Homoclinic orbit

A homoclinic orbit is a trajectory which loops around on itself in infinite time, i.e. if $𝑥$ lies in such an orbit $l i m 𝑡 \to \infty Φ 𝑡 𝑥 = l i m 𝑡 \to - \infty = 𝑥 *$ , where $𝑥 *$ is necessarily a saddle node called a homoclinic point.

Homoclinic orbits are likely to appear in a system with a Conserved quantity. In contrast a Heteroclinic trajectory flows from one saddle to another.

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