Fun with Non-Linear Dynamics
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Contents |
Introduction
What's the difference between
- f(x) = 2x
and
- f(x) = x^{2}?
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])
gives rise to the Feigenbaum diagram and the complex quadratic polynomial
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.,
for fixed points
- .
The method is a nice interface between
- analysis
and
- linear algebra.
The Setup
Non-linear dynamics given by
for the non-linear function with control parameter .
It holds that
using a Taylor expansion and the fact that for fixed points f_{i} = 0 and
- .
Combining Eqs. (1) and (2) yields
with the Jacobean matrix . Rephrasing this gives
which is an eigenvalue equation.
Solutions
For a 2-dimensional system the solution to Eq. (4) is given by
where
and
- .
The solution to Eq. (3) is hence
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
- saddles
- 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 . 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
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
- .
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].