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.

Remarks

The right side depends on the state $x$ only, not on the time $t$ directly. A non-autonomous system $x' = f(t, x)$ becomes autonomous with the extra state variable $s$ and the extra equation $s' = 1$. The equation $x' = k x$ of exponential growth and the logistic equation are autonomous. The equation $x' = t x$ is not. Solutions of an autonomous system can be translated in time.

Explanations

L1 human

In an autonomous system, the law of motion does not change with time. The velocity depends only on where the state is, not on when it is there.