Autonomous system

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be a function. The autonomous system with vector field $f$ is the system of ordinary differential equations $$x'(t) = f\left(x(t)\right).$$ A solution is a differentiable function $\gamma : I \to U$ on an interval $I$ with $\gamma'(t) = f(\gamma(t))$ for each $t \in I$. The set $U$ is the phase space, or state space, of the system.