In mathematics, a homoclinic orbit is a trajectory of a flow of a dynamical system which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium.
Homoclinic orbits and homoclinic points are defined in the same way for iterated functions, as the intersection of the stable set and unstable set of some fixed point or periodic point of the system.
Consider the continuous dynamical system described by the ODE
Suppose there is an equilibrium at, then a solution is a homoclinic orbit if
If the phase space has three or more dimensions, then it is important to consider the topology of the unstable manifold of the saddle point. The figures show two cases. First, when the unstable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a Möbius strip; in this case the homoclinic orbit is called twisted.
We also have the notion of homoclinic orbit when considering discrete dynamical systems. In such a case, if is a diffeomorphism of a manifold, we say that is a homoclinic point if it has the same past and future - more specifically, if it exists a fixed (or periodic) point such that
Read more about Homoclinic Orbit: Properties, Symbolic Dynamics
Famous quotes containing the word orbit:
“The Fitchburg Railroad touches the pond about a hundred rods south of where I dwell. I usually go to the village along its causeway, and am, as it were, related to society by this link. The men on the freight trains, who go over the whole length of the road, bow to me as to an old acquaintance, they pass me so often, and apparently they take me for an employee; and so I am. I too would fain be a track-repairer somewhere in the orbit of the earth.”
—Henry David Thoreau (1817–1862)