Chaotic systems — sensitivity to initial conditions, Lyapunov exponents

WHAT is chaos?

WHY does "deterministic" + "unpredictable" not contradict? The equations fix the future exactly. But we never know the present exactly — we have an error $\delta_0$. In a non-chaotic system that error grows slowly (linearly). In a chaotic system it grows like $e^{\lambda t}$, so even a microscopic $\delta_0$ blows up to order-1 in finite time. Determinism is about the rules; predictability is about our knowledge.

HOW fast do trajectories separate? — Deriving the Lyapunov exponent

We want a number that measures the separation rate. Build it from scratch.

Step 1 — Set up two nearby trajectories. Let $x(t)$ obey $\dot x = f(x)$. Start a neighbour at $x(0)+\delta_0$. Define the separation $\delta(t) = x'(t) - x(t).$ Why this step? We literally care about how the gap between two runs evolves — that gap is the thing that grows in chaos.

Step 2 — Linearise for small separation. $\dot\delta = f(x+\delta)-f(x) \approx f'(x)\,\delta.$ Why this step? While $\delta$ is small, only the first derivative (the local stretching rate) matters. We Taylor-expand and keep the leading term.

Step 3 — Solve the linear growth. If $f'(x)$ were a constant $\lambda$, then $\dot\delta=\lambda\delta\Rightarrow \delta(t)=\delta_0 e^{\lambda t}$. In general the local rate varies, so we average it along the orbit: $|\delta(t)| \sim |\delta_0|\, e^{\lambda t}, \qquad \lambda \equiv \lim_{t\to\infty}\lim_{|\delta_0|\to0}\frac{1}{t}\ln\frac{|\delta(t)|}{|\delta_0|}.$ Why this step? The exponential is the only function that turns a constant relative growth rate into a clean number $\lambda$. Taking $\ln$ then dividing by $t$ extracts that rate.

Why two limits, in this order? First $\delta_0\to0$ keeps the linearisation honest (we only want infinitesimal separation). Then $t\to\infty$ averages over the whole attractor so $\lambda$ is a property of the system, not of one lucky starting point.

For an $n$-dimensional system there are $n$ exponents (the Lyapunov spectrum) $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_n$, describing stretching along each independent direction. The maximal one $\lambda_1$ decides chaos.

The predictability horizon (the 80/20 takeaway)

Worked Example 1 — The logistic map (discrete chaos)

Map: $x_{n+1}=r\,x_n(1-x_n)$. For maps, $\lambda=\lim_{N\to\infty}\frac1N\sum_{n=0}^{N-1}\ln|f'(x_n)|$ with $f'(x)=r(1-2x)$.

Why this formula? Each iteration multiplies a small separation by $|f'(x_n)|$. After $N$ steps the product of these factors is $\prod|f'(x_n)|$; taking $\frac1N\ln$ turns the product into an average log-stretch — exactly $\lambda$.

• At $r=2$ the map has a stable fixed point $x^*=0.5$, $f'(x^*)=2(1-2\cdot0.5)=0$, so $\lambda=-\infty$ direction → strongly stable, no chaos.
• At $r=4$ it is fully chaotic with the exact value $\lambda=\ln 2\approx0.693$.

Why $\ln2$? At $r=4$ the map is conjugate to the tent map which doubles distances each step → factor 2 → $\lambda=\ln2$.

Worked Example 2 — Predictability time for weather

Suppose $\lambda \approx 0.4\ \text{day}^{-1}$, initial error $\delta_0=10^{-3}$ (fractional), tolerable error $\Delta=1$.

$t_{\text{pred}}=\frac{1}{0.4}\ln\frac{1}{10^{-3}}=\frac{\ln10^{3}}{0.4}=\frac{6.9}{0.4}\approx 17\ \text{days}.$

Why this step? We just invert $\Delta=\delta_0e^{\lambda t}$. Now improve the sensors 1000×, $\delta_0=10^{-6}$: $t_{\text{pred}}=\frac{\ln10^{6}}{0.4}=\frac{13.8}{0.4}\approx34\ \text{days}.$ A million-fold better data only doubled the horizon. That is the brutal logarithm.

Worked Example 3 — Sign of $\lambda$ tells the dynamics

Damped pendulum spiralling to rest: all $\lambda_i<0$ (errors die). Frictionless pendulum: $\lambda=0$ (periodic, errors stay bounded). Driven damped pendulum in its chaotic regime: $\lambda_1>0$ while $\sum_i\lambda_i<0$ (volume shrinks onto a strange attractor, yet stretches within it).

Why can volume shrink while $\lambda_1>0$? Dissipation contracts phase-space volume ($\sum\lambda_i<0$), but the flow simultaneously stretches in one direction ($\lambda_1>0$) and folds — stretch-and-fold is the engine of chaos.

Flashcards

Connections

• Phase space and flows
• Fixed points and linear stability analysis
• Strange attractors and fractal dimension
• The logistic map and period-doubling route to chaos
• Hamiltonian chaos and the KAM theorem
• Driven damped pendulum
• Liouville's theorem (phase-space volume)

Concept Map

Hinglish (regional understanding)