Mathematics / advanced
Nonlinear dynamics and bifurcations
Follow trajectories through phase space using stability, invariant sets, bifurcations, and deterministic chaos.
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Module 01 advanced
Nonlinear dynamics and bifurcations
Follow trajectories through phase space using stability, invariant sets, bifurcations, and deterministic chaos.
- 01
Flows, orbits, and phase portraits
Treat an autonomous differential equation as motion through state space, distinguishing a geometric orbit from its clock and its existence interval.
30 min - 02
Linearization and hyperbolic equilibria
Classify local motion with a Jacobian while keeping nonhyperbolic cases, coordinate changes, and global basins separate.
31 min - 03
Stable and unstable manifolds
Replace linear eigenspaces by nonlinear invariant geometry, derive a curved stable graph, and distinguish local sheets from their global continuations.
33 min - 04
Lyapunov functions and LaSalle invariance
Prove stability without explicit trajectories, use compact invariant sublevels, and identify the largest invariant subset of the zero-dissipation set.
34 min - 05
Planar dynamics and Poincaré–Bendixson
Use compact trapping regions to force periodic behavior, distinguish centers from limit cycles, and apply the divergence obstruction on its proper domain.
33 min - 06
Saddle-node, transcritical, and pitchfork bifurcations
Track equilibrium branches and their stability while distinguishing generic folds from branch-preserving and symmetry-constrained crossings.
34 min - 07
Hopf bifurcation and limit cycles
Connect a transversal imaginary-axis crossing with nonlinear amplitude saturation, distinguishing supercritical attraction from subcritical unstable cycles.
34 min - 08
Poincaré maps and Floquet multipliers
Reduce periodic-orbit stability to a transverse return map, distinguish multipliers from exponents, and account for the neutral flow direction.
35 min - 09
Chaotic maps and Lyapunov exponents
Quantify infinitesimal sensitivity along an orbit, analyze an expanding circle map, and distinguish positive exponents from a complete chaos claim.
35 min - 10
Symbolic dynamics and horseshoes
Encode a hyperbolic invariant set by bi-infinite itineraries, count periodic behavior, and state precisely what a horseshoe proves about a system.
36 min