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Exponential Growth Rates of Periodic Asymmetric Oscillators

Meirong Zhang and Zhe Zhou

Abstract

In this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + q+(t)x+ + q(t)x = 0, where q+; q ∊ L1(ℝ / 2πℤ) and x+ = max(x; 0), x = min(x; 0) for x ∊ ℝ. It will be proved that the exponential growth rate

does exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.

Published Online: 2016-3-10
Published in Print: 2008-11-1

© 2016 by Advanced Nonlinear Studies, Inc.