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.
Citation
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
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