Control of planar Bautin bifurcation

Abstract

On a versal deformation of the Bautin bifurcation it is possible to find dynamical systems that undergo Hopf or non-hyperbolic limit cycle bifurcations. Our paper concerns a nonlinear control system in the plane whose nominal vector field has a pair of purely imaginary eigenvalues. We find conditions to control the Hopf and Bautin bifurcation using the symmetric multilinear vector functions that appear in the Taylor expansion of the vector field around the equilibrium. The control law designed by us depends on two bifurcation parameters and four control parameters, which establish the stability of the equilibrium point and the orientation and stability of the limit cycles. Two examples are given.

Biophysical models of intrinsic homeostasis: Firing rates and beyond

“…Although such control can be safely assumed for firing rates, this is not obvious for more complex excitability characteristics. Indeed, control of several switches between excitability classes found in neurons (Box 1) has been treated mathematically [57][58][59], including a control strategy to decide a priori which bifurcations are traversed at locations in parameter space where bifurcations branch and several options are available (such as at the so-called Bogdanov-Takens (BT) point in Box 1, where Hopf, saddle-node, and saddlehomoclinic orbit (HOM) bifurcations coincide) [57]. These ideas have been applied to the control of bifurcations in conductance-based neuron models [54,60]*, offering a mathematical control perspective yet currently leaving potential neural implementations unaddressed.…”