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For general trajectories not confined exactly to a single stable or unstable manifold, the lecture argues that the largest positive Lyapunov exponent controls the long-time behavior. This is summarized on the slide as $\lambda_1$ dominating as $t\to\infty$.

Conditions: Trajectory experiences a mix of stable and unstable effects.; Looking at the long-time limit.

For general trajectories not confined exactly to a single stable or unstable manifold, the lecture argues that the largest positive Lyapunov exponent controls the long-time behavior. This is summarized on the slide as $\lambda_1$ dominating as $t\to\infty$.

Conditions: Trajectory experiences a mix of stable and unstable effects.; Looking at the long-time limit.

A negative Lyapunov exponent corresponds to compression of state space along stable manifolds, while a positive Lyapunov exponent corresponds to stretching of state space along unstable manifolds. The lecture uses these signs to distinguish contraction toward a fixed point from expansion away to infinity.

Conditions: Applies to the manifold directions discussed in the clip.; The clip does not treat mixed-sign spectra in detail before ending.

A negative Lyapunov exponent corresponds to compression of state space along stable manifolds, while a positive Lyapunov exponent corresponds to stretching of state space along unstable manifolds. The lecture uses these signs to distinguish contraction toward a fixed point from expansion away to infinity.

Conditions: Applies to the manifold directions discussed in the clip.; The clip does not treat mixed-sign spectra in detail before ending.

Lyapunov exponents are ordered by value from largest to smallest, not by absolute value. Thus $\lambda_1$ is the exponent closest to positive infinity and $\lambda_n$ is the exponent closest to negative infinity.

Conditions: n is the dimension of the dynamical system; Ordering is by real value, not magnitude

Lyapunov exponents are ordered by value from largest to smallest, not by absolute value. Thus $\lambda_1$ is the exponent closest to positive infinity and $\lambda_n$ is the exponent closest to negative infinity.

Conditions: n is the dimension of the dynamical system; Ordering is by real value, not magnitude