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2018 Existence and roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations
Chunmei Zhang, Meng Fan, Jimin Zhang
Rocky Mountain J. Math. 48(8): 2751-2783 (2018). DOI: 10.1216/RMJ-2018-48-8-2751

Abstract

The objective of this paper is to explore the existence and roughness of the nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations. We first propose a more general notion of trichotomies called the nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations. Then, we give a complete characterization of the notion of nonuniform $(h,k,\mu ,\nu )$-trichotomy for linear nonautonomous differential equations and prove that any linear nonautonomous differential equation admits a nonuniform $(h,k,\mu ,\nu )$-trichotomy if it has an $(H,K,L)$ Lyapunov exponent with different signs in a finite-dimensional space. Finally, we establish the roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomies in a very concise manner, which implies that the nonuniform $(h,k,\mu ,\nu )$-trichotomy persists under sufficiently small linear perturbations. This study exhibits some new interesting findings in trichotomy that extend the corresponding results for uniform and nonuniform trichotomies.

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Chunmei Zhang. Meng Fan. Jimin Zhang. "Existence and roughness of nonuniform $(h,k,\mu ,\nu )$-trichotomy for nonautonomous differential equations." Rocky Mountain J. Math. 48 (8) 2751 - 2783, 2018. https://doi.org/10.1216/RMJ-2018-48-8-2751

Information

Published: 2018
First available in Project Euclid: 30 December 2018

zbMATH: 06999283
MathSciNet: MR3895002
Digital Object Identifier: 10.1216/RMJ-2018-48-8-2751

Subjects:
Primary: ‎34D09, 34D10

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 8 • 2018
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