heteroclinic

Heteroclinic refers to a connection in dynamical systems between different invariant sets, typically between saddle-type equilibria or periodic orbits. A heteroclinic orbit is a trajectory that approaches one invariant set as time goes to negative infinity and a different invariant set as time goes to positive infinity. In the common case of equilibria p and q, a heteroclinic orbit γ satisfies α(γ)=p and ω(γ)=q, where α and ω denote the alpha- and omega-limit sets.

A key geometric condition is the intersection of the unstable manifold of p with the stable manifold of q. When these manifolds intersect transversely, the heteroclinic connection is robust to small perturbations, forming a structurally stable feature in the phase portrait. If a sequence of such connections links several invariant sets in a cycle, the system contains a heteroclinic cycle. More complex structures, called heteroclinic networks, consist of multiple interconnected heteroclinic connections.

Heteroclinic dynamics can organize phase space and produce slow switching between states, multi-stability, or, under certain conditions, chaotic behavior via Poincaré map constructions or related mechanisms. Nontransverse, or tangent, heteroclinic connections are more fragile and can undergo bifurcations that alter the global dynamics.

The concept applies to continuous-time flows and discrete-time maps and is relevant in diverse areas such as fluid dynamics, neuroscience, population dynamics, and Hamiltonian systems. It contrasts with homoclinic connections, where the trajectory begins and ends at the same invariant set.