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We provide a complete symmetry-breaking bifurcation control for equivariantsmooth differential systems with Bogdanov-Takens singularities. Controllercoefficient space is partitioned by critical controller sets into differentconnected regions. The connected regions provide a classification for allqualitatively different dynamics of the controlled system. Hence, a statefeedback controller design with four small controller coefficients is proposedfor an efficient and full singular symmetry-breaking control. Our approachworks well for nonlinear control systems with both controllable anduncontrollable linearizations. Origin is a primary equilibrium for theuncontrolled system. This gives rise to two secondary local equilibria for thecontrolled system. These equilibria further experience tertiary fold andhysteresis type bifurcations. The secondary and primary equilibria experienceHopf and Bautin bifurcations leading to the appearance of one limit cycle fromprimary equilibrium, and either one or two from each secondary equilibria. Thecollisions of limit cycles with equilibria lead to either a heteroclinic cycleor four different homoclinic cycles. Each pair of limit cycles may respectivelymerge together and disappear. This is a saddle-node bifurcation of limitcycles. Different combinations of these give rise to a rich list of bifurcationscenarios. Finite determinacy of each of these bifurcations has been thoroughlyinvestigated. This greatly influences their stabilization potential inapplications. We consider Chua system with a quadratic state-feedbackcontroller. Controlled Chua system experiences a pitchfork bifurcation, threeHopf bifurcations and two homoclinic bifurcations. There exist two differentregions of controller coefficient choices for feedback regularization and twonearby regions for supercritical Hopf stabilization approach.

Symmetry-breaking singular controller design for Bogdanov-Takens bifurcations with an application to Chua system