Poincare Maps and Hoft Bifurcations

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Poincare Maps and Hoft Bifurcations

• Presented by Tyler White

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• Let v be a point in Rn, let f be a map defined
  on Rn, 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.

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• 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.

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• Andronov-Hopf Bifurcation Theorem Let vdot
  f_sub_a(v) be a family of systems of differential
  equations in Rn 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 2pi/b as orbits
  approach (0,0).

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• 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.