Differential Equations with Multispiral Attractors

M. A. Aziz Alaoui
Labo. of Maths., Le Havre University

Int. Journ. of Bifurcation and Chaos, 9(6), 1009-1039, (1999)


This paper gives an improvement of both papers cited below (Ref-1, Ref-2), first by improving the idea and the method which allows me to find multispiral attractors, (indeed in this paper, only 2 new parameters are required to increase the number of cells (spirals) by two, unlike both papers cited below in which 4 new parametrs are necessary!). And secondly, by describing a collection of other systems which multispiral attractors are found.


A system of nonautonomous differential equations having Chua's piecewise-linearity is studied. A brief discussion about equilibrium points and their stability is given. It is also extended to obtain a system showing `multi-spiral' strange attractors, and some of the fundamental routes to `multi-spiral chaos' and bifurcation phenomena are demonstrated with various examples. The same work is done for other systems of autonomous or nonautonomous differential equations. This is achieved by modifying Chua's piecewise-linearity in order to have additional segments. The evolution of the dynamics and a mechanism for the development of multi-spiral strange attractors are discussed.

Interest and applications: for example in,
1) CNNs (Cellular Neural Networks): using weak linear coupling between chaotic cells, hyperchaos was obtained in a CNN array, this was demonstrated with the n-multispiral hypercube CNN, using n-multispiral as cells. (see the refs therin this paper).
2) secure communications: as the chaos is enhanced in such multispiral attractors, the latter could be used for communicating with chaos for increasing security when transmitting a useful inforamtion.

Ref-1: M.A. Aziz Alaoui, Rev. Mar. Sc. Phys. 1, 109-125, (1998)

Ref-2: M.A. Aziz Alaoui, Annales de l'ENIT -medélisation et calcul-, Vol.12(2) (1998)

Fig. 1. An example of amultispiral attractor with 21 spirals for modified Chua's circuit, generated by Eq.(2). See also a multispiral gallery and a multifold gallery.


A GIF animated view of this attractor is (asap) available.

Back to the previous page

Back to Aziz Alaoui's home page