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Why does the largest positive Lyapunov exponent λ1\lambda_1 dominate the long-term behavior?

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 λ1\lambda_1 dominating as t→∞t\to\infty.

Conditions

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

Reasoning, step by step

  1. Consider a sum of exponential terms representing the combined effects of different Lyapunov exponents.
  2. Take the limit as t→∞t \to \infty.
  3. Terms with negative exponents decay to zero and drop out.
  4. Among the remaining positive-exponent terms, the one with the largest coefficient in the exponent dominates the sum.
  5. Therefore, the largest positive Lyapunov exponent λ1\lambda_1 dictates the asymptotic separation growth.

Example

The whiteboard example demonstrates the rule: in a sum of exponentials like e0.1t+e10t+e−5t+e−15te^{0.1t}+e^{10t}+e^{-5t}+e^{-\frac{1}{5}t}, the negative-exponent terms decay away, and e10te^{10t} dominates e0.1te^{0.1t} as t→∞t\to\infty. The slide then states “biggest one (λ1\lambda_1) dominates as t→∞t \to \infty”.

Common misconceptions

  • Confusing largest value with largest absolute value.
  • Thinking that local contraction/expansion regions dictate the asymptotic exponent.

Watch the explanation

4:18 – 5:00Watch this moment ↗

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