Classical Dynamics of Particles and Systems · Problem 4-19

The Lyapunov exponent measures how fast nearby trajectories diverge—the fingerprint of chaos. For the tent map, you'll compute how the slope of the map amplifies small perturbations over many iterations. The key insight is that the tent map's piecewise-linear structure makes the derivative constant on each piece, so you can track the exponential growth of $|\Delta x_n|$ directly and take its logarithmic average. This reveals the threshold $\alpha = 1/2$ where the system transitions from stable to chaotic.

The Lyapunov exponent measures how fast nearby trajectories diverge—the fingerprint of chaos. For the tent map, you'll compute how the slope of the map amplifies small perturbations over many iterations. The key insight is that the tent map's piecewise-linear structure makes the derivative constant on each piece, so you can track the exponential growth of $|\Delta x_n|$ directly and take its logarithmic average. This reveals the threshold $\alpha = 1/2$ where the system transitions from stable to chaotic.