Control of planar Bautin bifurcation

Denis Carvalho braga (1) , Luis Fernando mello (2) , Carmen Rocşoreanu (3) , Mihaela Sterpu (4)

1 Instituto de Sistemas Elétricos e Energia
2 Instituto de Ciências Exatas [Itajuba]
3 Department of Economic Sciences
4 Department of Mathematics and Computer Sciences

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.

Mots clés: Hopf bifurcation; Bautin bifurcation; Stability; Limit cycle; Control system

Denis Carvalho braga, Luis Fernando mello, Carmen Rocşoreanu, Mihaela Sterpu. Control of planar Bautin bifurcation. Nonlinear Dynamics, 2010, 62 (4), pp.989-1000. ⟨10.1007/s11071-010-9779-2⟩. ⟨hal-00619934⟩