Poincare Maps and Hoft Bifurcations

1. Poincare Maps and Hoft Bifurcations Presented by Tyler White

2. Let v be a point in R^n, let f be a map defined on R^n, let gamma be a periodic orbit of the autonomous differential equation vdot = f(v). For a point v0 on gamma, the Poincare map T is defined on an n-1 dimensional disk D transverse to gamma at v0.

3. Definition: The eigenvalues of the (n-1) x (n-1) Jacobian matrix DvT(v0) are called the (Floquet) multipliers of the periodic orbit gamma. • If all the multipliers of gamma are inside (repectively, outside) the unit circle in the complex plane, then gamma is called an attracting periodic orbit. If gamma has some multipliers inside and some multipliers outside the unit circle, the gamma is called a saddle periodic orbit.

4. Andronov-Hopf Bifurcation Theorem: Let vdot = f_sub_a(v) be a family of systems of differential equations in R^n with equalibrium vbar = 0 for all a. Let c(a)+-ib(a) denote a complex conjugate pair of eigenvalues of the matrix Df_sub_a(0)that crosses the imaginary axis at a nonzero rate a = 0; that is, c(0) = 0, b = b(0) != 0, and c’(0) != 0. Further assume that no other eigenvalue of Df_sub_a(0) is an integer multiple of bi. Then a path of periodic orbits of vdot = f_sub_a(v) bifurcates from (a,v) = (0, 0). The periods of these orbits 2*pi/b as orbits approach (0,0).

5. Supercritical Bifurcation: At a supercritical Hoft bifurcation is seen as a smooth transition, the formally stable equilibrium starts to wobble in extremely small oscillations that grow into a family of stable periodic orbits as the parameter changes. • Subcritical bifurcation: this is seen as a sudden jump in behavior.