Chaotic behaviour of solutions to a perturbed Korteweg—de Vries equation

Blyuss, Konstantin B.

doi:10.1016/s0034-4877(02)80003-9

Cited by 25 publications

(12 citation statements)

References 12 publications

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“…First, solutions to this differential equation have been proved to blow up in finite time for initial conditions in an infinite-area subset of the phase plane [12], and this places useful limits on the regions where UPOs can be expected to be found. Second, the existence of Melnikov chaos has been proven [13] for solutions to this ODE, hinting that a nontrivial set of UPOs might be expected. Our results were obtained using the program detailed in [11], and they are considered to be close to complete, at least for p < 10.…”

Section: Worked Examplementioning

confidence: 96%

Unstable periodic orbit detection for ODEs with periodic forcing

Deane

Marsh

2006

Physics Letters A

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Section: Worked Examplementioning

confidence: 96%

Unstable periodic orbit detection for ODEs with periodic forcing

Deane

Marsh

2006

Physics Letters A

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“…In dealing with the initial-boundary value problems of NPDEs, some authors use the Galerkin method, that is, they assume the space distribution of motion or the form of mode to simplify the problems into the temporal chaos. In recent years, many researchers have been interested in the chaos solution of NPDEs and discussed the chaotic behavior of nonlinear periodic waves and solitary waves [5][6][7] by perturbing to the KdV equation, the KdV-Burgers equation, the sine-Gorden equation, the Schrödingers equation, etc. with a soliton solution.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Shan-yuan

Liu

2010

Appl. Math. Mech.-Engl. Ed.

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Based on the Timoshenko beam theory, the finite-deflection and the axial inertia are taken into account, and the nonlinear partial differential equations for flexural waves in a beam are derived. Using the traveling wave method and integration skills, the nonlinear partial differential equations can be converted into an ordinary differential equation. The qualitative analysis indicates that the corresponding dynamic system has a heteroclinic orbit under a certain condition. An exact periodic solution of the nonlinear wave equation is obtained using the Jacobi elliptic function expansion. When the modulus of the Jacobi elliptic function tends to one in the degenerate case, a shock wave solution is given. The small perturbations are further introduced, arising from the damping and the external load to an original Hamilton system, and the threshold condition of the existence of the transverse heteroclinic point is obtained using Melnikov's method. It is shown that the perturbed system has a chaotic property under the Smale horseshoe transform.

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“…The nature of this external perturbation can be different and varies from one physical problem to another. Due to the fact that many real physical systems which behave seemingly randomly turned out to be low-dimensional chaotic, a particular attention has been focused recently on the influence of soliton equations under external perturbations [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

“…A route to chaos via a period-doubling sequence has been investigated in a perturbed sine-Gordon system [7]. The conditions of their chaotic behavior for the KdV equation in the presence of external Hamiltonian perturbations have been studied by using the Melnikov theory [3]. Furthermore, chaos has also been found in the reduction of the perturbed KdV equation [8].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Behavior of the Forced Compound KdV-Burgers-type Equation with High-order Nonlinear Terms

Yu¹

2010

International Journal of Nonlinear Sciences and Numerical Simulation

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The dynamical behavior of the forced compound KdV-Burgers-type equation with high-order nonlinear terms is investigated. It is shown that chaos can occur when systematic parameters are appropriately chosen. Abundant bifurcation structures and different routes to chaos such as period-doubling bifurcations, intermittent bifurcations and crises are found by applying bifurcation diagrams, Poincare maps and phase portraits.

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Chaotic behaviour of solutions to a perturbed Korteweg—de Vries equation

Cited by 25 publications

References 12 publications

Unstable periodic orbit detection for ODEs with periodic forcing

Unstable periodic orbit detection for ODEs with periodic forcing

Nonlinear flexural waves and chaos behavior in finite-deflection Timoshenko beam

Dynamical Behavior of the Forced Compound KdV-Burgers-type Equation with High-order Nonlinear Terms

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