Heteroclinic orbit

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The phase portrait of the pendulum equation x'' + sin x = 0. The highlighted curve shows the heteroclinc orbit from (x, x') = (−π, 0) to (x, x') = (π, 0). This orbit corresponds with the pendulum starting upright, making one revolution through its lowest position, and ending upright again.
The phase portrait of the pendulum equation x'' + sin x = 0. The highlighted curve shows the heteroclinc orbit from (x, x') = (−π, 0) to (x, x') = (π, 0). This orbit corresponds with the pendulum starting upright, making one revolution through its lowest position, and ending upright again.

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

\dot x=f(x)

Suppose there are equilibria at x = x0 and x = x1, then a solution φ(t) is a heteroclinic orbit from x0 to x1 if

\phi(t)\rightarrow x_0\quad \mathrm{as}\quad t\rightarrow-\infty

and

\phi(t)\rightarrow x_1\quad \mathrm{as}\quad t\rightarrow+\infty

This implies that the orbit is contained in the stable manifold of x1 and the unstable manifold of x0.

[edit] See also

[edit] References

  • John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences Vol. 42), Springer
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