Abstract
The paper concerns the existence of affine-periodic solutions for discrete dynamicalsystems. This kind of solutions might be periodic, harmonic, quasi-periodic, even non-periodic.We prove the existence of affine-periodic solutions for discrete dynamical systems by using thetheory of Brouwer degree. As applications, another existence theorem is given via Lyapnovfunction.
Highlights
Introduction and main resultsGarrett Birkhoff introduced the concept of dynamical system [1], it vividly describes the physical background of differential equations
An example of a dynamical system in which time is a continuous variable is a system of m-dimensional, first-order, autonomous, ordinary differential equations dx = M (x), x ∈ Rm, dt where M : R × Rm → Rm is continuous
The conception of affine-periodic solutions was proposed, and the existence of solutions was studied for continuous dynamical system [11, 12, 16]
Summary
Garrett Birkhoff introduced the concept of dynamical system [1], it vividly describes the physical background of differential equations. The problem of periodic solution of continuous dynamical system has been a main subject of investigation. The conception of affine-periodic solutions was proposed, and the existence of solutions was studied for continuous dynamical system [11, 12, 16]. We are concerned with the existence of affine-periodic solutions for discrete dynamical systems. A basic topic is to investigate the existence of (Q, N )-affine-periodic solutions xn of system (1.3), i.e. Let I be identity matrix. The purpose of this paper is to investigate the existence of affine-periodic solutions for discrete dynamical system (1.3), where Q ∈ O(m).
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