to the weakly nonlinear Duffing oscillator, it has been possible to detect analytically some subharmonics that the standard Melnikov theory fails to detect, thus demonstrating the ability of the proposed technique to capture much more of the dynamics.

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