Chaos for a damped and forced KdV equation

Abstract The attractor for a damped and forced Korteweg–de Vries equation is studied. For large damping, it is shown that the attractor is trivial, with all solutions converging to a unique fixed point. For small damping, however, the dynamics can be complicated. It is investigated numerically how the attractor changes when the damping term decreases. In particular, a period-doubling cascade of periodic solutions is found, culminating with a chaotic regime, the ratio between consecutive points in this cascade converging to Feigenbaum’s constant. Beyond this cascade, the frequency spectrum of the solution is continuous and one Lyapunov exponent is positive. This seems to be the first work to obtain the onset of temporal chaos for this equation.

[1]  P. Bergé,et al.  L'ordre dans le chaos. , 1984 .

[2]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[3]  H. Herzel.  Chaotic Evolution and Strange Attractors , 1991 .

[4]  Y. Pomeau,et al.  L'ordre dans le chaos : vers une approche déterministe de la turbulence , 1988 .

[5]  R. Grimshaw,et al.  Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[6]  J. Ghidaglia.  Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time , 1988 .

[7]  E. Ott,et al.  Damping of Solitary Waves , 1970 .

[8]  I. Kevrekidis,et al.  Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations , 1990 .

[9]  Roger Temam,et al.  Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation , 1999 .

[10]  M. Feigenbaum.  Universal behavior in nonlinear systems , 1983 .

[11]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[12]  N. O. Weiss,et al.  Period doubling and chaos in partial differential equations for thermosolutal convection , 1983, Nature.

[13]  E. A. Cox,et al.  The evolution of resonant water-wave oscillations , 1986, Journal of Fluid Mechanics.

[14]  J. Ghidaglia.  A Note on the Strong Convergence towards Attractors of Damped Forced KdV Equations , 1994 .

[15]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[16]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[17]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[18]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[19]  D. Zheng,et al.  Period doubling in a perturbed sine-Gordon system, a long Josephson junction , 1989 .

[20]  Claudio Tebaldi,et al.  Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations , 1979 .

[21]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[22]  D. Crighton.  Applications of KdV , 1995 .

[23]  R. Russell,et al.  On the Compuation of Lyapunov Exponents for Continuous Dynamical Systems , 1997 .

[24]  G. Filatrella,et al.  Chaotic dynamics in the map model of fluxon propagation in long Josephson junctions , 1991 .