Physics / advanced
Deterministic motion beyond linearity
Analyze nonlinear phase portraits, bifurcations, driven oscillators, chaotic maps, and the limits of prediction in deterministic systems.
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Module 01 advanced
Deterministic motion beyond linearity
Analyze nonlinear phase portraits, bifurcations, driven oscillators, chaotic maps, and the limits of prediction in deterministic systems.
- 01
Fixed points and the geometry of motion
Locate equilibria, draw directional flow, and distinguish nodes, saddles, spirals, and centers without mistaking a sketch for a proof.
35 min - 02
What linear stability can and cannot prove
Use Jacobians, hyperbolicity, and Lyapunov functions to separate local decay, nonlinear stability, and the basin of attraction.
36 min - 03
When equilibria appear and exchange stability
Derive saddle-node and pitchfork normal forms, identify their nondegeneracy conditions, and recognize the role of symmetry.
38 min - 04
Hopf bifurcations and self-sustained oscillations
Separate radial amplitude selection from angular rotation and distinguish supercritical from subcritical Hopf behavior.
39 min - 05
Strobing a driven pendulum
Construct a return map by sampling at a fixed forcing phase and distinguish periodic, quasiperiodic, and potentially chaotic responses.
40 min - 06
The logistic map: stability one iteration at a time
Derive fixed-point and two-cycle stability, locate the first period doubling, and avoid treating every large parameter as chaotic.
40 min - 07
Lyapunov exponents and the horizon of prediction
Relate tangent-space growth to finite forecasting horizons while separating transient sensitivity from asymptotic chaos.
40 min - 08
Stretching, contraction, and strange attractors
Explain how local expansion can coexist with volume contraction and distinguish fractal geometry from evidence of chaos.
41 min - 09
Hamiltonian chaos and surviving invariant tori
Relate resonances and small divisors to the qualified persistence of quasiperiodic Hamiltonian motion under weak perturbations.
44 min - 10
Synchronization as a stable phase relationship
Use a reduced phase equation to derive locking thresholds, stable phase offsets, and the limits of entrainment.
39 min