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 |
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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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