Lyapunov functions for general nonuniform trichotomy with different growth rates

• Main Authors: Yongxin Jiang, Fang-Fang Liao
• Format: Article
• Language: English
• Published: Texas State University 2014-11-01
• Series: Electronic Journal of Differential Equations
• Subjects: Lyapunov functions; autonomous systems; nonuniform trichotomies; hyperbolicity
• Online Access: http://ejde.math.txstate.edu/Volumes/2014/244/abstr.html

In this article, we consider non-autonomous linear equations x' = A(t)x that may exhibit stable, unstable and central behaviors in different directions. We give a complete characterization of nonuniform $(\mu,\nu)$ trichotomies in terms of strict Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each given trichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential trichotomy, which admits different growth rates in the uniform and nonuniform parts.