Results

We have considered the affine feedback control problem

dx/dt = f(x, λ(εt)) + b u(x, λ(εt)),

where the uncontrolled vector field f(x, λ) undergoes a bifurcation at λ = 0, b is a fixed vector, and u(x, λ) is the scalar feedback, to be designed in such a way that there is an immediate exchange of stabilities.

• If f(x, λ) undergoes a transcritical or pitchfork bifurcation, a small constant control is usually sufficient to provoke an immediate exchange of stabilities. In this proceedings, we discuss how this constant depends on ε, and, for the pitchfork bifurcation, how it can be chosen to make the state track the upper or lower equilibrium created in the bifurcation.

• If f(x, λ) undergoes a Hopf bifurcation, the situation is more complicated because, as shown by Neishtadt, the system undergoes a bifurcation delay which is robust against analytic deterministic perurbations. In this paper, we design a feedback consisting of two parts: a linear part which pushes the imaginary parts of the linearization's eigenvalues to zero, and a nonlinear part which stabilizes the origin of the system at λ = 0. This produces a bifurcation with double zero eigenvalue, described by the cubic Liénard equation
  
  dx/dt = y
  dy/dt = μ(λ) x + 2 a(λ) y + γ(λ) x y + δ(λ) x² - x² y - x³
  
  which is codimension four unfolding of the singular vector field (-x² y, -x³). We prove that under certain conditions on f(x, λ), the associated adiabatic system displays an immediate transfer of stability to a stable equilibirum branch.