Poincaré sections: integrable tori vs perturbation-induced chaos
Poincaré Section (q, p)
Energy & Phase Portraits
ε=0: perfect KAM tori (integrable pendulum)
Liouville-Arnold theorem: for n degrees of freedom, n independent conserved quantities in involution guarantee integrability.
Action-angle variables (I, θ) reduce dynamics to linear flow on a torus: İ=0, θ̇=ω(I).
KAM theorem: most tori survive small perturbations; destroyed tori give rise to chaos (Birkhoff-Smale homoclinic tangles).
System: pendulum H₀ = p²/2 − cos(q), perturbed by ε·cos(q−t).