Chaotic motion

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In order to better understand the link between the concept of chaos and the waterwheel, it's perhaps useful to consider what 'chaotic motion' might mean both in the behaviour that can be observed and in mathematical terms. We've therefore given a short introduction to the difference between linear and nonlinear dynamics and then, by extension from nonlinear dynamics, chaotic motion.

Linear and nonlinear dynamics

 In the simplest terms, dynamical systems can be grouped into 2 sets:

  • Linear
  • Nonlinear

Linear systems can be described by a linear equation, i.e. all of the dependent variable terms are first order. An example of such a system would be a damped harmonic oscillator.

















Figure 1: Damped harmonic oscillator profile - courtesy Wikimedia Commons 

This system can be described by equation (1) below:

2nd order ODE for DHO

However, if we can also say:

linearisation

We can express it as shown in figure (2):

linearised

We can then see that the term in x1 on the right hand side are to the first power meaning that the equation is linear. This equation could then be solved analytically by integration to obtain an equation in which the position of the mass, x1, is described in terms of time, t.

Conversely, nonlinear systems - as the name suggests - are not linear by virtue of the fact that the equations that describe them contain dependent variable terms that are not first order, e.g. products, powers or functions of x1. An example of this type of system, in a similar area to the oscillator equation is that of a swinging pendulum.

pendulum






















Figure 2: A swinging pendulum - courtesy Wikimedia Commons

If we redefine x = θ, for ease of comparison with the linear system, the pendulum can be described by equation (3):

eq3

However, if we again say:

We can express it as shown in equation (4):

eq4

Now we can see the nonlinear nature as the x1 term is not first order.

This nonlinearity makes the equation difficult to solve analytically. The solution usually involves either making a simplifying assumption (in this case invoking the small-angle approximation which says that sin x = x for small angles) or by solving by iteration.

Chaotic dynamics

If we show the Lorenz equations, we can immediately see that they are of a nonlinear nature as described above (the quadratic terms xy and xz are the nonlinearities):

Lorenz Equations

Their chaotic nature stems from the fact that for certain critical values of the parameters (ρ, r and b) the equations exhibit great sensitivity to the intial conditions where slight errors that might be imperceptively small to begin with can grow exponentially with time and result in a system that is drastically different to that which might be predicted using similar values as starting conditions.