Journal of Statistical Physics, Vol. 67, Nos. 1/2, 1992 Resonance Trapping in Dissipative and Antidissipative Systems: An Ergodic Approach Gershon Wolansky 1 Received August 16, 1990; final October 10, 1991 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. KEY WORDS: Ergodic measures; resonance trapping; circle maps. 1. INTRODUCTION In general, resonant trapping in an oscillating nonlinear system of ordinary differential equations is associated with the dissipative nature of the system. The analysis is essentially local (resonance capture) and is concentrated in a neighborhood of a given resonance (see, e.g., refs. 4-7). In this paper we suggest a different approach which is global in nature, and introduce a definition for a resonant trapping in a generalized sense. Consider the model system = ~g(x, O, s) 0--~o(x) (1.1~) ~=1 Department of Mathematics, Technion 32000, Haifa, Israel. 33 0022-4715/92/0400-0033506.50/0 9 1992 Plenum Publishing Corporation