Heteroclinic orbit

In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit.

Consider the continuous dynamical system described by the ODE

Suppose there are equilibria at ${\displaystyle x=x_{0}}$ and ${\displaystyle x=x_{1}}$, then a solution ${\displaystyle \phi (t)}$ is a heteroclinic orbit from ${\displaystyle x_{0}}$ to ${\displaystyle x_{1}}$ if

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