Bifurcation Phenomena in Nonlinear Systems and Theory of Dynamical Systems

1990 ◽  
Author(s):  
H. Kawakami
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Bifurcation Phenomena

A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

scholarly journals Bifurcation in Mean Phase Portraits for Stochastic Dynamical Systems with Multiplicative Gaussian Noise

2020 ◽  
Vol 30 (11) ◽  
pp. 2050216
Author(s):  
Hui Wang ◽  
Athanasios Tsiairis ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Gaussian Noise ◽  
Stochastic Systems ◽  
Planck Equation ◽  
Phase Portraits ◽  
Stochastic Bifurcation ◽  
Stochastic Dynamical Systems ◽  
Solution Processes ◽  
Bifurcation Phenomena ◽  
Stability Type

We investigate the bifurcation phenomena for stochastic systems with multiplicative Gaussian noise, by examining qualitative changes in mean phase portraits. Starting from the Fokker–Planck equation for the probability density function of solution processes, we compute the mean orbits and mean equilibrium states. A change in the number or stability type, when a parameter varies, indicates a stochastic bifurcation. Specifically, we study stochastic bifurcation for three prototypical dynamical systems (i.e. saddle-node, transcritical, and pitchfork systems) under multiplicative Gaussian noise, and have found some interesting phenomena in contrast to the corresponding deterministic counterparts.

On the Dynamics of Large Systems With Localized Nonlinearities

1988 ◽  
Vol 55 (4) ◽  
pp. 946-951 ◽  
Author(s):  
P. Hagedorn ◽  
W. Schramm
Keyword(s):  
Integral Equation ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Dynamic System ◽  
Frequency Response ◽  
Time Domain ◽  
Numerical Procedure ◽  
Linear Subsystem ◽  
Large Systems ◽  
The Time Domain

In this paper, a certain class of dynamical systems is discussed, which can be decomposed into a large linear subsystem and one or more nonlinear subsystems. For this class of nonlinear systems the dynamic behavior is represented in the time domain by means of an integral equation. A simple numerical procedure for the solution of this integral equation is given. It is also shown how the decomposition of the system can be used in measuring the frequency response of the large linear subsystem, without actually separating it from the nonlinear subsystems. An elastostatic analogy is used to illustrate the ideas and a numerical example is given for a dynamic system.

scholarly journals DYNAMICAL SYSTEMS ON THREE MANIFOLDS PART II: THREE-MANIFOLDS, HEEGAARD SPLITTINGS AND THREE-DIMENSIONAL SYSTEMS

2007 ◽  
Vol 17 (06) ◽  
pp. 2085-2095 ◽  
Author(s):  
YI SONG ◽  
STEPHEN P. BANKS
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Systems Theory ◽  
Topological Structure ◽  
Three Dimensional ◽  
Local System ◽  
Operating Point ◽  
Global Behavior ◽  
Heegaard Splittings

The global behavior of nonlinear systems is extremely important in control and systems theory since the usual local theories will only provide information about a system in some neighborhood of an operating point. Away from that point, the system may have totally different behavior and so the theory developed for the local system will be useless for the global one. In this paper we shall consider the analytical and topological structure of systems on two- and three-manifolds and show that it is possible to obtain systems with "arbitrarily strange" behavior, i.e. arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We shall do this by considering Heegaard Splittings of these manifolds and the resulting systems defined on the boundaries.

A Theory of Index for Point Mapping Dynamical Systems

1980 ◽  
Vol 47 (1) ◽  
pp. 185-190 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Discrete Time ◽  
Difference Equations ◽  
Vector Fields ◽  
Time Difference ◽  
Point Mapping ◽  
Strongly Nonlinear

Dynamical systems governed by discrete time-difference equations are referred to as point mapping dynamical systems in this paper. Based upon the Poincare´ theory of index for vector fields, a theory of index is established for point mapping dynamical systems. Besides its intrinsic theoretic value, the theory can be used to help search and locate periodic solutions of strongly nonlinear systems.

SYMBOLIC DYNAMICS, COARSE GRAINING AND THE MONITORING OF COMPLEX SYSTEMS

2011 ◽  
Vol 21 (12) ◽  
pp. 3465-3475 ◽  
Author(s):  
VASILEIOS BASIOS ◽  
DÓNAL MAC KERNAN
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Complex Systems ◽  
Error Estimates ◽  
Time Scales ◽  
Symbolic Dynamics ◽  
Prediction Error ◽  
Probabilistic Approach ◽  
Coarse Graining ◽  
Complex Nonlinear Systems

Coarse graining techniques and their associated symbolic dynamics are reviewed with a focus on probabilistic aspects of complex dynamical systems. The probabilistic approach initiated by Nicolis and coworkers has been elaborated. One of the major issues when dealing with the dynamics of complex nonlinear systems, the fact that the inherent time-scales of the unfolding phenomena are not well separated, is brought into focus. Recent results related to this interdependence, which is one of the most characteristic aspects of complexity and a major challenge in prediction, error estimates and monitoring of nonlinear complex systems, are discussed.

ON AUTOPARAMETRIC ROUTE LEADING TO CHAOS IN DYNAMICAL SYSTEMS

2002 ◽  
Vol 12 (04) ◽  
pp. 819-826 ◽  
Author(s):  
S. N. VLADIMIROV ◽  
V. V. NEGRUL
Keyword(s):  
Numerical Simulation ◽  
Phase Space ◽  
Dynamical Systems ◽  
Equilibrium State ◽  
Dynamic Systems ◽  
Strange Attractor ◽  
Oscillatory Systems ◽  
Physical Experiments ◽  
Bifurcation Phenomena ◽  
Infinite Dimensionality

Features of transition from regular types of oscillations to chaos in dynamic systems with finite and infinite dimensionality of phase space have been discussed. It has been found that for some types of nonlinearity, transition to the chaotic motion in these systems occurs according to the identical autoparametric scenario. The sequence of bifurcation phenomena looks as follows: equilibrium state ⇒ limit cycle ⇒ semitorus ⇒ strange attractor. On the basis of the results of numerical simulation a conclusion was made about the typical nature of such a scenario. The results of numerical calculations are confirmed by results of physical experiments carried out on the base of radiophysical self-oscillatory systems.

SOME COMMON EXPRESSIONS AND NEW BIFURCATION PHENOMENA FOR NONLINEAR WAVES IN A GENERALIZED mKdV EQUATION

2013 ◽  
Vol 23 (03) ◽  
pp. 1330007 ◽  
Author(s):  
RUI LIU ◽  
WEIFANG YAN
Keyword(s):  
Dynamical Systems ◽  
Solitary Waves ◽  
Nonlinear Waves ◽  
Blow Up ◽  
Mkdv Equation ◽  
Bifurcation Method ◽  
The Third ◽  
Bifurcation Phenomena ◽  
Kink Waves ◽  
The Common

Using the bifurcation method of dynamical systems, we study nonlinear waves in the generalized mKdV equation ut + a(1 + bu2)u2ux + uxxx = 0. (i) We obtain four types of new expressions. The first type is composed of four common expressions of the symmetric solitary waves, the kink waves and the blow-up waves. The second type includes four common expressions of the anti-symmetric solitary waves, the kink waves and the blow-up waves. The third type is made of two trigonometric expressions of periodic-blow-up waves. The fourth type is composed of two fractional expressions of 1-blow-up waves. (ii) We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) We reveal two kinds of new bifurcation phenomena. The first phenomenon is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, blow-up waves, tall-kink waves and anti-symmetric solitary waves. The second phenomenon is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves. We also show that the common expressions include many results given by pioneers.

Sign in / Sign up

Export Citation Format

Share Document