1d dynamics 1 steady state:3 steady states: V(x)

1d dynamics: saddle-node and cusp bifurcations Similar systems: Langmuir, enzyme Exothermic reaction cut along  2 =const

General 2-variable system Dynamical system Stationary solution Jacobi matrix Stability conditions

Modified Volterra – Lotka system Modified prey–predator system accounting for saturation effects Stationary states Determinant of Jacobi matrix existence: k < 1 Trace of Jacobi matrix stability: c > 1 – 2k instability possible if k < 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,

k=0.3, c=0.7 k=0.3, c=0.5 k=0.3, c=0.39 k=0.3, c=0.3 Hopf bifurcation at c=0.4

Dynamics near Hopf bifurcation Jacobi matrix at the bifurcation point c = 1 – 2k eigenvalues eigenvectors U, U* Periodic orbit Slow dynamics: a =  u(t),  <<1  =   + i  du /dt = u(  –    |u| 2 ) Polar representation: u=(r/ )e i  dr/dt =   r(  – r 2 ) d  /dt =  Compute:

dr/dt = r(  1 +  2 r 2 – r 4 ) d  /dt =  1  +  2 r 2 Complex rep: u=r e i  du/dt = u(  1 +  2 |u| 2 – |u| 4 )  k =  k + i  k snp Hopf dr/dt = r(  – r 2 ) d  /dt =  Complex rep: du/dt = u(  – |u| 2 )  =  + i  Generic Hopf bifurcation supercritical subcritical Generalized Hopf bifurcation

Bifurcation diagrams supercritical Hopf subrcritical Hopf generalized Hopf pitchfork

Global bifurcations Saddle-Node Infinite PERiod (Andronov) Saddle-loop (homoclinic)

Dynamics with separated time scales oscillatory excitable biexcitable bistable