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The time average of an observable $J$ along a trajectory $u(t)$ is defined as the limit of the average of $J$ over a finite time window of length $T$ as $T$ approaches infinity. It represents the long-run average behavior of the observable.

Conditions: The system is described by an ODE with trajectory $u(t)$.; $J$ is a function of the state $u$.; The averaging interval starts at $t=0$ and has length $T$.; The limit $T\to\infty$ is taken.

The time average of an observable $J$ along a trajectory $u(t)$ is defined as the limit of the average of $J$ over a finite time window of length $T$ as $T$ approaches infinity. It represents the long-run average behavior of the observable.

Conditions: The system is described by an ODE with trajectory $u(t)$.; $J$ is a function of the state $u$.; The averaging interval starts at $t=0$ and has length $T$.; The limit $T\to\infty$ is taken.