René Lozi

EMERGENCE OF RANDOMNESS FROM CHAOS

In systems theory and science, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems [Aziz-Alaoui & Bertelle, 2009]. In this paper, we use the emergent property of the ultra weak multidimensional coupling of p 1-dimensional dynamical chaotic systems which leads from chaos to randomness. Generation of random or pseudorandom numbers, nowadays, is a key feature of industrial mathematics. Pseudorandom or chaotic numbers are used in many areas of contemporary technology such as modern communication systems and engineering applications. More and more European or US patents using discrete mappings for this purpose are obtained by researchers of discrete dynamical systems [Petersen & Sorensen, 2007; Ruggiero et al., 2006]. Efficient Chaotic Pseudo Random Number Generators (CPRNG) have been recently introduced. They use the ultra weak multidimensional coupling of p 1-dimensional dynamical systems which preserve the chaotic properties of the continuous models in numerical experiments. Together with chaotic sampling and mixing processes, ultra weak coupling leads to families of (CPRNG) which are noteworthy [Henaff et al., 2009a, 2009b, 2009c, 2010]. In this paper we improve again these families using a double threshold chaotic sampling instead of a single one. We analyze numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when very long series are computed. Moreover, a determining property of such improved CPRNG is the high number of parameters used and the high sensitivity to the parameters value which allows choosing it as cipher-keys. It is why we call these families multiparameter chaotic pseudo-random number generators (M-p CPRNG).

SECURE COMMUNICATIONS VIA CHAOTIC SYNCHRONIZATION II: NOISE REDUCTION BY CASCADING TWO IDENTICAL RECEIVERS

The signal recovered from the first reported experimental secure communication system via chaotic synchronization contains some inevitable noise which degrades the fidelity of the original message. In this letter we show by computer experiments that this noise can be significantly reduced by cascading the output of the receiver in the original system into an identical copy of this receiver.

Memfractance: A Mathematical Paradigm for Circuit Elements with Memory

Memristor, the missing fourth passive circuit element predicted forty years ago by Chua was recognized as a nanoscale device in 2008 by researchers of a H. P. Laboratory. Recently the notion of memristive systems was extended to capacitive and inductive elements, namely, memcapacitor and meminductor whose properties depend on the state and history of the system. In this paper, we use fractional calculus to generalize and provide a mathematical paradigm for describing the behavior of such elements with memory. In this framework, we extend Ohms law to the generalized Ohms law and prove it.

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincare maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

CONFINORS AND BOUNDED-TIME PATTERNS IN CHUA'S CIRCUIT AND THE DOUBLE SCROLL FAMILY

We study the patterns in the time waveforms of the spiral attractors of the double scroll equations for the theoretical Chuas parameters. We introduce several mathematical tools linked to these equations (half-Poincare maps, isochronic lines, selecting curves and antiselecting curves, spirals and antispirals, theorems of the turnstile, patterns and confinors). We prove that for given values of the parameters, only patterns in the time waveforms which are bounded in time can exist.

Fast chaotic optimization algorithm based on locally averaged strategy and multifold chaotic attractor

Recently, chaos theory has been used in the development of novel techniques for global optimization, and particularly, in the specification of chaos optimization algorithms (COAs) based on the use of numerical sequences generated by means of chaotic map. In this paper, we present an improved chaotic optimization algorithm using a new two-dimensional discrete multifold mapping for optimizing nonlinear functions (ICOMM). The proposed method is a powerful optimization technique, which is demonstrated when three nonlinear functions of reference are minimized using the proposed technique.

New nonlinear CPRNG based on tent and logistic maps

This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two coupled maps are studied: TTL^{RC} non-alternate, and TTL^{SC} alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim a new nonlinear CPRNG: MTTL_{2}^{SC} is proposed. In addition, we explore higher dimension and the proposed ring coupling with injection mechanism enables us to achieve the strongest security requirements.

SLOW MANIFOLDS OF SOME CHAOTIC SYSTEMS WITH APPLICATIONS TO LASER SYSTEMS

In this work we deal with slow–fast autonomous dynamical systems. We initially define them as being modeled by systems of differential equations having a small parameter multiplying one of their velocity components. In order to analyze their solutions, some being chaotic, we have proposed a mathematical analytic method based on an iterative approach [Rossetto et al., 1998]. Under some conditions, this method allows us to give an analytic equation of the slow manifold. This equation is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent systems left fast eigenvector. In this paper, we give another method to compute the slow manifold equation by using the tangent systems slow eigenvectors. This method allows us to give a geometrical characterization of the attractor and a global qualitative description of its dynamics. The method used to compute the equation of the slow manifold has been extended to systems having a real and negative eigenvalue in a large domain of the phase space, as it is the case with the Lorenz system. Indeed, we give the Lorenz slow manifold equation and this allows us to make a qualitative study comparing this model and Chuas model. Finally, we apply our results to derive the slow manifold equations of a nonlinear optical slow–fast system, namely, the optical parametric oscillator model.

Secure communications via chaotic synchronization in Chua's circuit and Bonhoeffer-Van der Pol equation: numerical analysis of the errors of the recovered signal

The signal recovered from the first reported experimental secure communication system via chaotic synchronization contains some inevitable noise which degrades the fidelity of the original message. By cascading the output of the receiver in the original system into an identical copy of this receiver, it had be shown by computer experiments that this noise can be significantly reduced. We discuss the heuristic laws governing the errors of the recovered signal which are observed in a new series of very careful computer experiments. This discussion is done comparing these laws to the results obtained using the linear filtering theory. Some discrepancy appears. In order to understand the origin of the discrepancy we consider another simpler model based on the Bonhoeffer-Van der Pol equation where no chaos occurs. In this case both two heuristic laws are in good agreement with the linear filtering theory.

Engineering of Mathematical Chaotic Circuits

We introduce the paradigm of chaotic mathematical circuitry which shows some similarity to the paradigm of electronic circuitry especially in the frame of chaotic attractors for application purpose (cryptography, generic algorithms in optimization, control, …).