Transition lines correspond to:
Hopf bifurcation (I II),
Period-doubling (II III),
Blow-up bifurcation (III IV).
4. Bifurcation Diagrams: Fixed Point Branching
By sweeping excitation amplitude B0B_0B0 or nonlinearity \lambda, we compute:
Amplitude of stationary field I0|I_0|I0 versus control parameter,
Eigenvalues of Jacobian at each branch point.
Key features:
Saddle-node bifurcation: sudden appearance/disappearance of solution branches,
Pitchfork bifurcation: symmetry-breaking leading to phase-patterning,
Limit cycle onset via Hopf instability.
Example:
Let B(x,t)=B0(x)(t)B(x,t) = B_0 \delta(x) \delta(t)B(x,t)=B0(x)(t)
Then the steady-state field amplitude exhibits:
I0B0(in weak damping regime)|I_0| \sim \sqrt{ \frac{B_0}{\lambda} } \quad \text{(in weak damping regime)}I0B0(in weak damping regime)
As B0B_0B0 crosses critical values, solutions become unstable or bifurcate into oscillatory/chaotic regimes.
5. Spatiotemporal Bifurcation Types Observed
These behaviors emerge from the interplay between dispersion, damping, nonlinearity, and external excitation profile.
