Introduction

What's the difference between

f(x) = 2x

and

f(x) = x2?

The little change in the latter equation gives rise to a whole new branch of physics: non-linear dynamics

Some topics include:

• fixed points, linear stability analysis and bifurcations
• logistic equations
• limit cycles
• chaos: strange attractors, fractals, sensitivity to initial conditions
• Liapunov exponents

with applications like

• population dynamics in ecosystems
• temporal-spatial pattern formations
• activator-inhibitor systems
• Belousov-Zhabotinsky reaction

Perhaps the most surprising feature of non-linear dynamics, is that perceptively simple deterministic equations yield an amazingly complex and self-similar patterns of solutions which cannot be forecast. E.g., iterative equations like the logistic map (see applets [1], [2])

$\qquad x_{n+1} = r x_n (1-x_n)$

gives rise to the Feigenbaum diagram and the complex quadratic polynomial

$\qquad z_{n+1} = z_n^2 +c$

yields the Mandelbrot (watch on youtube.com) and the Buddhabrot sets (watch on youtube.com).

Linear Stability Analysis

To deal with the complex evolution of a non-linear system, linear stability analysis looks at the behavior close to fixed points. I.e.,

$\dot x = \frac{dx}{dt}= 0$

for fixed points

$\overline x$.

The method is a nice interface between

• analysis

and

• linear algebra.

The Setup

Non-linear dynamics given by

$\dot \vec x = \vec f(\vec x, \vec u, t)$

for the non-linear function $\vec f$ with control parameter $\vec u$.

It holds that

$(1) \qquad \dot x_i = f_i(\overline x) + \frac{\partial f_i (\overline x)}{\partial x_i} (x_i - \overline x_i) + \dots \approx \frac{\partial f_i (\overline x)}{\partial x_i} (x_i - \overline x_i)$

using a Taylor expansion and the fact that for fixed points fi = 0 and

$(2) \qquad \delta x_i := x_i - \overline x_i \quad \Rightarrow \quad \dot x_i = \delta \dot x_i$.

Combining Eqs. (1) and (2) yields

$(3) \qquad \frac{\partial \vec x}{\partial t} = \vec f (\vec x, \vec u, t) = \delta \dot \vec x \approx \left. \mathcal{J}\right|_{\overline x} \delta \vec x$

with the Jacobean matrix $\mathcal{J}$. Rephrasing this gives

$(4) \qquad\dot \vec y = \mathcal{J} \vec y = \lambda \vec y$

which is an eigenvalue equation.

Solutions

For a 2-dimensional system the solution to Eq. (4) is given by

$\lambda_{1,2} = \frac{1}{2} \left( \tau \pm \sqrt{\tau^2 - 4 \Delta} \right)$

where

$\tau := \mathrm{det} ( \left. \mathcal{J}\right|_{\overline y} )$

and

$\Delta := \mathrm{det} ( \left. \mathcal{J}\right|_{\overline y} )$.

The solution to Eq. (3) is hence

$\delta \dot \vec x = \delta \vec x (0) \; \mathrm{exp} (\mathcal{J} t) = \delta \vec x (0)\; e^{\lambda t}$

determining the dynamics.

It should be noted that the properties of τ and Δ suffice to classify the fixed points. They determine the stability of the fixed points, i.e., if they are

• attractors and repellors
• attractive and repulsive focuses
• centers

of the systems trajectories.

Outlook

Fixed points can be created or destroyed, or their stability can change by tuning the control parameter $\vec u$. The parameter values for which this occurs are called bifurcation points.

It is the focus of catastrophe theory, how small perturbations of the potential describing the dynamics results in dramatic changes in the evolution of the system.

In essence, instabilities (via control parameters or feedback) give rise to new dynamics and are the source of evolution.

Diffusion-Driven Instabilities: Emergence of Spatial-Temporal Pattern Formation

The idea going back to Alan Turing (1952 "The Chemical Basis for Morphogenesis") is to combine stable non-linear dynamics with a diffusion term to generate new dynamics. This gives the system's dynamics a spatial component and explains how structure emerges from a homogeneous spatial distribution. Also known as Turing patterns.

Starting point is a stable system described by Eq. (3) and adding a diffusion term

$(5) \qquad \frac{\partial \delta \vec x}{\partial t} = \left. \mathcal{J}\right|_{\overline x} \delta \vec x + \vec D \triangle \delta \vec x$

This can be solved using Fourier transformations or a separation of variables ansatz. It is then seen that under certain conditions, the diffusion terms break the stability. In two dimensions a necessary condition for the diffusion constants is

$D_1 \neq D_2$.

This is a diffusion-driven instability.

This formalism has been used in the so called Schnakenberg model describing a (two-species tri-molecular) chemical reaction giving rise to pattern formation in a species' morphogenesis. The requirement of a finite spacial extension (location of the embryo) results in discrete solutions of stable excitations which, as the embryo grows, give rise to more and more stripes in the example of a Zebra fish.

Similar mechanisms have been used to explain mammalian coat patterns, see J. D. Murray Mathematical Biology.

Biological Pattern Formation via Activator-Inhibitor Systems

Cool applet (in German): [3].

Further information: [4].