Autonomous ordinary differential equations

Let $U\subseteq {\mathbb{R}}^n$ be open and let $V:U\to {\mathbb{R}}^n$ be a map. The equation

$x^{\mathrm{\prime }}(t)=V(x(t))$

is an autonomous ordinary differential equation: its right-hand side depends on the state alone and not explicitly on time. An initial value problem for this equation consists of a time $t_0\in \mathbb{R}$ and a state $x_0\in U$, written $x(t_0)=x_0$.

This is the special case of First-order systems, initial value problems, and solutions on intervals obtained from the time-dependent field $F(t,x):=V(x)$ on $\mathbb{R}\times U$. A solution on an interval $J$ is therefore a differentiable curve $x:J\to U$ with $t_0\in J$ satisfying the equation at every $t\in J$ and $x(t_0)=x_0$.

• Autonomous does not mean globally defined. The time variable ranges over all of $\mathbb{R}$, but the state space may be a proper open subset $U⊊{\mathbb{R}}^n$, and even on all of ${\mathbb{R}}^n$ a solution can fail to exist for all time if the vector field grows too fast.

• Initial time still matters. For an autonomous system the translated curve $t\mapsto x(t+t_1)$ is again a solution wherever it is defined, but the local existence theorem is still an initial value theorem at a stated time $t_0$.