Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows
Abstract
An equilibrium of a planar, piecewise-
C
1
, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to
λ
L
±i
ω
L
on one side of the discontinuity and
−
λ
R
±i
ω
R
on the other, with
λ
L
,
λ
R
>0
, and the quantity
Λ=
λ
L
/
ω
L
−
λ
R
/
ω
R
is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if
Λ<0
and subcritical if
Λ>0
.