Chaos and Sensitive Dependence

Definition Consider an autonomous system $\dot X = F(X)$, where $t$ is time, the dot means $d/dt$, the state $X$ lies in $\mathbb{R}^n$ and $F$ is continuously differentiable. Each initial state then has exactly one solution, and $\varphi_t(X_0)$ denotes the state at time $t$ of the solution with $X(0) = X_0$. Let $S$ be a set of states whose solutions exist and stay in $S$ for all $t \ge 0$. The system has sensitive dependence on initial conditions on $S$ if there is a distance $\varepsilon > 0$ with the following property. For every $X_0$ in $S$ and every $\delta > 0$ there are a state $Y_0$ in $S$ with $\lvert Y_0 - X_0 \rvert < \delta$ and a time $t \ge 0$ with

This does not conflict with determinism, because each exact state still has exactly one future. It limits prediction instead. If small errors grow roughly like $\delta e^{\lambda t}$ with $\lambda > 0$, an initial error $\delta$ reaches a size $\Delta$ after a time of about

so each tenfold improvement in the initial data extends this prediction horizon by only $(\ln 10)/\lambda$.

Definition With $t$ as time and a dot for $d/dt$, the Lorenz system for the state $X = (x, y, z)^{T}$ is

with constant parameters $\sigma, \rho, \beta > 0$. The classical values are $\sigma = 10$, $\rho = 28$ and $\beta = 8/3$. The right side $F(X)$ is a polynomial, so each initial state has exactly one solution, and the only nonlinear terms are $xz$ and $xy$. Replacing $(x, y, z)$ by $(-x, -y, z)$ carries solutions to solutions. The origin is an equilibrium for every choice of parameters. For $\rho \le 1$ it is the only one, and for $\rho > 1$ there are exactly two more,

The system also contracts volume. In general, for $\dot X = F(X)$ with $F$ continuously differentiable, carry a bounded region $D(0)$ of initial states along the solutions, let $D(t)$ be the region they fill at time $t$, and let $V(t)$ be its volume. While these solutions exist,

where $dX$ is the volume element. So a region loses volume while the divergence $\nabla \cdot F$ is negative throughout it, and the flow contracts volume when the divergence is negative everywhere. For the Lorenz field the divergence is the constant $-(\sigma + 1 + \beta)$, and therefore

Definition Let the solutions of $\dot X = F(X)$, with $F$ continuously differentiable, exist for all $t \ge 0$, and write $\varphi_t$ for the flow. A closed bounded set $A$ of states is an attractor if three conditions hold. First, the set is invariant, meaning $\varphi_t(A) = A$ for every $t \ge 0$. Second, it attracts an open set $U$ containing it, so every solution that starts in $U$ has distance from $A$ tending to $0$ as $t \to \infty$. Third, it is minimal, so no smaller closed subset of $A$ satisfies the first two conditions. The set of all initial states whose solutions approach $A$ is its basin of attraction. Definitions in the literature differ in the third condition, for example by asking for a solution that comes arbitrarily close to every point of $A$, so the one in use should be named. An attractor on which the motion is chaotic is called a chaotic attractor, and often a strange attractor. If $\nabla \cdot F \le -c$ everywhere for a constant $c > 0$, every closed bounded invariant set has zero volume, because its volume $V$ satisfies $V \le e^{-ct} V$ for every $t > 0$.

Definition Let $X(t)$ be a bounded solution of $\dot X = F(X)$ for $t \ge 0$, with $F$ continuously differentiable and Jacobian matrix $J(X)$. To first order, a perturbation $w$ of the state evolves by the variational equation $\dot w = J(X(t))\,w$. Let $\Phi(t)$ be its fundamental matrix with $\Phi(0) = I$. The largest Lyapunov exponent of the solution is

Every perturbation $w(0) \ne 0$ has the exponential growth rate $\limsup_{t \to \infty} \frac{1}{t} \ln \lvert w(t) \rvert$, which is at most $\lambda_{\max}$, and every perturbation outside a proper subspace has exactly the rate $\lambda_{\max}$. The number belongs to the solution being followed, and it describes infinitesimal perturbations over infinite time. A computation follows only a finite time $T$ with finite accuracy, so it returns a finite-time estimate

usually obtained by rescaling the perturbation at regular intervals, adding the logarithms of the growth factors, and discarding an initial transient.

Definition Chaos has several precise definitions in use, so a claim of chaos should name the one it relies on. A common working definition calls a deterministic system chaotic when three things hold. It has bounded solutions that never settle onto an equilibrium, a periodic orbit or a quasiperiodic motion. These solutions start from a set of initial states that is not negligible, such as an open set. Finally, nearby solutions separate in the sense of sensitive dependence on initial conditions, which in practice is detected by a positive largest Lyapunov exponent. A stricter definition concerns the motion on a closed bounded invariant set $\Lambda$ and asks for sensitive dependence on $\Lambda$, a solution in $\Lambda$ that passes arbitrarily close to every point of $\Lambda$, and periodic orbits passing arbitrarily close to every point of $\Lambda$. Smooth planar autonomous systems leave no room for chaos. By the Poincaré-Bendixson theorem, if $\dot X = F(X)$ in $\mathbb{R}^2$ has $F$ continuously differentiable and only finitely many equilibria, the omega-limit set of a solution that stays bounded for $t \ge 0$ is an equilibrium, a periodic orbit, or equilibria joined by solutions that tend to equilibria as $t \to \pm\infty$, and none of these carries chaotic motion. Chaos in a smooth autonomous system therefore needs at least three state variables. Three variables make chaos possible but never guarantee it.

Definition With $t$ as time and a dot for $d/dt$, the forced Duffing equation for a displacement $q(t)$ is

with damping $\delta \ge 0$, linear stiffness $\alpha$, cubic stiffness $\beta$, forcing amplitude $\gamma$ and forcing angular frequency $\omega > 0$. With the velocity $v = \dot q$ and $\gamma \ne 0$, the pair $(q, v)$ obeys a planar system whose right side depends on $t$, so the planar theorems for autonomous systems do not apply to it. Taking the forcing phase $\theta$ as a third state, read modulo $2\pi$, gives the autonomous system

A solution $q(t)$ of the forced equation corresponds to the solution $(q(t), \dot q(t), \omega t)$ of this system, the one with $\theta(0) = 0$. The divergence of the new field is $-\delta$, so for $\delta > 0$ the three-dimensional flow shrinks volume like $e^{-\delta t}$.