The Saddle-Node Separatrix-Loop Bifurcation : A Manifestation of Delayed Feedback

Gautam Sethia
Institute for Plasma Research

Abhijit Sen
Institute for Plasma Research

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     Last modified: March 30, 2006

Abstract
The differential equation for a pendulum with linear damping and constant applied torque exhibits a co-dimension 2 saddle-node separatrix-loop (SN-SL) bifurcation. The same equation also arises in the study of the DC current-driven resistively & capacitively shunted Josephson junction (RCSJ). This bifurcation scenario is also typical in a number of neuron models like Morris-Lecar, Chay-Cook, and Wilson-Cowan Models. We have recently been investigating an excitable neuron model consisting of a subcritical Hopf oscillator with a nonlinear time delayed feedback; and intrestingly enough the bifurcation scenario is the same in the two parameter space of the feedback strength and the time delay. In analogy with the damped pendulum and the Josephson junction, the larger time delays (i.e. more memory in our model) correspond to less damping; and the increasing self-feedback strength corrresponds to decreasing torque (pendulum) or DC bias (Josephson junction). The seemingly generic character of the bifurcation is the motivating factor for us to explore the response of our time delayed feedback model to external periodic and stochastic forcing and to study the resulting spiking and bursting events.