We deal with the following system of coupled asymmetric oscillators{x(1) + a(1)x(1)(+) - b(1)x(1)(-) + phi(1)(x(2)) = p(1)(t) x(2) + a(2)x(2)(+) - b(2)x(2)(-) + phi(2)( x(1)) = p(2)(t),where phi(i) : R -> R is locally Lipschitz continuous and bounded, p(i) : R -> R is continuous and 2 pi-periodic and the positive real numbers a(i), b(i) satisfy1/root a(i) + 1/root b(i) = 2/n, for some n is an element of N.We define a suitable function L : T-2 -> R-2, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincare map, in action-angle coordinates.

### Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

#### Abstract

We deal with the following system of coupled asymmetric oscillators{x(1) + a(1)x(1)(+) - b(1)x(1)(-) + phi(1)(x(2)) = p(1)(t) x(2) + a(2)x(2)(+) - b(2)x(2)(-) + phi(2)( x(1)) = p(2)(t),where phi(i) : R -> R is locally Lipschitz continuous and bounded, p(i) : R -> R is continuous and 2 pi-periodic and the positive real numbers a(i), b(i) satisfy1/root a(i) + 1/root b(i) = 2/n, for some n is an element of N.We define a suitable function L : T-2 -> R-2, appearing as the higher-dimensional generalization of the well known resonance function used in the scalar setting, and we show how unbounded solutions to the system can be constructed whenever L has zeros with a special structure. The proof relies on a careful investigation of the dynamics of the associated (four-dimensional) Poincare map, in action-angle coordinates.
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https://arxiv.org/abs/2103.06699
Systems of ODEs; Asymmetric oscillators; Unbounded solutions; Resonance
Boscaggin, A; Dambrosio, W; Papini, D
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1887975
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