Bifurcation Theory - Saddle-Node, Hopf, Transcritical, Pitchfork

Explore the intricacies of bifurcation theory in this comprehensive lecture from Dr. Shane Ross's course on 'Center manifolds, normal forms, and bifurcations'. Delve into local bifurcation theory near fixed points, examining qualitative changes in phase portraits as parameters vary. Focus on one-dimensional center manifolds and learn about saddle-node, transcritical, pitchfork, and period-doubling bifurcations. Gain insights into the Hopf bifurcation for vector fields and maps, understanding how periodic orbits emerge from fixed points. Discover which bifurcations are robust to perturbations and how they apply to equilibrium points of vector fields and periodic orbits.

Local Bifurcation Theory.
System in n dimensions, only look at center manifold directions.
saddle-node bifurcation,.
transcritical bifurcation,.
pitchfork bifurcation,.
period-doubling bifurcation for maps, related to the period doubling cascade (it's like a pitchfork bifurcation for maps).
Hopf bifurcation for vector fields and maps, the generation of periodic orbits out of fixed points.