A Melnikov Method for Homoclinic Orbits with Many Pulses

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

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Application of the Random Melnikov Method for Single-Degree-of-Freedom Vessel Rolls Motion

29th International Conference on Ocean, Offshore and Arctic Engineering: Volume 3 ◽

10.1115/omae2010-20182 ◽

2010 ◽

Author(s):

Kaiye Hu ◽

Yong Ding ◽

Hongwei Wang ◽

Jide Li

Keyword(s):

Global Stability ◽

Global Bifurcation ◽

Melnikov Method ◽

Threshold Value ◽

Homoclinic Orbits ◽

Nonlinear Damping ◽

Degree Of Freedom ◽

Bounded Noise ◽

Single Degree Of Freedom ◽

Single Degree

Basing on the nonlinear dynamics theory, the global stability of ship in stochastic beam sea is researched by the global bifurcation method. In this paper, bounded noise is first briefly introduced. Bounded noise is a harmonic function with constant random frequency and phase. It has finite power and its spectral shape can be made to fit a target spectrum, such as Pierson-Moskowitz spectrum, by adjusting its parameters. This paper considered the stochastic excitation term as bounded noise and the influence of nonlinear damping and nonlinear righting moment, setup the random single degree of freedom nonlinear rolling equation. Then the random Melnikov process for the nonlinear system with homoclinic orbits under both dissipative and bounded noise perturbations is derived. The random Melnikov mean-square criterion is used to analysis the global stability of this system. The research indicates that the bounded noise can approximately simulate the wave excitation and if the noise exceeds the threshold value, the ship will undergo stochastic chaotic motion. That will lead ships to instability and even to capsizing.

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Subharmonic bifurcation and chaos of a carbon nanotube supported by a Winkler and Pasternak foundation

International Journal of Modern Physics B ◽

10.1142/s0217979219502072 ◽

2019 ◽

Vol 33 (19) ◽

pp. 1950207

Author(s):

Liangqiang Zhou ◽

Fangqi Chen ◽

Ziman Zhao

Keyword(s):

Carbon Nanotube ◽

Melnikov Method ◽

Homoclinic Orbits ◽

Dynamic Responses ◽

Pasternak Foundation ◽

Stable And Unstable Manifolds ◽

Subharmonic Bifurcation ◽

Global Dynamic ◽

Critical Curves ◽

Subharmonic Bifurcations

In this paper, by employing both analytical and numerical methods, global dynamic responses including subharmonic bifurcations and chaos are investigated for a carbon nanotube supported by a Winkler and Pasternak foundation. The criteria of chaos arising from transverse intersections for stable and unstable manifolds of homoclinic orbits are proposed with the Melnikov method. The critical curves separating the chaotic and nonchaotic regions are plotted in the parameter plane. The parameter conditions for subharmonic bifurcations are also obtained by the subharmonic Melnikov method. It is proved rigorously that the route to chaos for this model is infinite subharmonic bifurcations. The stability of subharmonic bifurcations is also studied by the characteristic multipliers. Numerical simulations are given to confirm the analytical results.

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Structural Stability of Perturbed mKdV Solitons

Australian Journal of Physics ◽

10.1071/ph910495 ◽

1991 ◽

Vol 44 (5) ◽

pp. 495 ◽

Cited By ~ 1

Author(s):

J Roessler

Keyword(s):

Structural Changes ◽

Lattice Models ◽

Melnikov Method ◽

Homoclinic Orbits ◽

Travelling Wave ◽

Mkdv Equation ◽

Reduced Form ◽

Stable And Unstable Manifolds ◽

Standard Application ◽

Numerical Generation

Periodic perturbations are applied to the homoc\inic orbits corresponding to solitons of the modified Korteweg-de Vries (mKdV) equation, which is significant in plasma physics and lattice models. It is observed that for certain distinct frequencies the homoclinic orbits do not split into stable and unstable manifolds, which means absence of horseshoes and chaos. The analysis is performed on a travelling wave reduced form of the mKdV equation both by standard application of the Melnikov method as well as numerical generation of POincare maps. In particular, the geometry of the homoclinic orbits and their structural changes under perturbations is investigated.

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Saddle-Node Bifurcation and Homoclinic Persistence in AFMs with Periodic Forcing

Mathematical Problems in Engineering ◽

10.1155/2019/8925687 ◽

2019 ◽

Vol 2019 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Alexánder Gutiérrez Gutiérrez ◽

Daniel Cortés Zapata ◽

Diego Alexánder Castro Guevara

Keyword(s):

Local Existence ◽

Melnikov Method ◽

Homoclinic Orbits ◽

Conservative System ◽

Nonlinear Damping ◽

Periodic Forcing ◽

Nonconservative System ◽

Lennard Jones ◽

Atomic Force ◽

Saddle Node

We study the dynamics of an atomic force microscope (AFM) model, under the Lennard-Jones force with nonlinear damping and harmonic forcing. We establish the bifurcation diagrams for equilibria in a conservative system. Particularly, we present conditions that guarantee the local existence of saddle-node bifurcations. By using the Melnikov method, the region in the space parameters where the homoclinic orbits persist is determined in a nonconservative system.

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LOOKING FOR CHAOS.

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127496000205 ◽

1996 ◽

Vol 06 (03) ◽

pp. 485-496 ◽

Cited By ~ 2

Author(s):

HARRY DANKOWICZ

Keyword(s):

Melnikov Method ◽

Homoclinic Orbits ◽

Melnikov Function ◽

Higher Order ◽

Second Order ◽

Order Analysis ◽

First Order ◽

Order Calculation ◽

Alternative Approach

This paper derives an alternative approach to the Melnikov method, which greatly reduces the amount of algebra involved in higher-order calculations. To illustrate this, a particular system is studied for which such a higher-order analysis is necessary, due to an identically vanishing first-order Melnikov function. The results of a second-order calculation imply the existence of transverse homoclinic orbits and, consequently, the existence of a horseshoe.

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Homoclinic Orbits of a Buckled Beam Subjected to Transverse Uniform Harmonic Excitation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419500615 ◽

2019 ◽

Vol 29 (05) ◽

pp. 1950061

Author(s):

D. M. Zhang ◽

Y. L. Jin ◽

F. Li

Keyword(s):

Melnikov Method ◽

Homoclinic Orbits ◽

Harmonic Excitation ◽

Slow Manifold ◽

Singular Perturbation Method ◽

Smale Horseshoe ◽

Buckled Beam ◽

Resonance Band ◽

The One ◽

Geometric Singular Perturbation Method

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

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Detecting Hidden Chaotic Regions and Complex Dynamics in the Self-Exciting Homopolar Disc Dynamo

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417300087 ◽

2017 ◽

Vol 27 (02) ◽

pp. 1730008 ◽

Cited By ~ 49

Author(s):

Zhouchao Wei ◽

Irene Moroz ◽

Julien Clinton Sprott ◽

Zhen Wang ◽

Wei Zhang

Keyword(s):

Vector Fields ◽

Eddy Currents ◽

Complex Dynamics ◽

Equilibrium Points ◽

Melnikov Method ◽

Homoclinic Orbits ◽

Three Dimensions ◽

Chaotic Attractors ◽

Polynomial Vector Fields ◽

Poincaré Compactification

In 1979, Moffatt pointed out that the conventional treatment of the simplest self-exciting homopolar disc dynamo has inconsistencies because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo. Here we return to the simple dynamo system proposed by Moffatt, and demonstrate previously unknown hidden chaotic attractors. Then we study multistability and coexistence of three types of attractors in the autonomous dynamo system in three dimensions: equilibrium points, limit cycles and hidden chaotic attractors. In addition, the existence of two homoclinic orbits is proved rigorously by the generalized Melnikov method. Finally, by using Poincaré compactification of polynomial vector fields in three dimensions, the dynamics near infinity of singularities is obtained.

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