Dynamical Systems

by Chris Clarke

Key Concepts

A continuous dynamical system is any physical thing that can be regarded as having states which can be specified (in a small enough region of the space of all states) by a collection of numbers, and where the change in the system with time is given, in terms of these numbers, by a definite rule. For example, a pendulum is a dynamical system, and its state at any time is specified by the position of the pendulum (a point on a circle, if the pendulum is allowed to go all the way round) and the speed of the pendulum bob at that time, which is a number. If we decide on one fixed direction round the circle in which to measure the speed, then this number can vary over all negative and positive numbers (negative numbers meaning that the bob is moving backwards relative to the chosen direction). It we draw the circle representing position in one plane and the line representing the speed at right angles to this plane, then we can represent the state space of the pendulum by a cylinder.

If the rule depends only on the current state, and not on the time, the system is called autonomous. Note that a dynamical system is, mathematically speaking, deterministic because it is governed by precise rules, though practically it may be impossible to predict its motion. By contrast a quantum system is not deterministic, because of the measurement phase of its evolution (see the paper on Quantum Theory).

As the state of a dynamical system varies in time it traces out a path in state-space called a trajectory. The only exception is when the state is at an equilibrium point, which is a special case when the rule specifying the dynamical system requires that the state does not change at all with time. For the pendulum the equilibrium points are the states with zero speed at the top and bottom of the circle of positions. In these cases the trajectory consists of just a single point. Thinking of dynamics in this pictorial way moves the subject from the arena of numbers and equations into the arena of shapes and forms.

A system in some region of its space of states is called dynamically unstable if a small variation in the initial conditions can lead to a difference which, on some region of  the trajectory, starts growing exponentially. (Strictly this is called Liapoounov instability). In cases like this the system is practically unpredictable , even if theoretically its motion is determined by a precise rule. The Lorenz System (see below) provide a good example of this. The combination of dynamical instability with quantum theory demonstrates that, however one interprets quantum theory, the universe is fundamentally unpredictable on a large scale as well as a small scale. Dynamical instability is sometimes called “the butterfly effect”: a butterfly flapping its wings in Brazil can cause a hurricane in Bengal.

Chaos. The term was first used of a dynamical system whose behaviour showed no predictable pattern. Various attempts have been made to specialize the term to say something more positive – for instance, to describe behaviour where a trajectory wanders round a region of the space of all states, coming arbitrarily close to any specified sate (a situation also called ergodic)

An attractor is a trajectory in the state-space of a dynamical system which has the property than any state sufficiently close to the attractor will move towards it. If the attractor is neither a (stable) equilibrium point nor a closed loop (called a limit cycle) then it is called a strange attractor.

The Lorentz System 

The system here is a very simplified model of convection in a gas, which Lorentz was studying in order to understand weather system. It produced the first example of a strange attractor to be identified and explored in detail. While convection is in reality a complicated motion of the entire gas which would involve millions of parameters to specify the position of each part of the gas at all accurately, in this model the state of the system is described by just three parameters, and the equations governing them are reduced to a very simple form. The picture below (generated by Dynamics solver by Juan M Aguirregabiria) shows the way in which two of these parameters very with time.

Two trajectories are shown, in red and blue, starting close together in the inner part of the loops on the left hand side. The system illustrates the “butterfly effect” in which a small variation in the initial conditions produces a large difference in the eventual behaviour. The trajectories stay close together as they pass through a stable region. After performing three loops of a quasi-cyclical motion (nearly returning to their starting point, but drifting away slightly) they enter an unstable region. Thereafter the two behave quite differently, even though they started close together. The red trajectory veers off to the right hand region, where it completes one loop and then returns to the left and makes a further loop there. The blue trajectory completes one more loop in the left region before moving to the right, where it takes up a quasi-cyclical motion in the right hand region.

Emergence of Structure 

Dynamical systems have the property that structure can spontaneously emerge from them. Though this happens in continuous dynamical systems, it is easiest to study by reducing the dynamical system to a discrete one, in which time progresses by steps instead of changing continuously. We can get a discrete system out of a continuous one by taking “snapshots” at either a regular spacing, or when a trajectory cuts through a chosen surface. In the examples we are about to look at, the space of states is also discrete, being made up of the set of possible patterns on a grid. Again, this is for the sake of reducing the problem to something simple enough to handle. A system called “life” is described next. A simpler system called “Langton’s ant” is described elsewhere.

Conway’s Life (described in more detail on this page)

The “game” (for one player, with no choice at any stage!) is played on an infinite grid of squares, each one of which can be either black or white. The picture below (Pictures of Conway’s “Life” are generated by the programme Life32, by Johan G. Bontes) shows a part of a random pattern, imagined to continue indefinitely in all directions.

The pattern evolves one step at a time according to a deterministic rule:  At each step, if a black square is surrounded by more than three other black squares it “dies” (turns white) through overcrowding, while if it is surrounded by less than two black cells it dies from loneliness. If a blank square is surrounded by exactly three black squares then a new black square is “born” there.

After 21 steps the above random has evolved to

– which is starting to manifest the sort of grouping that we might call a pattern. After 62 steps quite stable structures have formed: static rocks, forms that oscillate regularly between two shapes, and forms that move steadily across the board. The picture below shows a snapshot of some of these.