scholarly journals Existence of Perpetual Points in Nonlinear Dynamical Systems and Its Applications

2015 ◽  
Vol 25 (02) ◽  
pp. 1530005 ◽  
Author(s):  
Awadhesh Prasad
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Critical Points ◽  
Nonlinear Dynamical Systems ◽  
Transient Dynamics ◽  
Coexisting Attractors ◽  
Nonlinear Dynamical ◽  
New Class ◽  
Perpetual Points ◽  
Bifurcation Behavior

A new class of critical points, termed as perpetual points, where acceleration becomes zero but the velocity remains nonzero, are observed in dynamical systems. The velocity at these points is either maximum or minimum or of inflection behavior. These points also show the bifurcation behavior as the parameters of the system vary. These perpetual points are useful for locating the hidden oscillating attractors as well as coexisting attractors. Results show that these points are important for a better understanding of transient dynamics in the phase space. The existence of these points confirms whether a system is dissipative or not. Various examples are presented, and the results are discussed analytically as well as numerically.

UNIFORMITY AND SPATIAL CHAOS OF SPATIAL PHYSICS KINEMATIC SYSTEM

2010 ◽  
Vol 24 (28) ◽  
pp. 5495-5503
Author(s):  
SHUTANG LIU ◽  
FUYAN SUN ◽  
JIE SUN
Keyword(s):  
Dynamical Systems ◽  
Lattice Model ◽  
Nonlinear Dynamical Systems ◽  
Coupled Map Lattice ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Uniform System ◽  
Spatial Chaos ◽  
Kinematic System ◽  
Bifurcation Behavior

This article summarizes the uniformity law of spatial physics kinematic systems, and studies the chaos and bifurcation behavior of the uniform system in space. In particular, it also fully explains the relation among the uniform system, the coupled map lattice model which has attracted considerable interest currently, and one-dimensional nonlinear dynamical systems.

Effect of noise on the bifurcation behavior of nonlinear dynamical systems

2007 ◽  
Vol 33 (3) ◽  
pp. 914-921 ◽  
Author(s):  
Apostolos Serletis ◽  
Asghar Shahmoradi ◽  
Demitre Serletis
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Bifurcation Behavior

Estimating Ljapunov-Exponents Using the Generalized Cell Mapping Method

1995 ◽  
Author(s):  
Swen Schaub ◽  
Werner Schiehlen
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Mapping Method ◽  
Cell Mapping ◽  
Coexisting Attractors ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Global Stability Analysis ◽  
Probabilistic Description ◽  
Stability Of Dynamical Systems

Abstract Ljapunov-Exponents are widely used to characterize the local stability of dynamical systems. On the other hand, Cell Mapping methods provide an effective numerical tool for global study by a probabilistic description of the time evolution. Using this description together with powerful interpolation techniques, an iterative method for global stability analysis with estimated Ljapunov-Exponents for all coexisting attractors of nonlinear dynamical systems is presented.

CONTROLLING CHAOTIC SYSTEMS VIA PHASE SPACE RECONSTRUCTION TECHNIQUE

2003 ◽  
Vol 13 (02) ◽  
pp. 467-471 ◽  
Author(s):  
Y. J. CAO ◽  
P. X. ZHANG ◽  
S. J. CHENG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Linear Part ◽  
Phase Space Reconstruction ◽  
Control Law ◽  
Reconstruction Technique ◽  
Nonlinear Dynamical ◽  
Novel Approach

A novel approach to control chaotic systems has been developed. The approach employs the technique of phase space reconstruction in nonlinear dynamical systems theory to construct a linear part in the reconstructed system and design a feedback control law. The effectiveness of the proposed approach has been demonstrated through its applications to two well-known chaotic systems: Lorenz chaos and Rössler chaos.

CLOSURES OF FRACTAL SETS IN NONLINEAR DYNAMICAL SYSTEMS WITH SWITCHED INPUTS

2001 ◽  
Vol 11 (08) ◽  
pp. 2205-2215 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Nonlinear Dynamical ◽  
Fractal Sets ◽  
Fractal Set

This paper studies closures of fractal sets observed in nonlinear dynamical systems excited stochastically by switched inputs. The Duffing oscillator and the forced dumped pendulum are analyzed as examples. The dynamics of the system is characterized by a fractal set in the phase space. We can numerically construct a closure that encloses the fractal set. Furthermore, it is shown that the closure is a limit cycle attractor of a dynamical system defined by the switching manifold.

NONEXISTENCE OF PERIODIC SOLUTIONS IN A CLASS OF DYNAMICAL SYSTEMS WITH CYLINDRICAL PHASE SPACE

2005 ◽  
Vol 15 (04) ◽  
pp. 1423-1431 ◽  
Author(s):  
YING YANG ◽  
ZHISHENG DUAN ◽  
LIN HUANG
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Periodic Solutions ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Linear Matrix ◽  
Specific Kind ◽  
Nonlinear Dynamical ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

This paper investigates the nonexistence of a specific kind of periodic solutions in a class of nonlinear dynamical systems with cylindrical phase space. Those types of systems can be viewed as an interconnection of several simpler subsystems with the interconnecting structure specified by a permutation matrix. Frequency-domain conditions as well as linear matrix inequalities conditions for nonexistence of limit cycles of the second kind are established. The main results also define the frequency range on which cycles of the second kind of the system cannot exist. Based on this LMI approach, an estimate of the frequency of cycles of the second kind can be explicitly computed by solving a generalized eigenvalue minimization problem. Numerical results demonstrate the applicability and validity of the proposed method and show the effect of nonlinear interconnections on dynamical behavior of entire interconnected systems.

scholarly journals Perpetual Points: New Tool for Localization of Coexisting Attractors in Dynamical Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750063 ◽  
Author(s):  
Dawid Dudkowski ◽  
Awadhesh Prasad ◽  
Tomasz Kapitaniak
Keyword(s):  
Dynamical Systems ◽  
Critical Points ◽  
Basins Of Attraction ◽  
Physical Significance ◽  
Stationary Points ◽  
Coexisting Attractors ◽  
Potential Applicability ◽  
Stable Equilibria ◽  
Perpetual Points ◽  
Damped Oscillator

Perpetual points (PPs) are special critical points for which the magnitude of acceleration describing the dynamics drops to zero, while the motion is still possible (stationary points are excluded), e.g. considering the motion of the particle in the potential field, at perpetual point, it has zero acceleration and nonzero velocity. We show that using PPs we can trace all the stable fixed points in the system, and that the structure of trajectories leading from former points to stable equilibria may be similar to orbits obtained from unstable stationary points. Moreover, we argue that the concept of perpetual points may be useful in tracing unexpected attractors (hidden or rare attractors with small basins of attraction). We show potential applicability of this approach by analyzing several representative systems of physical significance, including the damped oscillator, pendula, and the Henon map. We suggest that perpetual points may be a useful tool for localizing coexisting attractors in dynamical systems.

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