Dynamical Systems Induced by Continuous Time Difference Equations and Long-time Behavior of Solutions

The paper considers the continuous time difference equation x(t + 1) = f(x(t)), t∊R ⁺ (∗) with f being a continuous interval map. Although the long-time behavior of continuous solutions of Eq. (∗) has been extensively described under certain added conditions, there are a number of relatively simple and yet pivotal results for Eq. (∗) with no restrictions on f , that were not published. The paper is to compensate for this gap. Herein, properties of the solutions are derived from that of the ω -limit sets of trajectories of the dynamical system induced by Eq. (∗) . In particular, if the ω -limit set corresponding a solution x(t) is a cycle or the closure of an almost periodic trajectory, x(t) tends (in the Hausdorff metric for graphs) to a certain upper semicontinuous function.

Keywords:

• Dynamical System
• Difference Equation With Continuous Time
• 1d Maps
• Upper Semicontinuous Functions
• Gradient Catastrophe