What is the geometric meaning of negative versus positive Lyapunov exponents?
Conditions
- Applies to the manifold directions discussed in the clip.
- The clip does not treat mixed-sign spectra in detail before ending.
Reasoning, step by step
- Take the sign rule from the slide: negative exponents compress, positive exponents stretch.
- If every exponent is negative, compression occurs in all directions, so trajectories converge to a fixed point.
- If every exponent is positive, stretching occurs in all directions, so trajectories diverge to infinity.
Example
The slide states negative compress state space along stable manifolds and positive stretch it along unstable manifolds. The speaker explains all-negative exponents shrink the fabric in all directions and trajectories converge to a fixed point; all-positive exponents stretch in all directions and trajectories diverge to infinity.
Common misconceptions
- Confusing local linear information with global geometry.
- Mixing Lyapunov exponents with eigenvalues.
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