2010 Multi-folded toroidal chaos

As pointed out by Otto E. Rössler, to produce chaos, it is sufficient to combine an oscillator with a nonlinear switch. [1] Here the oscillator is a relaxation oscillator containing a storage element represented by a capacitor, which is charged gradually through a resistor from the voltage source, and then discharged
rapidly through a threshold element, such as a neon lamp. This element has to have in fact two distinct thresholds, one for ignition, and another, lower the extinguishing voltage. When the decreasing voltage across the capacitor reaches the extinguishing level, the element switches off, and the capacitor starts charging again, repeating the cycle. Here the nonlinear switch is replaced with a threshold for the excitation of oscillations - the ignition threshold - and some lower parameter level for reducing the oscillations - the extinguishing threshold. [2] In this way, Sergey Kuznetsov, Alexander Kuznetsov and Nataliya Stankevich proposed such a three-dimensional system in the form of

For appropriate parameter values,

this system produces a beautiful multi-folded toroïdal chaos shown in Fig. 1. The surface of section reveals the toroïdal structure and numerous foldings. The route to chaos is clearly a Curry-Yorke scenario according to which the torus is progressively folded, thus switching from quasi-periodic behavior to toroïdal chaos. [3]

When parameter mu is decreased from a value greater than 1 for which the system produces a period-1 limit cycle, there is first a quasi-periodic regime. Many periodic windows are observed as shown in Fig. 2. Progressively, the attractor approaches the rotation axis - the z-axis -, some folding occurs, inducing chaos.

The synchronizability of this 3D toroïdal oscillator was also investigated. [4]