Period T: —
M[1,1]: — M[1,2]: —
M[2,1]: — M[2,2]: —
Trace(M): —
μ₁: —
μ₂: —
—
Floquet theorem: For ẍ + f(x,ẋ) periodic orbit of period T, linearize: δẍ = Jδx. Propagate the fundamental matrix Φ(T) = M — the monodromy matrix.
Floquet multipliers: eigenvalues of M. For Hamiltonian systems, det(M)=1, so μ₁μ₂=1.
Stability: |μ| < 1 → stable; |μ| > 1 → unstable; μ real on unit circle → neutral.
Simple pendulum: ẍ = −sin(x). Linearized: δẍ = −cos(x_orbit)·δx. For small amplitudes M→ identity; near separatrix (θ₀→π) → instability.
Circle chart: Floquet multipliers on complex plane — inside unit circle = stable.
Floquet multipliers: eigenvalues of M. For Hamiltonian systems, det(M)=1, so μ₁μ₂=1.
Stability: |μ| < 1 → stable; |μ| > 1 → unstable; μ real on unit circle → neutral.
Simple pendulum: ẍ = −sin(x). Linearized: δẍ = −cos(x_orbit)·δx. For small amplitudes M→ identity; near separatrix (θ₀→π) → instability.
Circle chart: Floquet multipliers on complex plane — inside unit circle = stable.