Rossler system

So far we have used two-dimensional maps to help us understand how stretching and folding can generate strange attractors. Now we return to differential equations. 

In theculinary spirit ofthe pastry map and the baker s map, Otto Rossler (1976) found inspiration in a taffy-pulling machine. By pondering its action, he was led to a system of three differential equations with a simpler strange attractor than Lorenz s. The Rossler system has only one quadratic nonlinearity xz  

We first met this system in Section 10.6, where we saw that it undergoes a period- 

Numerical integration shows that y this system has a strange attractor for 

Let s consider the schematic picture in more detail, following the visual approach of Abraham and Shaw (1983). Our goal is to construct a geometric model of the Rdssler at- 

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Analysis) Find the fixed points of the Rossler system, and state when they exist. Try to classify them. Plot a partial bifurcation diagram of x vs. c, for fixed a, b. Can you find a trapping region for the system ... 

The Rossler system has only one nonlinear term, yet it is much harder to analyze than the Lorenz system, which has two. What makes the Rossler system less tractable ... 

Numerically integrate the Rossler system for a = 0.4, 6=2, c = 4, and obtain a long time series for x(f). Then use the attractor-reconstruction method for various values of the delay and plot (x(f), x(f + t)). Find a value of t for which the reconstructed attractor looks similar to the actual Rossler attractor. How does that T compare to typical orbital periods of the system ... 

The Rossler system lacks the sym metry of the Lorenz system. 

S. Yanchuk, Yu. Maistrenko, and E. Mosekilde. Loss of synchronization in coupled Rossler systems. Physica D, 154 26-42, 2001. 

Fig. 15.10. Standard deviation of frequencies

The logistic map, referred to previously, is a now famous, but simple, quadratic function that displays the same period-doubling cascade into chaos exhibited by the Rossler system (and by quite a few experimental examples, as well). A map is an equation that gives an iterative rule for generating a sequence of points x, Xj, X3,... given an initial value Xq. The logistic map is given by... 

The Rossler system and the new Lorenz system (C.2,19) are remarkable in that both have a doubly degenerate equilibrium state with characteristic exponents equal to (0, iij). The feature of this bifurcation is that the unfolding may contain a torus bifurcation curve along with curves corresponding to homoclinic loops to saddle-foci, and therefore non-trivial dynamics may emerge instantly in a neighborhood of the bifurcating equilibria. ... 

C.7. 82. Apply the Shilnikov theorem and explain what kind of behavior one should anticipate in the Rossler system [172, 188]... 

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