Chaotic Behavior of the Biharmonic Dynamics System

Vladimir S. Aslanov
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In the book: Elhadj Z., Models and Applications of Chaos Theory in Modern Sciences, Science Publishers, [Science Publishers(USA), Jersey, British Isles, Enfield, New Hampshire], (2011) pp. 627-644, (ISBN:  9781578087228)
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Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like: a sin θ + b sin 2θ. The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes under small perturbations. Melnikov method gives analytical criterion for heteroclinic chaos in terms of system parameters. A transition from chaotic to regular motion of the biharmonic oscillator was found as the heteroclinic chaos can be removed by increasing the coefficient of a damping force. The analytical results obtained using Melnikov method has been confirmed by a good match with numeric research.

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