Nonautonomous Ordinary Differential Equations

Abstract

The solutions of the autonomous ordinary differential equation

$$ \dot{x} = f(x) $$

(1.1)

(where ẋ stands for dx/dt) give rise to a semidynamical (even dynamical) system on IR^d provided f: W → IR^d is continuous on the open subset W ⊂ IR^d and the solutions of Equation (1.1) through any point (x₀,t₀) ∈ W × IR are uniquely defined and remain in W for all time. In fact, if Φ(x₀;t) denotes the solution of Equation (1.1) through (x₀,0) evaluated at time t ∈ IR⁺, it can be verified that (W,Φ) is a semidynamical system.