1d dynamics

1
1d dynamics
V(x)
1 steady state
3 steady states
2
1d dynamics saddle-node and cusp bifurcations
Similar systems
Langmuir, enzyme
cut along m2const
Exothermic reaction
3
General 2-variable system

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

4
Modified Volterra Lotka system

• Modified preypredator system accounting for
  saturation effects
• Stationary states
• Determinant of Jacobi matrix
• existence k lt 1
• Trace of Jacobi matrix
• stability c gt 1 2k
• instability possible if k lt 0.5
• Hopf bifurcation c 1 2k

0, 0, 1, 0, k, (1 - k) (c k)
k, 1 k, (1 k) k
1 k, k,
5
k0.3, c0.5
k0.3, c0.7
Hopf bifurcation at c0.4
k0.3, c0.3
k0.3, c0.39
6
Dynamics near Hopf bifurcation

• Jacobi matrix at the bifurcation point c 1
  2k
• eigenvalues
• eigenvectors U, U
• Periodic orbit

Compute

• Slow dynamics
• a ??u(t), ? ltlt1
• ? m?? iw
• du /dt u(? ????u2)
• Polar representation
• u(r/?)eiq??????
• dr/dt ??r(m? r2)
• dq/dt w

7
Generalized Hopf bifurcation
dr/dt r(m1 m2 r2 r4) dq/dt w1 w2
r2 Complex rep ur eiq du/dt u(?1 ?2
u2 u4) ?????????????????k mk iwk
snp
subcritical
dr/dt r(m? r2) dq/dt w Complex rep du/dt
u(? u2) ?? m iw
Generic Hopf bifurcation
supercritical
Hopf
8
Bifurcation diagrams
pitchfork
supercritical Hopf
generalized Hopf
subrcritical Hopf
9
Global bifurcations
Saddle-loop (homoclinic)
Saddle-Node Infinite PERiod (Andronov)
10
Dynamics with separated time scales
excitable
oscillatory
bistable
biexcitable