Topic (elapsed time)
Lecture 1: Planar ODEs	• Solutions of planar autonomous ODE systems (22:25), Orbits (28:20) and phase portraits (32:32). • Equilibria and cycles. Homo- and heteroclinic orbits to equilibria (36:50). • Equivalence of planar ODEs (59:00). • Classification of generic equilibria (1:07:40). • Poincaré return maps and classification of cycles (1:38:16). • Types of homoclinic orbits (1:50:00).
Lecture 2: Planar ODEs	• Remarks on topological equivalence (3:25). Orbitally equivalent systems (10:00). Topologically conjugate systems; diffeomorphic systems and substitution of variables (14:50) . • Classification of some non-hyperbolic equilibria (33:25) and cycles (1:07:20). • Poincare- Bendixson Theorem (1:14:50). Dulac criteria (1:23:25). • Planar Hamiltonian systems (1:30:40) and their dissipative perturbations (1:36:23).
Lecture 3: One-parameter equilibrium bifurcations in planar ODEs	• Bifurcation points (01:20). Bifurcations sets and diagrams (03:48). • Classification of bifurcations (09:00). Codimension of a bifurcation (14:05). • Hyperbolic equilibria do not bifurcate (23:00). Simplest equilibrium bifurcations (29:47). • Fold/saddle- node (40:00) and Andronov-Hopf (59:28) bifurcations of equilibria and their normal forms. • Efficient computation of complex equations (1:22:15). • Derivation of the normal form for Andronov-Hopf bifurcation (1:36:19).
Lecture 4: One-parameter global bifurcations in planar ODEs	• Fold bifurcation of cycles (3:30) and the normal form for its Poincaré return map (30:00). • Saddle homoclinic bifurcation (57:20). • Bifurcation of a homoclninc orbit to a saddle-node (1:37:30). • Saddle heteroclinic bifurcation (1:44:45). • Structural stability of planar ODEs (1:46:11).
Lecture 5: Two-parameter bifurcations in planar ODEs	• Curves of fold and Andronov-Hopf bifurcations in the parameter plane (03:12). • List of codim 2 equilibrium bifurcations in generic systems (19:50). • Cusp (30:28), Bogdanov-Takens (1:04:13), and Bautin (1:29:50) bifurcations, and their topological normal forms. • Remarks on global codim 2 bifurcations: Triple cycle (1:48:54), neutral saddle homoclinic orbit (1:55:31), noncentral homoclininc orbit to a saddle-node (1:57:50), saddle heteroclinic cycles (1:59:47), "figure-of-eight" (2:04:42), and saddle to saddle-node connection (2:05:00).
Lecture 6: One-parameter bifurcations of equilibria in n-dimensional ODEs	• Solutions, orbits, equilibria, cycles, connecting orbits, and chaotic invariant sets of n-dimensional ODEs (02:42). • Phase portraits and topological equivalence (13:30). Bifurcations of n- dimensional ODEs (14:38). • Hyperbolic equilibria (22:40) and Grobman-Hartman Theorem (24:25). Stable and unstable manifolds of hyperbolic equilibria (27:15). • Center- manifold reduction for bifurcations of equilibria (38:25). • Codim 1 local bifurcations in n-dimensional systems: Fold (1:17:37) and Andronov-Hopf (1:26:10). • Practical computation of the critical normal form coefficients for fold (1:47:20) and Andronov- Hopf (1:58:27) bifurcations.
Lecture 7: One-parameter bifurcations of cycles in n-dimensional ODEs	• Periodic orbits in n-dimensional ODEs: Poincaré maps (2:19), multipliers (13:00), hyperbolic cycles (30:00) and Grobman- Hartman Theorem for maps (32:20). • Center- manifold reduction for bifurcations of limit cycles (59:00). Simplest critical cases (1:16:24). • Codim 1 bifurcations of cycles in n-dimensional systems and normal forms for their Poincaré return maps: fold bifurcation (1:23:10), period- doubling (1:36:50), and Neimark-Sacker bifurcation (1:55:20 + the first 9 minutes of Lecture 8)
Lecture 8: Some global one- parameter bifurcations in n-dimensional ODEs	• Homoclinic orbits in n-dimensional ODEs (09:20). Leading eigenvalues (18:23). Stable and unstable invariant sets (24:30). Homoclinic orbits to hyperbolic equilibria in 3D, simple and twisted saddle cases (31:48). • Homoclinic center-manifold theorem (37:36). • Codim 1bifurcations of homoclinic orbits to hyperbolic equilibria (38:06). Shilnikov's theorems: Saddle (1:06:03), saddle-focus (1:28:38), and focus- focus (1:36:10) cases. • Bifurcations of homoclinic orbits to the saddle-node (1:38:36) and saddle-saddle (1:40:10) nonhyperbolic equilibria.