Many neural systems generate stable stereotyped behavior with regions of localized slowing that resemble what is seen near a homoclinic or heteroclinic bifurcation. Sensitivity to perturbations from sensory input and noise may be crucial for adapting the behavior to external demands, but the nature of this type of sensitivity in the heteroclinic limit is still under active investigation. The animations show two one-parameter families of dynamical systems containing a limit cycle that disappears in a heteroclinic bifurcation. The first animation shows a smooth system, and the parameter is adjusted to show both the heteroclinic and Hopf bifurcations in this system. The second animation shows a piecewise linearization of the smooth system around the saddle points, which now contains a heteroclinic bifurcation and a limit cycle fold bifurcation. The third animation and the accompanying image show the isochrons of the piecewise linear system (all points which converge onto the limit cycle with a particular phase as time advances).

For more information on these systems and their phase sensitivity, please see:

Shaw KM, Park Y-M, Chiel HJ, Thomas PJ. Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit. SIAM J. Appl. Dyn. Sys. 2012;11:350–391.