The Lyapunov Exponent Test and the 0-1 Test for Chaos compared

In this paper we will discuss two methods to measure chaos for dynamical systems; the Lyapunov Exponent test and the 0-1 test. The Lyapunov Exponent test requires phase space reconstruction and has been used for a longer time, whereas the 0-1 test is quite new and works directly with the time series. To make a comparison, we will use the logistic map, fa(x) = ax(1-x), to show advantages and disadvantages of the two methods. In chapter 1, we will introduce the notion of chaos and see why the logistic map is a very good example when discussing chaos. After this chapter, we will introduce the two methods to distinguish between regular, i.e. periodic, dynamics and chaotic dynamics. Next to that, we will see the implementation of the tests with regard to the logistic map. A comparison between the two tests can be read in the last chapter. Here we see that, although the 0-1 test seemed a better test at the start, the Lyapunov Exponent test is much easier to understand and to implement, provided that the map f is known explicitly. It is also able to determine the bifurcation points and the super attractive points, if present, whereas the 0-1 test is not able to find those. However if a phase space reconstruction is not possible, the 0-1 test can still be used and is therefore a more general test to measure chaos.