Nonlinear Dynamics: Attractors, Bifurcations, and Chaos

A deterministic system with simple nonlinear rules can behave unpredictably — this discovery in the mid-20th century completely changed our understanding of nature. Chaos is not disorder, but deterministic unpredictability caused by sensitivity to initial conditions.

Phase Space and Attractors

Phase space: the space of all possible states of a system. For a system with variables $x_1,...,x_n$ — this is $\mathbb{R}^n$. The trajectory of the system is a curve in phase space.

Attractor: a set of states towards which all nearby trajectories tend. Types:

Point attractor: the system tends to a stationary state. A mathematical pendulum with friction: tends to the resting position $x=0$.

Limit cycle: stable periodic oscillations. Biological oscillators (heartbeat, circadian rhythms). Lotka-Volterra equations (predator-prey) for certain parameters.

Strange attractor: non-periodic, bounded, motion sensitive to initial conditions. Lorenz attractor, Rössler attractor.

Lorenz Attractor and Deterministic Chaos

Edward Lorenz (1963) studied a simplified model of atmospheric convection:

$ \frac{dx}{dt} = \sigma(y − x), \quad \frac{dy}{dt} = x(\rho − z) − y, \quad \frac{dz}{dt} = xy − \beta z $

Decoding of variables and parameters:

$x$ — intensity of convective flow
$y$ — temperature difference between ascending and descending flows
$z$ — deviation of the vertical temperature profile from linearity
$\sigma$ (Prandtl) $=10$ — ratio of viscosity to thermal conductivity
$\rho$ (Rayleigh) $=28$ — thermal gradient
$\beta=8/3$ — geometric parameter

At $\sigma=10, \rho=28, \beta=8/3$: the system behaves chaotically — the trajectory "flutters" between two "wings" of the butterfly attractor, never repeating itself.

Butterfly effect: initial conditions $x_0=(1,1,1)$ and $x_0'=(1.0001,1,1)$ (difference 0.01%) $\rightarrow$ exponential divergence after ~30 units of time. Lyapunov exponent $\lambda_1 \approx 0.9$: initial error $\varepsilon$ grows as $\varepsilon e^{\lambda t}$.

Bifurcations: Qualitative Changes in Behavior

Bifurcation — a qualitative change in the behavior of a system with the smooth variation of a parameter. Types:

Saddle-node (turning point): with changing parameter, two equilibrium states approach each other and disappear. Example: population declines below extinction threshold with reduced birth rate.

Pitchfork: one stationary state branches into three. When heating a magnet: below Curie point — two stable magnetization states ($\uparrow$ or $\downarrow$). Above — one unstable.

Hopf bifurcation: with changing parameter, the stationary state loses stability and a limit cycle is born. Cardiac arrhythmia as a transition through Hopf bifurcation.

Feigenbaum scenario (period-doubling): logistic map $x_{n+1} = r \cdot x_n(1 − x_n)$. For $r$ from 1 to 4: $r < 3$ — stable state, $r \in [3, 3.45]$ — period 2, $r \in [3.45, 3.54]$ — period 4, $r > 3.57$ — chaos. Feigenbaum constant $\delta = 4.669...$ — universal constant, identical for all systems of this class.

Hausdorff Dimension and Fractals

Attractors of chaotic systems have a non-integer "fractal" dimension. Lorenz attractor: dimension $\approx 2.06$ (not a 2D surface and not a 3D volume).

Fractals — objects with self-similarity at various scales. Norwegian coastline: measured with step 50 km gives one length, with step 1 km — 2.4 times greater length. Dimension $\approx 1.52$.

Applications in finance: stock returns — a "fractal" time series, self-similar at various scales (Mandelbrot). The classic "random walk" (Brownian motion) is a special case with dimension 1.5.

Numerical Example: Logistic Map

$r=3.9$, $x_0=0.2$:

$x_1=3.9 \times 0.2 \times 0.8=0.624$
$x_2=3.9 \times 0.624 \times 0.376=0.916$
$x_3=3.9 \times 0.916 \times 0.084=0.300$

For $x_0'=0.2001$: the divergence from $x_0=0.2$ doubles every $\approx 2.1$ steps. After 30 steps, initial error $0.0001$ has turned into an error $\approx 0.5$ (the entire range [0,1]).

Assignment: Implement the Lorenz system (scipy.integrate.odeint). For $\sigma=10, \rho=28, \beta=8/3$: (1) plot the 3D trajectory of the attractor; (2) calculate the divergence of two trajectories with initial conditions differing by $10^{-6}$ and estimate Lyapunov exponent $\lambda_1$; (3) investigate what happens at $\rho=20$ (subcritical regime) and $\rho=99$ (different chaotic regime).