Amplitude expansion

1
Amplitude expansion
eigenvectors (Jacobi) .UeU, e?????(near
a bifurcation) ?????????????????????(Jacobi)
.V?V, ? O(1)
Deviation from stationary point u(t) U v(t)V
du/dt e u ??? u2 ??? uv ??? v2 ???? u3
slow ? v ??? u2 ??? uv
fast quasistationary
u O(e), v ??? u2 / ? O(e??)
du/dt e u ??? u2 (??? ??? / ? ???? ??u3

Saddle-node du/dt e?u
??? u2 u? u e /(2???? du/dt
e?/(2???? ??? u2 Cusp du/dt e? u e ???
u2 (??? ??? / ? ???? ??u3
2
Double zero eigenvalue
Det(Jacobi) Trace(Jacobi) 0
(find the condition parameterized by s.s)
Transformed Jacobi matrix Jordan normal form
One eigenvector only! U (1 0)
Complete the coordinate frame by another vector
(Jacobi) .VU
Deviation from stationary point u(t) U v(t)V
du/dt v dv/dt 0
Transformed linear system
u O(e), d/dt O(e?), v O(e??)
3
Double zero eigenvalue nonlinear expansion
Deviation from stationary point u(t) U v(t)V
du/dt v ??? u2 ??? uv ???? u3 dv/dt
e2 e??u ??? u2 ??? uv
deviation from double-0
Denote p v ??? u2 ??? uv ???? u3
Transformed system
Unfolding of double zero
du/dt p dp/dt f(u) pg(u)
f(u), g(u) polynomials
conservative subsystem
dissipative correction
4
Weakly dissipative system local analysis
dy/dt p dp/dt f(y) ? p g(y)

• Dynamical system
• ????????????
• Stationary solution
• Jacobi matrix
• Stability conditions

f(ys)0, p0
5
Weakly dissipative system global analysis
Trajectories at d? 0
Conserved energy
energy change
At small d compute the rate of energy change by
averaging over the period T
integration limits are values of y at turning
points where p vanishes
6
Unfolding of bifurcation at double zero eigenvalue
sub-Hopf
saddle u-node
dy/dt p dp/dt m? y2 ? p(m? y)
m?
SN
Det ys Tr? (m? ys)
SL
no static solutions
??
saddle s-node
7
Unfolding of bifurcation at double zero eigenvalue
subcritical Hopf
pitchfork
saddle loop
dy/dt p dp/dt m1y y3 p(m2 y2)
supercritical Hopf
(symmetric to the change of signs)
subcritical Hopf