Electron. J. Differential Equations, Vol. 2016 (2016), No. 233, pp. 1-32.

Dynamical bifurcation in a system of coupled oscillators with slowly varying parameters

Igor Parasyuk, Bogdan Repeta

Abstract:
This paper deals with a fast-slow system representing n nonlinearly coupled oscillators with slowly varying parameters. We find conditions which guarantee that all omega-limit sets near the slow surface of the system are equilibria and invariant tori of all dimensions not exceeding n, the tori of dimensions less then n being hyperbolic. We show that a typical trajectory demonstrates the following transient process: while its slow component is far from the stationary points of the slow vector field, the fast component exhibits damping oscillations; afterwards, the former component enters and stays in a small neighborhood of some stationary point, and the oscillation amplitude of the latter begins to increase; eventually the trajectory is attracted by an n-dimesional invariant torus and a multi-frequency oscillatory regime is established.

Submitted May 2, 2016. Published August 25, 2016.
Math Subject Classifications: 34C15, 34C23, 34C46, 37D10.
Key Words: Dynamical bifurcations; transient processes; invariant tori; multi-frequency oscillations; coupled oscillators; fast-slow systems.

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Igor Parasyuk
Faculty of Mechanics and Mathematics
Taras Shevchenko National University of Kyiv
64/13, Volodymyrska Street
City of Kyiv, 01601, Ukraine
email: pio@univ.kiev.ua
Bogdan Repeta
Faculty of Mechanics and Mathematics
Taras Shevchenko National University of Kyiv
64/13, Volodymyrska Street
City of Kyiv, 01601, Ukraine
email: bogdan.repeta@gmail.com

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