## Chaotic Motion

The 18^{th} century astronomer and mathematician Pierre Simon de Laplace was a great believer in the predictability of motion:

“The present state of nature is obviously a consequence of what it was in the preceding moment, and if we conceive of an intelligence which at a given instant comprehends all the relations of the entities of the Universe, it could state the respected positions, motions and general effects of all these entities at any time in the past or future.”

Deterministic Chaos concerns systems that are governed by deterministic equations, yet are *not* predictable. A deterministic system is one for which there are well defined rules, or equations, governing changes at all times. If these rules are known, they can be used to determine the state of the system at a later time, from information about the state of the system at an earlier time. If a deterministic system has the additional property of being chaotic, then the way in which it evolves with time will be sensitive to the initial conditions that eventually the expected predictability will, in practice, break down.

A linear map is a linear rule for turning one set of values into another. The process of repeatedly applying a given rule, using the output of one application as the input of another is known as iteration. The systems we are considering may be described as an iterated linear map.

In the 1970s, ecologists studied populations of biological species using a modification of a linear map. The map that they studied takes account of a limit to growth, and is specified by the equation,

*p*_{n+1} = *kp*_{n}[(*p*_{max} – *p*_{n})/*p*_{max}]

Where *p*_{n} is the population in year *n*, and *p*_{max} is some maximum population above which growth is supposed to be impossible.

This equation can be simplified using *x*_{n} = *p*_{n}/*p*_{max} giving,

*x*_{n+1} = *kx*_{n}(1 – *x*_{n})

This equation is called the logistic map.

In a limit cycle, the sequence of values produced by an iterated map settles down to an oscillation that repeats itself after a certain number of steps.

A linear system is described by a linear map, or a linear equation of motion, whereas a non-linear system is described by a non-linear map, or a non-linear equation of motion.

The doubling period of the limit cycle when the parameter *k* is increased is known as period doubling.

In the limit where the periods of the limit cycles are large, the intervals between period doubling get exponentially smaller as k increases, each interval being smaller than the previous interval by a factor known as the Feigenbaum constant δ = 4.669201609102990671853…

The number of iterations of a logistic map that occur before a change in the initial value leads to significant differences in the predicted sequence is known as the horizon of probability.

A map in which each value has two possible predecessors is called a quadratic map. Every quadratic map approaches chaos in the same manner. They all show the period doubling slide into chaos, with intervals between period doubling getting exponentially smaller as k increases. The intervals between successive period doublings are reduced by a factor of δ, the Feigenbaum constant, just as they are in the logistics map.

Universality refers to the observation that different systems may exhibit common behavioural features.

A fractal pattern is a pattern for which enlarging a small part produces a pattern that is similar to the original pattern, i.e. fractal patterns are self similar.

A state space is a way of representing the state of a system. If *m* quantities are required to completely specify the state, then the state space is *m*-dimensional, and any state of the system is represented by a point in that *m*-dimensional space. The path that a system traces out in a state space is known as a trajectory.

A strange attractor is a region of state space into which trajectories are attracted and within which they exhibit chaotic behaviour.

The characteristics of deterministic chaos are;

*Deterministic rules*: There are rules, equations or laws that describe how the state of the system at a particular time determines the state of the system at a slightly later time.*Unpredictable outcomes*: a small uncertainty in the initial state of a system will increase exponentially with time, so after a sufficiently long time the state will be unpredictable.