Homoclinic orbits in the dynamics of resonantly driven coupled pendula

Kovačič, Gregor; Wettergren, Thomas A.

doi:10.1007/bf00916826

Cited by 33 publications

(29 citation statements)

References 22 publications

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“…(See also the exposition in [18], and the references to original works cited there.) An easy calculation, similar to that in [15], shows that for our model…”

Section: The N-pulse Melnikov Functionsupporting

confidence: 56%

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Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Holm¹,

Kovačič²

1997

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“…(See also the exposition in [18], and the references to original works cited there.) An easy calculation, similar to that in [15], shows that for our model…”

Section: The N-pulse Melnikov Functionsupporting

confidence: 56%

“…(The details of the calculation are similar to, but simpler than, those in [15].) By varying the parameter I in the formulas (3), (4), and (5), we conclude that all the heteroclinic solutions (3) corresponding to a fixed value of the angle φ 0 lie in a plane, and that their endpoints (4) and (5) …”

Section: The Ideal Cavitymentioning

confidence: 74%

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Holm¹,

Kovačič²

1997

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Section: The Ideal Cavitymentioning

confidence: 92%

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

et al. 1997

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“…where symbol cc and NST, respectively, denote the parts of the complex conjugate of the functions on the right-hand side of Equations (19) and the non-secular terms.…”

Section: Equations Of Motion and Perturbation Analysismentioning

confidence: 99%

“…Using this method, they gave an application to the forced and damped sine-Gordon equation. Later on, Kovacic and Wettergren [19] employed a modified Melnikov method to study the existence of multi-pulse jumping of homoclinic orbits and chaotic dynamics in resonantly forced coupled pendula. Furthermore, Kaper and Kovacic [20] investigated the existence of several types of multi-bump homoclinic orbits to resonance bands for completely integral Hamiltonian systems subjected to small amplitude Hamiltonian and damped perturbations.…”

Section: Introductionmentioning

confidence: 99%

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

2005

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This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance-primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.

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Homoclinic orbits in the dynamics of resonantly driven coupled pendula

Cited by 33 publications

References 22 publications

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

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