by M. A. Aziz-Alaoui

updated july 2001

Key words and phrases~:

Ordinary Diff- Equations, Mappings,
Dynamical systems,
Chaos,
Strange attractors, multispirals,
Synchronization,
Chaotification,
Bifurcation,
Dynamic of populations,
Numerical tools borrowhed from nonlinear dynamical systems theory.

Research Area}

My research is in the area of Mathematics (nonlinear dynamics, ODE, mappings)
including :

Chaotic dynamical systems (ODE),

Chaotic mappings,

Strange Attractors,

Bifurcations leading to chaotic behaviour,

Phase transitions and critical phenomena,

Synchronization of chaos (and anti-synchronization),

Analysis and control of nonlinear dynamics and chaos,

Populations dynamics.
Chronologically, they can be separated into two periods ~:

Before 1993:

Work on my first thesis were essentially centered
around the theory of confinors, whose goal is
the study of chaos and the transition toward chaos
in the nonlinear differential systems in low dimension.
We privileged in this work techniques that take
into account the geometry
of the signal (that is, the shape of certain projections of solution
in one under space of the phase space) when this one presents patterns.
The objective was to give a mathematical form to the gait of scientists,
chemists or electricians and to describe signals gotten in the experimental
systems in the shape of pattern bifurcations.

After 1993:

My research activities (completely different from the
previous and centered on relatively new themes) are varied
more and articulate at the present time on the following points (theoretical
and numerical) for which the essential goal
is the understanding of certain nonlinear phenomena.
But I am especially interested myself in the
applications of chaos and strange attractors.
This work can be presented in five parts~:

Synchronization of chaos in continuous dynamical systems.

Reconstruction of orbits and recuperation of parameters
of discrete dynamical systems.

Chaotification, anti-control of chaos, research of new strange
attractors and multispirales ones.

Nonlinear study of some ecosystem models
(populations dynamics), stability, chaos and topological
characterization of the strange attractors that appear there.

Study of certain discrete chaotic maps.

The either achieved or in-progress work pursues a triple goal~:

BEFORE, DURING and AFTER CHAOS.
Indeed they can be inscribed in a general thematic in order
to better understand phenomena occurring~:

BEFORE chaos, that is to say the
transition toward chaos [P1-P3, P5-P6,P8, S1, S4, S5]

DURING ~ chaos, that is to say the
understanding of structures of some chaotic
attractors [P1, P2, P8, S3-S5].

AFTER chaos, that is to say the possible
applications and the
utilization of chaos (synchronization, anti-synchronization,
control, applications in communication security systems
for example).

One can add to this last point, the new
notions of chaotification, [P3, P5, P6, P8, S4-S5].

--------------------------

Partie 1

Chaotic synchronisation for continuous dynamical systems

Abstract of part 1

The use of chaotic signals to transmit information has been
a very active research topic in the last decade.
In all proposed schemes for secure communications using the idea of
synchronization, there is an inevitable noise degrading the fidelity
of the original message.
In this part, the errors on the recovered signal in the synchronized
Bonh\"oeffer-Van Der Pol and Chua's system are analyzed.
We give a noise-reduction algorithm and we study intensively the
the possibility to reduce this inevitable
noise. This is possible by connecting two identical receivers in cascade.
While other noise reduction methods could conceivably be introduced,
the technique reported there, has the advantage that it is easy to
implement in practise.
Furthermore cascade chaotic synchronization technique offers a good
improvement over the single-stage chaotic result reported in Kocarev
et al. (1992).

--------------------------

--------------------------

Partie 2~

Recovering trajectories and parameters of chaotic piecewise-linear
dynamical systems

Abstract of part 2~

Over the last few years,
there has been a growing interest in developing communications using
chaotic signals.
Such a problem generates many mathematical problems which are, in particular,
connected with the problems of recovering trajectories of chaotic dynamical
systems from their epsilon-perturbed trajectories, (this is connected
to the theory of shadowing which is important from the numerical point of view).
Here, we have used a new notion of synchronization
which is different from the classical one
used in the part 1 of this document, (indeed, we have no
subsytems to synchronize, but we construct a sequence of intervals (or polygons)
containing a sequence of points which tends towards the unknown true
trajectory).
This kind of mathematical problem appears to be equivalent
to the extraction of an information bearing signal from an input
signal in order to reduce
the noise in an input signal, to encode/decode electronic messages
for secure communications or to increase and stabilize the power of
lasers.
%There are different approaches to investigate this type of task.

The purpose of this part is to give results on estimating
trajectories and parameters of families of piecewise linear chaotic
dynamical systems from their \varepsilon- perturbed trajectories.

--------------------------

Partie 3~

New strange attractors, multispiral chaos and chaotification

Abstract of part 3~

In addition to the rapid development of chaotic synchronization and chaos control,
anticontrol of chaos, by means of making a nonchaotic dynamical system chaotic
or enhancing the existing chaos (chaotification) of chaotic systems,
has attracted increasing attention due to its great potential in nontraditional
applications such as those found within
the context of electronic or biological systems for example.
In this part, various systems of differential equations having Chua's
piecewise-linearity are extended to obtain a system showing `multispiral'
strange attractors, then with enhanced chaotic behavior.
These attractors appear as a result
of the combination of several `one-spiral' attractors similar to Rossler's attractor
or to Chua's double scroll.
The evolution of the dynamics, bifurcation phenomena and a mechanism for the
development of multispiral strange attractors are discussed.
The same work is done for other systems of autonomous or nonautonomous
differential equations.
The notion of multi-folds in some mappings, such as Hénon-
or Lozi-type map, is also given.
Our computer results showed that, for systems we have studied,
the following conjecture is sensible:
%%%

%

%----------%%%%%%%%%%%%%%%%
Conjecture :
For any n in IN^* , n >=2 , there exists a nonempty set
of parameters B_n for which each one of these systems presents strange
attractors with n spirals,
with the relationship, B_n subset of B_{n+2}.
Furthermore,
we report the finding of a new chaotic attractor in a new
piecewise-linear continuous-time three-dimensional autonomous system.
The dynamical behavior of is system is studied.
System equilibria and their stabilities
are discussed. Routes to chaos and bifurcations of the system are
demonstrated
with various numerical examples, where the chaotic features are
justified
numerically via computing the system fractal dimensions, Lyapunov
exponents, and power spectrum.

--------------------------

--------------------------

Partie 4~

Stability, chaos and analysis of the dynamics of
a three species food chain model

Abstract of part 4~

A fairly realistic three-species food chain ecological model, using
Leslie-Gower scheme, in
which the generalist predator is assumed to reproduce mostly sexually,
is considered.
%We give sufficient conditions for .
We estabilish and prove theorems on the boundedness
of the trajectories of the system, existence and
stability of equilibria.
We have also explored numerically the chaotic behavior of the system by
plotting various phase portraits, timewave forms or power spectra.
Even if the selection of biologically realistic parameter values for the
numerical simulation of ecological models is difficult and our parameter
range is narrow, a very rich and complex dynamic has appeared, presenting
various sequences of period doubling leading to chaos or sequences of
period-halving leading to limit cycles.
A type-I intermittency has also been observed.
It suggests that, without any external factor such as epidemic or weather
conditions, populations of preys and predators may evolve regularly and then,
abruptly start to evolve in a chaotic manner. After a given time duration,
the population evolves regularly again.
When the growth-rate of the generalist predator is large enough, a
bistability can be observed, i.e. two different attractors with their
respective attraction basins may co-exist. In such a case, an epidemic or
significant climate change may provoke a transition from one dynamical behavior
to another one. For instance, the time evolution of the populations can be
regular and become irregular after an epidemic, or vice versa .
Another particular behavior is when the trajectory of the time evolution of the
system describes a homoclinic orbit. In such a case, the populations are
almost constant during a finite time interval and, suddenly, a large
oscillation is observed. Since the time duration of such a ``nearly static''
phase is highly dependent on initial conditions, the large oscillations
seem to appear randomly.
On the oher hand,
the quality of a phase portrait reconstructed from a single
scalar time series is investigated. Such a situation simulates a real
study where only a single species is counted. It is found that the analysis can
be safely performed when a single specy involved in the food chain is counted.
Using the concept of knot-holder a template synthetizing the topological
properties of the attractor is built.
A brief comparison with the topology of the
attractor found for some enough close system, and in which the
top predator Z is assumed to be a vertebrate, is given.

--------------

--------------------------

Partie 5~

Dynamics of a chaotic mapping of the type Hénon-Lozi

Abstract of part 5~

The principal motivation for this work is to develop a map which is better
amenable for an analytical treatment as compared to the Hénon map and is one that still
possesses the characteristics of a Hénon-type dynamics.
In order to prove the existence of an attractor for a given system, we
often analyze oversimplified alternatives of that system which allow us to
carry out
proofs, but by doing so, it leads us to lose the essentials of the
dynamics of the original system. We wanted to go a step further by introducing a map
L(a,b,epsilon), which is
a smooth version of the piecewise linear Lozi map.
We derived some of its properties and compared them
with the Lozi map.
We obtained very good agreement with Lozi's when we take the limit epsilon ---> 0.
This paper is a first step.
It does the comparison of the Lozi map (which is a piecewise linear version
of the Hénon map) with the map that we introduce.
This comparison is done for fixed parameters and also through global bifurcation
by changing a parameter. If epsilon measures the degree of
smoothness, we prove that, as epsilon ---> 0, the stability and the
existence of the fixed points are the same for both maps.
We also numerically compare the chaotic dynamics, both in the form of an attractor and
of a chaotic saddle.
The next step would be to do a similar comparison with the
Hénon map.
It would be interesting to check whether, for sufficiently small
epsilon, some weak form of quasi-hyperbolicity could be established. Finally, an
important part of this program would be
to study (analytically and numerically) the notorious problems
of stability islands and breakdown of hyperbolicity.
These problems form the main obstacles to an analytical
treatment of continuous maps.

--------------------------

--------------------------

Some Applications

Communicating with Chaos (synchronization and control of chaos)

Multiple access communication (chaotification)

Secure optical communications (Recovering parameters of chaotic
systems)

Dynamics of populations

Other applications
- Combustion
- Instability of burst
- Study and compression of mixing processus

--------------------------

--------------------------

{Work in progress}

Currently, my research is continuing along the same lines.\\ In particular, the mathematical theory which is hidden behind the synchronization of chaos, requires further studies. Moreover, the heuristic laws found numerically [P11, S6] on the errors on the recovered signal when we study secure communications via the chaotic synchronization, must be refined and studied theoretically.\\ In the domain of the reconstruction of orbits and recovering parameters of discrete dynamical systems, (synchronization and anti-synchronization), a number of questions remain unanswered. In particular, in the first stage, results ned to be extended to dimension-two systems, and sufficiently powerful algorithms need to b writen.\\ It would also be very interesting, concerning the problem that I am studing in population dynamics, to better understand the dynamics of systems and to give a better determination of homoclinical orbits than the numercial one presented in [P2]. A precise study of bifurcations leading to chaos is on-going with my PhD student. It would then appear natural to study perturbed versions of these models, versions taking into account the fluctuations and the existing variations in natural habitats as well as effects of seasonality. Several constant parameters of the system are then replaced by others which vary periodically with respect to time.\\ It is very exciting to think that a theoretical proof of the existence of chaos in the application of Hénon and for a nonzero measure set of parameters could one day be found. Progress might be made using ideas as those brought up in [P1] at least, in a first time, while establishing theoretical results of the type weak quasi-hyperbolicity, as given by L.S. Young or of breakdown of hyperbolicity.\\ Finally, the notions of chaotification (enhancing chaos in chaotic systems or to make chaotic systems that are not chaotic) are again relatively new and multispiral attractors open new fields of research and applications.

*
Multi-Attractors
Demonstrations*
(Gallery of Multi-spiral Strange attractors)

Back to Aziz Alaoui's home page