# Christian MiraFree researcher, after retirement, no affiliation · No official affiliation to a department after retirement (1999)

Christian Mira

PhD

## About

312

Publications

36,997

Reads

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3,407

Citations

Introduction

Additional affiliations

June 1974 - June 1976

**COPPE (Federal University of Rio de Janeiro)**

Position

- Senior Researcher

Description

- The COPPE is an institute for Graduate Studies and Research in Engineering. With Prof. Liu Hsu, I had the task to create a research team on nonlinear dynamics and applications.

Education

September 1954 - June 1957

**Université de Toulouse, Faculté des Sciences**

Field of study

- Mathematics

September 1954 - June 1957

**ENSEEHT**

Field of study

- Electrotechnics, Electronics, Hydraulics, Fluid Dynamics

## Publications

Publications (312)

Nonivertible map meeting in Minneapolis in 1996 (?)

The author of this article has no expertise in cardiology and its different branches, including rhythmology, field initially based on a closed co-operation between cardiologists and signal processing specialists. However, it is on the setting of a similarity of dynamic behaviors, common to all nonlinear oscillating systems, that this paper takes th...

This article deals with some characteristical situations in the phase space and in a parameter space associated with a dynamical system giving rise to complex nonlinear phenomena. Examples are given

This Research Gate article deals with a bifurcation resulting from the contact of two singularities of different nature: a chaotic attractor and an unstable periodic orbit generated by a Dim1 quadratic map. The topic was the purpose of two fundamentally different approaches leading to results independently obtained in 1976 and 1982. In the setting...

This paper is the first part of an article devoted to a complex dynamical behavior called "intermittency". This word was introduced in the famous Pomeau & Manneville's paper: "Intermittent Transition to Turbulence in Dissipative Dynamical Systems" (Commun. Math. Phys. 74, 189-197, 1980), in order to explain intermittent transition to turbulence in...

In its first part, this article is devoted to the Giraud's pioneering results described in a Phd Thesis (1969). A second part deals with recent additional results obtained via programs for scanning the phase and the parameter planes. Some 50 years later, the Giraud's contribution remains an original complete qualitative study of systems (in the the...

From 1975, the sudden and explosive growth of publications in English on the nonlinear dynamics field, the most part written without knowledge of previous results in another languages, has revealed the existence of two schools of researchers, resulting from two ways of approaching the problems. The first is a contemporary "Prevailing School", essen...

This Part IIa is a continuation of a first one "Hybrid systems. (I) A former step toward a theory" (Research Gate article, Nov. 2017). It is devoted to a modeling of rectifiers with voltage feedback, the purpose being to facilitate a rectifier synthesis, by defining parameter space regions giving rise to the smallest amplitude of residual ripple at...

In 1966, I proposed to Alain Giraud, young engineer of the "Ecole des Arts et
Métiers", a research topic related to modeling control systems with switches, at that time called "Systèmes à Commutations", with application to the synthesis of rectifiers with current, and voltage feedback. The "Systèmes à Commutations" associated a continuous part desc...

The article object is only to show how certain models of dynamic systems may generate strange shapes, some reminding possible known forms, or communicating an "artistic" feeling. In this context the properties of such models are essential for understanding the mathematical sense of generated pictures. This gives the opportunity of an incursion into...

The notion of "a priori" and "posteriori" models, terminology inspired by a Tomovic’s paper (1963), was introduced in chapter 1 of the book "Optimization in Control Theory and Practice" by I.
Gumowski & C. Mira (Cambridge University Press, 1968). From large extracts of this chapter, this report provides basic elements of thought that can guide the...

Revised preface of the book "Histoire de la théorie des oscillations non linéaires" by Jean-Marc Ginoux. ". In an original form, combining many illustrations, publications extracts, author biographies, so far unknown documents and correspondence copies, the whole accompanied by an extensive bibliography, the great merit of this contribution is to m...

This unpublished article deals with "Chaos", as a notion that has overflowed the purely scientific context. The result is an abuse of concepts and terms from the "hard sciences", intellectual confusions about the content of scientific discourse and its philosophy. It is shown that its association with a "theory" is highly questionable. Indeed it is...

This text is a critical analysis of what is called "Theory" about the chaos researches.

This report is the first part dealing with the notion of "germinal dynamics" via embedding of a Dim(p-1) noninvertible map into a Dim (p) invertible map, when p=2, and 3. For p=3, the simplest situation is considered, that of an embedded noninvertible map generating a simply connected basin.

This text is devoted to the fractal box-within-the-box bifurcation structure ("structure boîtes-emboîtées" in French, 1975), which describes the bifurcation organization generated by an unimodal map. With respect to the general Sharkovskji paper "Coexistence of cycles of a continuous mapping of the line into itself" (1964), giving the cycles orderi...

This publication compares the characteristic properties of two different types of chaotic areas generated by Dim2 noninvertible maps: non mixed chaotic areas and mixed chaotic areas. The first ones are bounded by critical arcs of finite rank. The second ones are bounded by critical arcs, associated with arcs of the unstable manifold of saddle point...

Most of papers dealing with chaotic attractors, observed in the phase plane of a Dim2 map via a numerical simulation, don't consider the fundamental nature of the map: its invertibility, or its non invertibility. Being unaware of this fact, some basic properties of the map solutions so are unseen when the map is noninvertible. Indeed noninvertibili...

Most of papers dealing with maps don't consider which might be the first question about a fundamental characteristic of their nature: are they invertible, or not? This even though noninvertible maps generate a very large set of continuous map classes, invertible maps constituting only a small subset among the continuous maps set. This "Research Rep...

The effect of a delay on the stability of the synchronized steady state is purely quantitative when the delay is small, leading merely to a small reduction of the state influence domain. If the delay exceeds a critical value, the effect on this domain increases rapidly and the latter becomes zero when the delay reaches a second critical value. At t...

A class of differential equations with pure delay and a hyperbolic nonlinearity, analogous to the Michaelis-Menten term in chemical reaction kinetics, is examined. Conditions for the existence of periodic solutions are established. The amplitude and period dependence on the equation parameters is estimated analytically. A mixed analytico-numerical...

Solutions of "limit cycle" type are analytically considered for autonomous differential equations with variable delay. In the non autonomous case, with a periodic forcing term, resonance and subharmonic resonance situations are examined.

The first paper deals with a delayed nonlinear phase control system. It is shown that an arc of the stability boundary, in the parameter plane, is non-dangerous in the Bautin sense (i.e. crossing through the boundary, the solution qualitative change is continuous).
The second paper is devoted to the analytical determination of solutions of the Cher...

From the Goodwin formulation, an approximate model has the form of a Dim2 autonomous ODE with variables x(t) the mRNA which codes for the protein y(t), the latter acting as a repressor. This model, which neglects delays of synthesis and transport of (x,y) from the place of production to the place of effects, does not give rise to periodic solutions...

Considering a first order nonlinear differential equation with delay, the first text deals with bifurcations giving rise to "limit cycle" types from an asymptotically stable constant state. By means of error-controlled high-accuracy numerical computations, it is found that such an approach is an adequate substitute for analytically unavailable theo...

This Gumowski's paper shows that a small delay c, neglected in a first step, may lead to large qualitative changes for the ODE solution, i.e. the dynamics system is sensitive with respect to small delay. In the non autonomous case the periodic and almost periodic solutions are of more varied types than for analogous ODE without pure delay.

The present Technical Report is a continuation of two previous ones, dealing with families of maps, studied in the 1970 years under the appellations "conservative maps with bounded nonlinearities" and "quasi-conservative maps with bounded nonlinearities". After they have been called "Gumowski-Mira map" by several authors. Now, in this report the bo...

Written in French, in "Comptes Rendus à l'Académie des Sciences, Paris", at a time when, in Western countries, nonlinear researches were still in an underdeveloped state after Poincaré's results, these fundamental results have remained quasi-ignored in spite of their interest.

With respect to a preceding one, this report adds informations about the mathematical properties of the exhibited "aesthetical" images.

The term "Gumowski Mira map" was never used by the two authors who have studied the equations so designated. Such an attitude would have been contrary to scientific ethics. A web search based on operative words "Gumowski Mira map" shows that several authors have found here a subject of articles, mainly related to "esthetical" images of chaotic solu...

From the 1962s, a group of researchers located in Toulouse (France) devoted a large part of its activity to the qualitative methods of nonlinear dynamics, in this framework specially to nonlinear maps. In 1973, the group organized the first international conference dedicated to nonlinear maps and their applications. The purpose of this chapter is t...

The first researches in the nonlinear dynamics field began in Toulouse from the year 1958. This paper relates the first period (i.e. the “prehistoric” one ending in 1976, the emergence of the word chaos due to May, and the beginning of the “historic” times) of the Toulouse group activity in complex nonlinear dynamics. This period was characterized...

This paper deals with bifurcation structures related to families of fractional harmonics (or ultra-subharmonics) solutions generated by the Duffing–Rayleigh equation with a nonsymmetrical periodic external force. It presents some results on fractional reducible harmonics and their bifurcations. In particular a new type of contact of fold curves wit...

This text considers the embedding of a Dim1 piecewise continuous and piecewise linear map family, studied by Leonov in the years 1960. The embedding is of Hénon's map type. After having reminded the Leonov's results, the existence domains of different attracting sets are determined in a parameter plane for positive and negative values of the embedd...

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixe...

Talks presented in the framework of the "Institut Poincaré" in Paris, for the 100th anniversary of the Henri Poincaré's death.

In a parameter plane, crossroad areas and spring areas are two typical organizations of fold and flip bifurcation curves centred at a fold cusp point. Till now only spring areas in a “symmetrical” configuration have been described. This letter introduces another type of spring area for which such a “symmetry” does not exist. It is called a dissymme...

In this paper we give an example of transition to fractal basin boundary in a two-dimensional map coming from the applicative context, in which the hard-fractal structure can be rigorously proved. That is, not only via numerical examples, although theoretically guided, as often occurs in maps coming from the applications, but also via analytical to...

We investigate the properties of recurrence of the type $x_{n+1}=(a+\sum_{i=0}^{k-2}x_{n-i})/x_{n-(k-1)}$, known as Lyness iterations from [Lyness, 1942, 1945, 1961] and recently analyzed by several authors in the case a > 0, see e.g. [Kocic et al., 1993; Csornyei & Laczkovich, 2000]. We reconsider Lyness recurrences at the light of some recent res...

Properties of the basins of noninvertible maps of a plane are studied using the method of critical curves. Different kinds of basin bifurcation, some of them leading to basin boundary fractalization are described. More particularly the paper considers the simplest class of maps that of a phase plane which is made up of two regions, one with two pre...

In this chapter we briefly reconsider some of the main properties of noninvertible maps which characterize the related dynamics. We shall recall how the effects of noninvertibility may be important in understanding global bifurcations both in the structure of the invariant attracting sets and in the structure of the basins of attraction.

The first part is devoted to a presentation of specific features of noninvertible maps with respect to the invertible ones.
When embedded into a three-dimensional invertible map, the specific dynamical features of a plane noninvertible map are the
germ of the three-dimensional dynamics, at least for sufficiently small absolute values of the embeddi...

Part I of this paper has been devoted to properties of the different Julia set configurations, generated by the complex map TZ: z′ = z² - c, c being a real parameter, -1/4 < c < 2. These properties were revisited from a detailed knowledge of the fractal organization (called "box-within-a-box"), generated by the map x′ = x² - c with x a real variabl...

This paper is Part II of an earlier paper dealing with the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, and an external periodical excitation of period τ = 2π/ω (amplitude E). In the absence of this excitation, this equation of Duffing type does not give rise to self-oscillation...

Properties of the different configurations of Julia sets J, generated by the complex map TZ: z′ = z² - c, are revisited when c is a real parameter, -1/4 < c < 2. This is done from a detailed knowledge of the fractal bifurcation organization "box-within-a-box", related to the real Myrberg's map T: x′ = x² - λ, first described in 1975. Part I of this...

Article published on the Website "Scholarpedia" [Mira C., 2 (9): 2328]. It gives a general presentation of the matter for Dim2 noninvertible maps, associated with references and with illustrative examples of basin evolutions described in its "First Subpage" : transition from a simply connected basin to non connected one, or multiply connected one,...

Some fractal sets generated by noninvertible maps are identified. Two topics are presented: the fractal "box-within-a-box" bifurcation structure (1975), and the fractalization of basin boundaries.

Two-dimensional (Z₁-Z₃-Z₁) maps are such that the plane is divided into three unbounded open areas: one Z₃ generating three real rank-one preimages, bordered by two regions Z₁ generating only one real rank-one preimage. The paper aim is essentially devoted to the basin structures generated by such maps, more particularly when they are fractal, and...

The present paper focuses on the two time scale dynamics generated by 2D polynomial noninvertible maps T of (Z0 - Z2) and (Z1 - Z3 - Z1) types. This symbolism, specific to noninvertible maps, means that the phase plane is partitioned into zones Zk, where each point possesses the k real rank-one preimages. Of special interest here is the structure o...

This paper is devoted to the numerical study of a two-dimensional nonautonomous ordinary differential equation with a strong cubic nonlinearity, submitted to an external periodical excitation of period τ = 2π/ω , having an amplitude E. In absence of this excitation the equation of Duffing type does not give rise to self oscillations. It may be a mo...

This paper continues the study of the global dynamic properties specific to maps of the plane characterized by the presence of a denominator that vanishes in a one-dimensional submanifold. After two previous papers by the same authors, where the definitions of new kinds of singularities, called focal points and prefocal sets, are given, as well as...

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z0) each point having no real preimage, and (ii) (denoted by Z2) each point having two real preimages. The regions Z0, Z2 are separated by the critical curve LC, locus of points having two coincident preimages. Z2 is mad...

The paper deals with the solutions generated by a two-dimensional noninvert-ible map defined by a cubic polynomial. The map is of the (Z 1 − Z 3 − Z 1) type, i.e. the plane is divided into three unbounded open areas: one Z 3 generating three real rank-one preimages, bordered by two regions Z 1 generating only one real rank-one preimage. This short...

The object of the present work is to study a one-dimensional nonautonomous equation (a Mann iteration) with geometric weights. Its study amounts to that of a two-dimensional autonomous rational map having a vanishing denominator in a non classical case. A map with a vanishing denominator possesses a set of points, called the prefocal set, with part...

This paper is devoted to the study of a family of two-dimensional noninvertible maps, depending on two parameters. The family is topologically semiconjugated to the complex quadratic map Z for a particular parameter value. The variation of this parameter value permits a full identification of one of the possible bifurca-tion mechanisms leading to t...

This paper is the second part of an earlier work devoted to the properties specific to maps of the plane characterized by the presence of a vanishing denominator, which gives rise to the generation of new types of singularities, called set of nondefinition, focal points and prefocal curves. A prefocal curve is a set of points which are mapped (or "...

This paper concerns the description of some properties of p-dimensional invertible real maps Tb, turning into a (p-1)-dimensional noninvertible ones T0, p=2,3, when a parameter b of the first map is equal to a critical value, say b=0. Then it is said that the noninvertible map is embedded into the invertible one. More particularly, properties of th...

Two applications of discrete models in the form of noninvertible maps are presented. The first one is related to the now popular problem of communicating via chaos synchronization in the signal processing field. The second topic concerns a method of information storage in periodic solutions (called cycles) of piecewise continuous noninvertible maps...

This paper concerns some results on global dynamical properties and bifurcations of two-dimensional maps, invertible or noninvertible, presenting a vanishing denominator. This last characteristic may give rise to specific singularities of the phase plane, called focal points and prefocal curves. The presence of these sets may cause new types of bif...

In this paper, sufficient conditions of global synchronization in
finite-time, which can be a minimum, are presented for the generic class
of piecewise linear maps. The conditions of synchronization are based
upon general results of the robust control theory, the observability
theory, and the specificity of the chaotic motion generated by the map....

This text deals with the domain of existence of the solution of a Schröder's equation, related to a two-dimensional real iteration process, defined by functions which do not satisfy the Cauchy-Riemann conditions. Its purpose is limited to the identification of the difficulties generated by the determination of this domain. When the Cauchy-Riemann c...

For models in the form of noninvertible maps we propose a numerical method to calculate a class of basin bifurcation sets in a parameter space. It is known that basin bifurcations may result from the contact of a basin boundary with the critical curve (locus of points having two coincident rank-one preimages) segment. Therefore, when the map is smo...

We show that unbounded chaotic trajectories are easily observed in the iteration of maps which are not defined everywhere, due to the presence of a denominator which vanishes in a zero-measure set. Through simple examples, obtained by the iteration of one-dimensional and two-dimensional maps with denominator, the basic mechanisms which are at the b...

This paper deals with bifurcation structures related to families of fractional harmonics (or ultra-subharmonics) solutions generated by the Duffing-Rayleigh equation with a nonsymmetrical periodic external force. It presents some results on fractional reducible harmonics and their bifurcations. In particular, a new type of contact of fold curves wi...

Many classes of discrete dynamical systems give rise to models in the form of noninvertible maps. With respect to invertible maps, noninvertible maps introduce a singularity of a different nature: the critical set of rank-one, as the geometrical locus of points having at least two coincident preimages. Such new singularities play a fundamental role...

History of the Toulouse research group and its contribution to nonlinear and chaotic dynamics from 1962. Basin Boundaries of Two-Dimensional Noninvertible Maps. Non connected basins and multiply connected basins. Chaotic Attractors of Dim2 Noninvertible Maps. Normal Forms for Resonant Bifurcations. Dim2 Conservative Maps. Dim1 Non-invertible Maps,...

Consider the Henon's map Tb: (x→ 1-ax2+y, y→bx), the parameters (a, b) being such that |b|<1, with the existence of an attracting set A. This paper deals with an approximate implicit analytical representation of the stable manifold WS(q1) of the saddle fixed point q1 belonging to the basin boundary of the attracting set A. A method of successive ap...

The CP-PLL is a typical mixed signal device; there is no general
theory to describe exactly the dynamics of its nonlinear transient. Many
models have been presented to model the behavior of the PLL. A discrete
linear model, a discrete nonlinear model and an event-driven model of
the second order PLL have been proposed. A comparison of these models...

This paper deals with some properties of bifurcation structures in the parameter space related to the Duffing equation in the presence of an external periodical excitation B + B0cos t. So global qualitative modifications of structures in the parameter plane (B, B0) are considered, when a third parameter ε (the damping term of the equation) varies....

This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of focal points and of prefocal curves are introduced in order to characterize some new kinds of contact bifurcations specific to...

The presence of focal points in Dim2 maps is due to a vanishing denominator.

This paper deals with Dim1 and Dim2 maps with a vanishing denominators and the conditions for generating unbounded but not diverging sets of attraction.

This paper deals with a new coding scheme in digital implementation for secure communications. It is based on specific dynamic features generated by noninvertible maps. The main results are a global chaos synchronization, an exact synchronization without a residual error generated by the classical methods, a robustness with respect to channel distu...

This paper presents a general survey about bifurcations specific to noninvertible maps. These bifurcations result from a contact of a non classical singularity, the "critical set" . More particularly:
- The contact of a "critical set" with a simply connected basin boundary can generate non-connected basins, ormutipli-connected ones, or mixed non-c...

This paper deals with Dim2 maps with a vanishing denominators and the conditions for generating unbounded but not diverging sets of attraction. Through simple examples, the basic mechanisms which are at the basis of the existence of unbounded but not diverging chaotic trajectories are explained.

Two dynamical systems with chaotic behaviors generated by piecewise linear noninvertible maps are synchronized. The paper describes a special class of such maps, interesting for secure communications, and presnting advantages compared with the use of continuous ODE.

This paper deals with chaos synchronization for secure
communications, the chaotic signals being generated by piecewise linear
noninvertible maps. With respect to the solutions described until now,
such maps permit specific dynamic features improving significantly the
system characteristics. Chaos synchronization is obtained by introducing
an obser...

This paper does not pretend to present a comprehensive history of nonlinear dynamics. Its purpose is more modest and limited to some historical aspects of this topic. The first part of this paper deals with the early foundations of nonlinear dynamics (essentially the Poincaré and Lyapunov results). The succeeding sections cover the period 1910–1970...

Simple electronic oscillators were at the origin of many studies related to the qualitative theory of dynamical systems. Chua's circuit is now playing an equivalent role for the generation and understanding of complex dynamics.

Simple electronic oscillators were at the origin of many studies related to the qualitative theory of dynamical systems. Chua's circuit is now playing an equivalent role for the generation and understanding of complex dynamics.
In honour of my friend Leon Chua on his 60th birthday.

Dim2 noninvertible maps generate new types of chaotic behaviors, and basin structures with respect to the invertible case (invited conference)

The paper presents a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters ma...