Synthesis of my Research activity
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].

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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).

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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.

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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: %%%
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%----------%%%%%%%%%%%%%%%% 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.

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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.

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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.

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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

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{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)