Resonance trapping in dissipative and antidissipative systems: An ergodic approach

Abstract

We propose a weak definition for a resonance trapping in oscillating systems. This definition requires the convergence of orbits, in the sense of measures convergence, to ah ergodic invariant measure, supported in a small neighborhood of the resonance zone. Then we apply this definition to a simplified, single-frequency oscillating system which admits a finite number of resonance points. It turns out that, under some assumptions, this generalized concept of resonance trapping may include the case where all resonances are repelling in the classical sense. The analysis is reduced to the investigation of the integrability of the logarithmic singularity with respect to an invariant measure of a reduced mapping.