Contents Next: Example II: The Up: Lyapunov Exponents Previous: Lyapunov Exponents

Consider the function f defined by , where a>0. Given any initial point , it is clear that the dynamical system will have orbit

Introduce a slight perturbation to the initial condition, say, . The resulting orbit is:

So, after k iterations, these two orbits that started at neighbouring points and have separated by . Clearly, if 0<a<1, all orbits converge to zero. For a=1, all points are fixed points of the map (f is the identity map). For a>1, orbits diverge to infinity and, choosing k sufficiently large, neighbouring initial points have orbits that become arbitrarily far apart.

Since for any x, it follows that the Lyapunov exponent of the linear map is the same for any orbit, namely . The Lyapunov exponent is negative if 0<a<1, in which case all orbits converge to zero. When a>1, the Lyapunov exponent is positive and the distance between neighbouring orbits becomes arbitrarily large. Hence, is related to the sensitivity of orbits to perturbations. For , neighbouring orbits remain close. However, when a>1, even though , the system is not chaotic. This is due to the fact that the system is unbounded for a>1. (Boundedness is another requirement for chaos; this is better illustrated by another example.)

Contents Next: Example II: The Up: Lyapunov Exponents Previous: Lyapunov Exponents