Nonlinear Dynamics & Chaos

Most real mechanical systems are nonlinear, and nonlinearity can produce chaos — deterministic yet unpredictable motion with exponential sensitivity to initial conditions. Understanding chaos requires new tools: Lyapunov exponents, Poincaré sections, and bifurcation diagrams.

Key Concepts

Sensitive Dependence

Two nearby trajectories in a chaotic system diverge exponentially:

|\delta x(t)|\approx|\delta x_0|e^{\lambda t}

, where

\lambda>0

is the Lyapunov exponent. This makes long-term prediction impossible even with excellent initial knowledge.

Fixed Points & Stability

A fixed point satisfies

\dot{\vec x}=0

. It is stable (attractor) if nearby trajectories converge, unstable (repeller) if they diverge. Stability is determined by the eigenvalues of the Jacobian

\partial\vec f/\partial\vec x

at the fixed point.

Period Doubling

Many systems transition to chaos through a cascade of period-doubling bifurcations. The ratio of successive bifurcation parameters converges to the Feigenbaum constant

\delta\approx4.669

Poincaré Section

A 2D slice through the phase space taken stroboscopically (once per forcing period for driven systems). Periodic orbits appear as points; chaotic orbits fill a fractal strange attractor.

Key Equations

Lyapunov exponent

\lambda = \lim_{t\to\infty}\frac{1}{t}\ln\frac{|\delta x(t)|}{|\delta x_0|}

Positive λ indicates chaos; largest Lyapunov exponent characterizes the system.

Logistic map

x_{n+1} = rx_n(1-x_n)

Simplest chaotic map; chaos appears for r > 3.57... Feigenbaum constant δ ≈ 4.669.

Pendulum (nonlinear)

\ddot{\theta} + \frac{g}{l}\sin\theta = -\beta\dot{\theta} + A\cos(\omega_d t)

Driven damped pendulum: exhibits chaos for large driving amplitude A.

Worked Example

Stability of a Fixed Point

Problem

The Lorenz system has a fixed point at the origin. The Jacobian there has eigenvalues $\lambda_1=-22.8$ , $\lambda_2=11.8$ , $\lambda_3=-2.67$ (for standard parameters $\sigma=10$ , $\rho=28$ , $\beta=8/3$ ). Is the origin stable?

Solution

A fixed point is stable only if all eigenvalues have negative real parts.

$\lambda_2=11.8>0$ : at least one eigenvalue is positive.

Therefore the origin is an unstable saddle point — trajectories near it will be repelled in the direction of the positive eigenvalue.

Answer Unstable — the positive eigenvalue

\lambda_2=11.8

causes nearby trajectories to diverge from the origin.

Practice

Exercises

7 problems

1 of 7

Two chaotic trajectories start $|\delta x_0|=10^{-10}$ m apart with Lyapunov exponent $\lambda=2.0$ /s. After $t=10$ s, find $|\delta x|$ (in m). Round to one decimal.

Your answer

2 of 7

In the logistic map $x_{n+1}=rx_n(1-x_n)$ , find the non-zero fixed point $x^*$ for $r=3.0$ . ( $x^*=1-1/r$ )

Your answer

(dimensionless)

3 of 7

The first period doubling of the logistic map occurs at $r_1=3.0$ , the second at $r_2=3.449$ , the third at $r_3=3.544$ . Estimate the Feigenbaum constant $\delta=(r_2-r_1)/(r_3-r_2)$ .

Unlock Exercise 3