Bifurcation of periodic solutions and invariant tori for a four-dimensional system

Liu, Xuanliang; Han, Maoan

doi:10.1007/s11071-008-9421-8

Cited by 15 publications

(7 citation statements)

References 10 publications

(7 reference statements)

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“…With these buckling conditions, the homoclinic phenomena for each case will be investigated by the extended Melnikov-type method, respectively. The existence of transverse homoclinic orbits and homoclinic bifurcation is of significance since they imply the occurrence of chaotic motion by the symbolic dynamics and Smale-Birkhoff theorem [55][56][57].…”

Section: Hamilton Formulation By Double-mode Truncationmentioning

confidence: 99%

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

Wang

2016

Nonlinear Dyn

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The homoclinic phenomena in doublelayered nanoplates (DLNP) are investigated. Based on the nonlocal continuum theory, the nonlinear dynamical equations for DLNP subjected to in-plane excitation are derived by double-mode Galerkin truncation. The extended Melnikov method is utilized to discuss the homoclinic phenomena and chaotic motion for the buckled DLNP system. The criterions for the existence of transverse homoclinic orbits are established under different four buckling cases (i.e., the first-and second-type synchronous buckling as well as asynchronous buckling). And the results derived by the abovementioned analysis are verified by molecular dynamics and Lyapunov exponent spectrum. Small-scale effect on the homoclinic motion is mainly inspected. From the result, it is rather novel that the transversality condition is independent of the nonlinear terms in the equations. This fact means that it is not necessary to distinguish whether the boundaries are movable or immovable for the in-plane parametric excitation DLNP. The parametric regime where homoclinic phenomena appear shrinks with the augment of nonlocal parameter for the first-type synchronous buckling case. However, this trend is just opposite for the other three buckling cases.Finally, it can be seen that homoclinic phenomena more likely occur on Mode I (i.e., lower mode) for the second-type synchronous and asynchronous buckling cases. However, the homoclinic motion for the firsttype asynchronous buckling most likely takes place on Mode III.

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Section: Hamilton Formulation By Double-mode Truncationmentioning

confidence: 99%

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

Wang

2016

Nonlinear Dyn

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“…(2)(3)(4)(5)(6)(7)(8)(9) For small 0 ≠ ε , there generates a periodic orbit of system in the neighborhood of…”

Section: Sufficient Condition For the Existence Of Periodic Solutionmentioning

confidence: 99%

The sufficient condition for the existence of periodic solution about iced cable with two degrees of freedom

Wei

2011

2011 International Conference on Multimedia Technology

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In this paper, with symbolic computation software of Maple, Melnikov function and sufficient condition for the existence of periodic solutions about iced cable with two degrees of freedom are investigated. The conclusion not only enriches the behavior of nonlinear dynamics about iced cable, but also provides the reference to the study of controlling the icing disaster, which is caused by large amplitude low frequency vibration of iced cable.

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“…Liu and Han [19] investigated the bifurcation of the periodic solutions for a four-dimensional nonlinear dynamic system by use of the Poincaré map and the integral manifold theory. Boinn [20] illustrated that the bifurcations of subharmonic orbits in perturbed nonlinear systems can be detected by using the subharmonic Melnikov method.…”

Section: Introductionmentioning

confidence: 99%

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

et al. 2015

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The existence and bifurcations of the subharmonic orbits for four-dimensional non-autonomous nonlinear systems are investigated in this paper. The improved subharmonic Melnikov method is presented by using the periodic transformations and Poincaré map. The theoretical results and the formulas are obtained, which can be used to analyze the subharmonic dynamic responses of four-dimensional nonautonomous nonlinear systems. The improved subharmonic Melnikov method is used to investigate the subharmonic orbits of a simply supported rectangular thin plate under combined parametric and external excitations for verifying the validity and applicability of the method. The theoretical results indicate that the subharmonic orbits can occur in the rectangular thin plate with 1:1 and 1:2 internal resonances. The results of numerical simulation also indicate the existence of the subharmonic orbits for the rectangular thin plate, which can verify the analytical predictions.

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Bifurcation of periodic solutions and invariant tori for a four-dimensional system

Cited by 15 publications

References 10 publications

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

Nonlocal effect on the nonlinear dynamic characteristics of buckled parametric double-layered nanoplates

The sufficient condition for the existence of periodic solution about iced cable with two degrees of freedom

Subharmonic Melnikov method of four-dimensional non-autonomous systems and application to a rectangular thin plate

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