How can separation grow exponentially on a bounded attractor?
Conditions
- The trajectory lies on an attractor
- Separation is measured transversely with renormalization
Reasoning, step by step
- Recognize the apparent paradox: attractors are bounded sets, so how can a separation vector grow exponentially?
- Understand that the definition of Lyapunov exponents involves measuring local stretching in the transverse direction while moving along the attractor.
- Visualize the process as repeatedly picking up a small local picture, placing it along a short segment of the attractor, measuring the crosswise expansion, and then renormalizing and moving forward.
- Conclude that the exponential growth refers to this local transverse stretching with renormalization, not unrestricted global separation.
Example
The speaker asks how a bounded attractor can have exponentially growing separation, then answers that renormalization and transverse local measurement resolve the issue. The picture analogy is to pick up the drawing, put it down along a small piece of the attractor, move it farther forward, and repeat.
Common misconceptions
- Thinking exponential separation contradicts bounded attractors.
- Assuming the separation vector grows indefinitely in the ambient space without being reset or renormalized.
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