Development of stochastic oscillations in a one-dimensional dynamical system described by the Korteweg-de Vries equation

1999 ◽  
Vol 88 (1) ◽  
pp. 182-195 ◽  
Author(s):  
A. V. Gurevich ◽  
K. P. Zybkin ◽  
G. A. Él’
Keyword(s):  
Dynamical System ◽  
Vries Equation ◽  
One Dimensional ◽  
De Vries ◽  
Dimensional Dynamical System ◽  
Stochastic Oscillations

scholarly journals Forced Oscillations of a Damped Korteweg-de Vries Equation in a Quarter Plane

2003 ◽  
Vol 05 (03) ◽  
pp. 369-400 ◽  
Author(s):  
Jerry L. Bona ◽  
S. M. Sun ◽  
Bing-Yu Zhang
Keyword(s):  
Dynamical System ◽  
Limit Cycle ◽  
Small Amplitude ◽  
Vries Equation ◽  
Initial Condition ◽  
Infinite Dimensional ◽  
De Vries ◽  
Exponentially Stable ◽  
Dimensional Dynamical System ◽  
Infinite Dimensional Dynamical System

Laboratory experiments have shown that when nonlinear, dispersive waves are forced periodically from one end of an undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. It is our purpose here to establish this as a fact at least in the context of a damped Korteweg-de Vries equation. Thus, consideration is given to the initial-boundary-value problem [Formula: see text] For this problem, it is shown that if the small amplitude, boundary forcing h is periodic of period T, say, then the solution u of (*) is eventually periodic of period T. More precisely, we show for each x > 0, that u(x, t + T) - u(x, t) converges to zero exponentially as t → ∞. Viewing (*) (without the initial condition) as an infinite dimensional dynamical system in the Hilbert space Hs(R+) for suitable values of s, we also demonstrate that for a given, small amplitude time-periodic boundary forcing, the system (*) admits a unique limit cycle, or forced oscillation (a solution of (*) without the initial condition that is exactly periodic of period T). Furthermore, we show that this limit cycle is globally exponentially stable. In other words, it comprises an exponentially stable attractor for the infinite-dimensional dynamical system described by (*).

scholarly journals BISTABILITY, BIFURCATION AND CHAOS IN A LASER SYSTEM

1995 ◽  
Vol 05 (04) ◽  
pp. 1021-1031 ◽  
Author(s):  
F. O'CAIRBRE ◽  
A. G. O'FARRELL ◽  
A. O'REILLY
Keyword(s):  
Dynamical System ◽  
Laser Physics ◽  
Laser System ◽  
Bifurcation And Chaos ◽  
One Dimensional ◽  
Fabry Perot ◽  
Dimensional Dynamical System ◽  
Parameter Ranges

In this paper we study a one-dimensional dynamical system that provides a model for the evolution of a Fabry–Perot cavity in Laser Physics. We determine the parameter ranges for bistability and chaos in the system. We also examine the bifurcations of the system and the occurrence of various types of cycles.

scholarly journals Markov covers and finiteness of periodic attractors for diffeomorphisms with a dominated splitting

2000 ◽  
Vol 20 (1) ◽  
pp. 1-14
Author(s):  
MASAYUKI ASAOKA
Keyword(s):  
Dynamical System ◽  
Dominated Splitting ◽  
One Dimensional ◽  
Surface Diffeomorphisms ◽  
Periodic Attractors ◽  
Dimensional Dynamical System

In this paper, we show the existence of Markov covers for $C^2$ surface diffeomorphisms with a dominated splitting under some assumptions. Using a Markov cover, we can reduce the dynamics to a one-dimensional dynamical system having a good metric property. As an application, we show finiteness of periodic attractors for the above diffeomorphisms.

scholarly journals Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative

1989 ◽  
Vol 9 (4) ◽  
pp. 737-749 ◽  
Author(s):  
M. Yu. Lyubich
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Critical Points ◽  
Schwarzian Derivative ◽  
Dimensional Manifold ◽  
Generic Point ◽  
One Dimensional ◽  
Manifold With Boundary ◽  
Dimensional Dynamical System

AbstractIt is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

EXACT TRAVELING-WAVE SOLUTIONS FOR ONE-DIMENSIONAL MODIFIED KORTEWEG–DE VRIES EQUATION DEFINED ON CANTOR SETS

Fractals ◽  
2019 ◽  
Vol 27 (01) ◽  
pp. 1940010 ◽  
Author(s):  
FENG GAO ◽  
XIAO-JUN YANG ◽  
YANG JU
Keyword(s):  
Traveling Wave ◽  
Vries Equation ◽  
Traveling Wave Solutions ◽  
Mathematical Physics ◽  
Cantor Sets ◽  
Wave Transformation ◽  
One Dimensional ◽  
Wave Solutions ◽  
De Vries ◽  
The One

The one-dimensional modified Korteweg–de Vries equation defined on a Cantor set involving the local fractional derivative is investigated in this paper. With the aid of the fractal traveling-wave transformation technology, the nondifferentiable traveling-wave solutions for the problem are discussed in detail. The obtained results are accurate and efficient for describing the fractal water wave in mathematical physics.

Sign in / Sign up

Export Citation Format

Share Document