Do Lyapunov Exponents on a Map Reflect Those of the Flow?

[Joppy]

I only have a basic understanding of this theorem at the moment, and may make another post regarding some of the details at another time. Currently I just want to check if I'm interpreting it correctly:

Loosely speaking, the theorem enables us to say that the Lyapunov exponents exist almost everywhere in the phase space under some measure theoretic/ergodic requirements (?).

I'm curious about what happens when we study the Lyapunov exponents on a subspace of our phase space. I guess a typical example would be studying the map induced by a flow for some system (perhaps through a Poincare section).

Now, suppose we can identify the Lyapunov exponents for the map readily, but not for the flow. Are the exponents on our map totally independent to those of the flow? I assume that they would almost never be equal, but perhaps related to each other in some way. Can we verify that exponential expansion or contraction on the map is also true for the flow? And does this have anything to do with Oseledet's...

 

[jvicens]

Hello,

Thank you for your post and for your interest in the Lyapunov exponents theorem. Your interpretation is mostly correct. The theorem states that for a dynamical system, under certain conditions, the Lyapunov exponents exist almost everywhere in the phase space. This means that for almost all initial conditions, the Lyapunov exponents can be calculated.

When studying the Lyapunov exponents on a subspace of the phase space, such as a Poincare section, the exponents may be different from those of the entire system. This is because the dynamics on the subspace may be different from the dynamics of the entire system. However, there are cases where the exponents on the subspace can be related to those of the entire system. This is known as the Oseledets theorem, which gives a way to calculate the Lyapunov exponents on a subspace using the exponents of the entire system.

Regarding your question about verifying exponential expansion or contraction on the map and the flow, it is possible for the exponents to be different between the two. In fact, the exponents on the map may be easier to calculate compared to those of the flow. However, there are ways to relate the exponents of the map to those of the flow, such as using the Oseledets theorem.

I hope this helps clarify your understanding of the Lyapunov exponents theorem. If you have any other questions or would like to discuss this further, please feel free to make another post. Thank you.