Can anyone help me in understanding the contraction and the expansion of the phase space? what are Lyapunov exponents? and how come one understand this concept intuitively?
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$\begingroup$ One thing that may help: they do not contract or expand due to Lyapunov exponents. Lyapunov exponents are a tool with which to measure the contraction or expansion of a system. $\endgroup$– Cort AmmonCommented Mar 5, 2019 at 0:35
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$\begingroup$ then what causes the contraction and the expansion of the phase space?&i am sorry i meant to say some thing else i think i will edit my post & thank you $\endgroup$– akhil krishnanCommented Mar 5, 2019 at 0:39
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$\begingroup$ Rather than "expanding and contracting" a more descriptive terminology is "stretching and folding." $\endgroup$– Lewis MillerCommented Mar 5, 2019 at 15:48
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$\begingroup$ Thanks! I personalized the phenomena of attractors..I will use the technical term next time. $\endgroup$– akhil krishnanCommented Mar 5, 2019 at 15:51
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1 Answer
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Consider a given phase space point. Now imagine a tiny hyper-ball around it and let this point move through the phase space according to the system's dynamics.
The Lyapunov exponents will tell you how this hyper-ball deforms itself, in average, along this phase space trajectory. It may, for instance, converge to a point, explode, spaghettify, etc.
The number of exponents is the same as the number of phase space dimensions. If the system's a flow, then one of the exponents is always zero, corresponding to the local direction of the flow; negative exponents indicate exponential attraction, and positive exponents exponential repulsion of nearby trajectories - the defining characteristic of chaos.
So, it's not the phase space itself which contracts or expands, but rather balls of trajectories (neighborhoods) on this space, as the system dynamics pushes nearby trajectories close together or further apart in certain directions (called stable/unstable, respectively) at any given point.
You can even estimate the exponents numerically by placing initial conditions very close to the point you're considering and measuring how they approach or depart from it as the system evolves.
And, with the help of Lyapunov vectors, you get not only the deformation rates (i.e., Lyapunov exponents) for the tiny hyper-ball, but also the directions along which the expansions/contractions take place at given point:
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$\begingroup$ Is there any online video showing how does this hyper ball gets deformed and so on?? $\endgroup$ Commented Mar 5, 2019 at 1:55
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$\begingroup$ " negative exponents indicate exponential attraction, and positive exponents exponential repulsion - one of the hallmarks of chaos." what about zero Lyapunov exponent $\endgroup$ Commented Mar 5, 2019 at 2:17
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$\begingroup$ @akhilkrishnan A zero exponent means neither exponentially stable, nor unstable, but neutral. $\endgroup$– stafusaCommented Mar 5, 2019 at 2:20
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$\begingroup$ @akhilkrishnan I can't find right any animation of hyper-balls being deformed, sorry. $\endgroup$– stafusaCommented Mar 5, 2019 at 2:23
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$\begingroup$ So does a zero lyapunov indicates that there does exist an attractor but it's not a stable ? $\endgroup$ Commented Mar 5, 2019 at 2:30