The Lorenz map

Now let us compare the results for the Rossler system with those of the one-dimensional map. To do that we will plot the Lorenz map for the Rossler system. The Lorenz map (not to be confused with the 'Lorenz system'!) is a way of obtaining a one-dimensional map from a time-series of a dynamical system. For the Rossler system in the chaotic state, say $c=5$, the time series for the $x$ variable is shown in Fig(3.12b). Label the successive local maxima of that graph as $x_n$, that is the $n$-th maxima is at $x_n$. The Lorenz map is then a plot of $x_{n+1}$ versus $x_n$ for various $n$. The Lorenz map for the Rossler system is shown below, Fig.(3.14).

Fig.(3.14)

The points lie on an almost one-dimensional unimodal curve. Thus there should be an approximate relationship of the form $x_{n+1}=f(x_n)$ with $f$ a unimodal map. From Feigenbaum's work it then follows that the Rossler system will also show a period-doubling route to chaos and the same Feigenbaum constant, as is indeed the case ! However not all systems have a one-dimensional Lorenz map, since that requires the system's strange attractor to be almost flat. In physical terms this requirse that the system have only one or two degrees of freedom that are dominant. In fact the experiments quoted above had that feature. In other words, the universality theory is developed only for low-dimensional chaotic systems.