Tangent Dynamics, Chaos, and Lyapunov Exponents
Stub. How a small perturbation grows (or shrinks) along a trajectory: tangent dynamics and finite-time growth rates, chaos as the generic outcome once some directions grow exponentially, the repeated-QR algorithm for tracking tangent dynamics numerically without overflow, and the convergence of the resulting Lyapunov spectrum — the full generalization of a single Lyapunov exponent to a high-dimensional system.
Corresponds to lectures: Tangent dynamics and finite-time growth (9/8); Chaos (9/10); Tangent dynamics and repeated QR (9/15); Convergence of Lyapunov spectra (9/17) — the second half of the “Foundations and computational tools” unit. Tangent dynamics tracks how an infinitesimal perturbation to the state evolves under the (possibly time-varying, for a trajectory rather than a fixed point) Jacobian — the direct high-dimensional generalization of the scalar Liapunov exponent from a 1D map. Chaos is what happens when some perturbation directions grow exponentially almost everywhere: nearby trajectories separate exponentially fast, making long-term prediction impossible despite fully deterministic dynamics. Numerically, naively multiplying many Jacobians together overflows almost immediately, which is why the repeated QR algorithm periodically re-orthonormalizes the perturbation directions (an application of the QR factorization) — the accumulated diagonal scaling factors converge to the full Lyapunov spectrum, one exponent per phase-space dimension, characterizing expansion/contraction along every independent direction at once.
Outline (TODO — flesh out each)
- Tangent dynamics
- Finite-time growth rates
- Chaos
- Repeated QR for tracking tangent dynamics numerically
- The Lyapunov spectrum and its convergence