Safieddine Bouali

Assistant Professor in Economics

Management Institute,

University of Tunis,

41, rue de la Liberté,

2000 le Bardo, Tunisia




PhD. in Theoretical Economics, University of Rennes 1, France, February 2001.

Master Degree in Economics, University of Tunis, Tunisia, December 1989. 

 Courses presently taught:

*First Semester, 2015-2016


-Safety and Risk Control

*Second Semester, 2015-2016

-Globalization and International Economics

-Labor Economics



Current Research Interests:

-Chaos Theory:


Historically, Edward Norton Lorenz established  in 1963 the first strange attractor ever discovered. These amazing mathematical objects report chaotic dynamical patterns from deterministic equations. The related oscillations in a finite portion of time aren’t predictable and never resume a same path.

Strange attractors are they three-dimensional objects? Rather mathematical behaviors ! These are the displays of deterministic dynamics and cannot be summarized by a given frequency. 

Explicitly oriented to the simplest exhibition of these dynamics, with very teensy mathematical tools, the purpose of this web-page aims to oversimplify to the young students the Chaos Theory.

In this e-page, we present our own research which led to new classes of 3D & 4D Strange Attractors.

See menu for further technical details and also beautiful pictures created by the Mathematics & Art pathfinder : Jos Leys.



I.  3D Strange Attractors 

The chaotic dynamics are non-standard flots, but can be represented in a finite phase spaces.

1/ Strange Attractor Type I :

Monarch Safye.jpg









The paper published in the International Journal of Bifurcation and Chaos (1999), 9, 4, 745-756:








The PDF version available here: Bouali Safi-Chaos Bouali Safi-Chaos  


 Simulation and Animation by the wizard of Mathematical Imagery Jos Leys :



2/ Strange Attractor Type II :









The research paper was published in Nonlinear Dynamics (2012), 70, pp. 2375–2381.

DOI 10.1007/s11071-012-0625-6









The PDF version available here:nody2.pdf nody2.pdf 

Simulation and Animation by the Mathematical Imagery pathfinder Jos Leys :


3/ Strange Attractor Type III :










The paper :

A 3D Strange Attractor with a Distinctive Silhouette. The Butterfly Effect Revisited 


Mathematical Imagery and Simulation Designed by the Maths & Arts creator Jos Leys :


4/ Strange Attractor Type IV : 

Attractor bouali iv

The paper :

Basins of Attraction Plasticity of a Strange Attractor with a Swirling Scroll

Beautiful simulations made by Jos Leys :



II. Hyperchaotic Attractors

The Science of Process mixing order and disorder can be extended to the space of dimension four. Here too there are strange attractors ! 

1/ Hyperchaotic 4D Strange Attractor A:


 Bax(a)Lac 107(b)

Lac 109(c) Lac 1088(d)

       3D projections of the 4D Hyperchaotic Attractor

   (a), (b), (c), and (d) are part views of the Attractor since the overall

                representation of the 4D space is unrealizable

The paper:

A New Hyperchaotic Attractor with Complex Patterns 


An artistic animation by the maths & Arts pathfinder Jos Leys  :


2/ Hyperchaotic 4D Strange Attractor B:


(a)Bouali attractor 44 (b)Bouali attractor 46

(c)Bouali attractor 42(d)Bouali attractor 41

            3D projections of the phase portrait of the Attractor

                      (a), (b), (c), and (d) are the different representations

                                            of the Hyperchaotic Attractor


 The paper :

A Novel 4D Hyperchaotic Attractor with Typical Wings

Awesome simulation and Animation by Jos Leys :


The article "Hidden Structure and Complex Dynamics of Hyperchaotic Attractors",

analyzing the two hyperchaotic systems is published in:

Annual Review of Chaos Theory, Bifurcations and Dynamical Systems, Vol. 6, pp. 48-58.




Cover images, figures and graphics of the paper are kindly provided by Jos Leys.





-Complex Systems:

In a wide range of nonlinear phenomena, dynamical behavior can be suitably formulated with differential equations.

We explored also theoretical fields far from our Economics Education.


Tropical Cyclone Dynamics:



Idealized Tropical Cyclone; Aerological circulation follows a quasi-torus structure

(front slice removed to display its internal structure).

Coauthored paper by: 

Safieddine Bouali and Jos Leys (2013): "Tropical Cyclone Genesis: A Dynamician’s Point of View", pp. 187-192, in:

Proceedings of the 4th International Interdisciplinary Chaos Symposium

Stavrinides, S.G., Banerjee, S., Caglar, S.H., Ozer, M. (Eds.)

2013, XV, 581 p. 236 illus.











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