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In dynamical systems, what is the distinction between a homoclinic orbit and a limit cycle? It seems to me like a homoclinic orbit is effectively just a limit cycle with a particular fixed point along the way, but in both cases can't we describe the object as an attractor (or source, depending on stability) curve in phase space, periodically traversed?

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  • $\begingroup$ And why might we prefer one or the other these in our system? Is there an advantage to making the fixed-point pit-stop? $\endgroup$ Commented Sep 18, 2023 at 18:34

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There are two crucial differences:

  • A homoclinic orbit is traversed just once. The dynamics cannot start before the fixed point or progress beyond it. Once it reaches the fixed point, it’s … well … fixed. The dynamics starts at the fixed point at $t→-∞$ and returns to it at $t→∞$.

  • Homoclinic orbits are not necessarily attractive. In particular, you can have homoclinic orbits in Hamiltonian systems, which do not feature attractors (or repellors) at all.

For example, consider a ball on some geography without friction: The ball starts with zero velocity at a hill top accelerates through a valley, slows down on the other (higher) side of the valley, and then returns through the valley to its starting point, where it arrives with zero velocity at $t→∞$. This is a homoclinic orbit. There is no periodicity. Moreover, the scenario is Hamiltonian and thus cannot have limit cycles at all.

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