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In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit 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 x = x 0 {\displaystyle x=x_{0}} and x = x 1 {\displaystyle x=x_{1}} , then a solution ϕ {\displaystyle \phi } is a heteroclinic orbit from x 0 {\displaystyle x_{0}} to x 1 {\displaystyle x_{1}} if
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