Clusters in an assembly of globally coupled bistable oscillators

We study the dynamics of an assembly of globally coupled bistable elements. We show that bistability of elements results in some new features of clustering in the assembly when there is global coupling. We provide conditions for the existence of stable amplitude-phase clusters and splay-phase states. Received 12 June 1998 and Received in final form 30 November 1998     

Spike-burst and other oscillations in a system composed of two coupled, drastically different elements

The dynamics of a system composed of two nonlinearly coupled, drastically different nonlinear and eventually oscillatory elements is studied. The rich variety of attractors of the system is studied with the help of phase space analysis and Poincare maps. Received 19 March 1999 and Received in final form 1 November 1999     

Probing viscosity of a polymer melt at the nanometre scale with an oscillating nanotip

The Navier-Stokes equation is used to analyze the additional phase delay when an oscillating nanotip touches intermittently an entangled polymer melt. Even when the tip oscillates at frequencies of several hundred kilohertz, it is shown that the inertial contributions are negligible as long as the indentation depth is no more than a few ten nanometers. Consequently, a stationary solution can be used leading to the simple Stokes formula. Two simple geometries of the tip are investigated. A smooth tip apex with a spherical shape and an elongated tip apex that aims at mimicking a single asperity. The tip shape has a drastic influence on the measured viscosity at the local scale. A simple calculation indicates that the viscous force acting against the tip motion may exhibit several different behaviors as a function of the indentation depth. Using the variational principle of least action, we derive the corresponding phase variation of the oscillator as a function of the indentation depth. It is shown that there exist situations for which an absolute value of the local viscosity could be measured. Received 13 April 2001 and Received in final form 1 August 2001     

Birth and long-time stabilization of out-of-equilibrium coherent structures

We study an analytically tractable model with long-range interactions for which an out-of-equilibrium very long-lived coherent structure spontaneously appears. The dynamics of this model is indeed very peculiar: a bicluster forms at low energy and is stable for very long time, contrary to statistical mechanics predictions. We first explain the onset of the structure, by approximating the short time dynamics with a forced Burgers equation. The emergence of the bicluster is the signature of the shock waves present in the associated hydrodynamical equations. The striking quantitative agreement with the dynamics of the particles fully confirms this procedure. We then show that a very fast timescale can be singled out from a slower motion. This enables us to use an adiabatic approximation to derive an effective Hamiltonian that describes very well the long time dynamics. We then get an explanation of the very long time stability of the bicluster: this out-of-equilibrium state corresponds to a statistical equilibrium of an effective mean-field dynamics. Received 28 February 2002 / Received in final form 24 July 2002 Published online 31 October 2002 RID="a" ID="a"e-mail: Thierry.Dauxois@ens-lyon.fr RID="b" ID="b"UMR-CNRS 5672 RID="c" ID="c"UMR 5582     

Atomic force microscopy investigation of chemically stabilized pericardium tissue

Native and chemically stabilized porcine pericardium tissue was imaged by the contact mode atomic force microscopy (AFM), in air. Chemically stabilized pericardium is used as a tissue-derived biomaterial in various fields of the reconstructive and replacement surgery. Collagen type I is the main component of the fibrous layer of the pericardium tissue. In this study, the surface topography of collagen fibrils in their native state in tissue and after chemical stabilization with different cross-linking reagents: glutaraldehyde (GA), dimethyl suberimidate (DMS) and tannic acid (TA) was investigated. It has been found that chemical stabilization causes considerable changes in the surface topography of collagen fibrils as well as in the spatial organization of the fibrils within the tissue. The observed changes in the D-spacing pattern of the collagen fibril correspond to the formation of intrafibrilar cross-links, whereas formation of interfibrilar cross-links is mainly responsible for the observed tangled spatial arrangement of fibrils and crimp structure of the tissue surface. The crimp structure was distinctly seen for the GA cross-linked tissue. Surface heterogeneity of the cross-linking process was observed for the DMS-stabilized tissue. SDS-PAGE electrophoresis was performed in order to evaluate the stabilization effect of the tissues treated with the cross-linking reagents. It has been found that stabilization with DMS, GA or TA enhances significantly the tissue resistance to SDS/NaCl extraction. The relation between the tissue stability and changes in the topography of the tissue surface was interpreted in terms of different nature of cross-links formed by DMS, GA and TA with collagen.     

Wave propagation along interacting fiber-like lattices

A fiber-like lattice with resistively coupled electronic elements mimicking a 1-D discrete reaction-diffusion system is considered. The chosen unit or element in the fiber is the paradigmatic Chua's circuit, capable of exhibiting bistable, excitable, oscillatory or chaotic behavior. Then the dynamics of a structure of two such interacting parallel active fibers is studied. Suitable conditions for the interaction to yield synchronization and other forms of collective behavior involving both fibers are obtained. They include wave front propagation, pulse reentry and pulse propagation failure, overcoming of propagation failure, and the appearance of a source of synchronized pulses. The possibility of designing controlled dynamic contacts by means of one or a few inter-fiber couplings is also discussed. Received 12 December 1998     

Quasi-periodic and periodic solutions for dynamical systems related to Korteweg-de Vries equation

We consider quasi-periodic and periodic (cnoidal) wave solutions of a set of n-component dynamical systems related to Korteweg-de Vries equation. Quasi-periodic wave solutions for these systems are expressed in terms of Novikov polynomials. Periodic solutions in terms of Hermite polynomials and generalized Hermite polynomials for dynamical systems related to Korteweg-de Vries equation are found. Received 15 October 2001 / Received in final form 6 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: nakostov@ie.bas.bg     

Time delay as a key to apoptosis induction in the p53 network

A feedback mechanism that involves the proteins p53 and mdm2, induces cell death as a controlled response to severe DNA damage. A minimal model for this mechanism demonstrates that the response may be dynamic and connected with the time needed to translate the mdm2 protein. The response takes place if the dissociation constant k between p53 and mdm2 varies from its normal value. Although it is widely believed that it is an increase in k that triggers the response, we show that the experimental behaviour is better described by a decrease in the dissociation constant. The response is quite robust upon changes in the parameters of the system, as required by any control mechanism, except for few weak points, which could be connected with the onset of cancer. Received 8 May 2002 / Received in final form 9 July 2002 Published online 17 September 2002     

Spatial orders appearing at instabilities of synchronous chaos of spatiotemporal systems

Various spatial orders introduced by the instabilities of synchronous chaotic state of spatiotemporal systems are investigated by considering coupled map lattice and chaotic partial differential equation. In particular, the motions of on-off intermittent states at the onset of the instabilities are studied in detail. The chaotic desynchronized patterns can be described by a simple universal form, including three parts: the synchronous chaos; a spatially ordered pattern, determined by the unstable mode of the reference synchronous chaos; and on-off intermittency of the scale of this given pattern. Received 31 July 2002 / Received in final form 20 November 2002 Published online 31 December 2002     

Dynamics of twists on antiferromagnetic spin chains: Theory

The equation of motion of twists on classical antiferromagnetic Heisenberg spin chains are derived. It is shown that twists interact via position- and velocity-dependent long-range two-body forces. A quiescent regime is identified wherein the system conserves momentum. With increasing kinetic energy the system exits this regime and momentum conservation is violated due to walls annihilation. A bitwist system is shown to be integrable and its exact solution is analysed. Many-twist systems are discussed and novel periodic twist lattice solutions are found on closed chains. The stability of these solutions is discussed. Received 12 June 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: rbbll@phy.cam.ac.uk     

Complexifications and real forms of Hamiltonian structures

We consider generalizations of the standard Hamiltonian dynamics to complex dynamical variables and introduce the notions of real Hamiltonian form in analogy with the notion of real forms for a simple Lie algebra. Thus to each real Hamiltonian system we are able to relate several nonequivalent ones. On the example of the complex Toda chain we demonstrate how starting from a known integrable Hamiltonian system (e.g. the Toda chain) one can complexify it and then project onto different real forms. Received 18 October 2001 / Received in final form 24 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: gerjikov@inrne.bas.bg     

Thermodynamics and transport in an active Morse ring chain

We investigate the stochastic dynamics of an one-dimensional ring with N self-driven Brownian particles. In this model neighboring particles interact via conservative Morse potentials. The influence of the surrounding heat bath is modeled by Langevin-forces (white noise) and a constant viscous friction coefficient γ. The Brownian particles are provided with internal energy depots which may lead to active motions of the particles. The depots are realized by an additional nonlinearly velocity-dependent friction coefficient γ ₁(v) in the equations of motions. In the first part of the paper we study the partition functions of time averages and thermodynamical quantities (e.g. pressure) characterizing the stationary physical system. Numerically calculated non-equilibrium phase diagrams are represented. The last part is dedicated to transport phenomena by including a homogeneous external force field that breaks the symmetry of the model. Here we find enhanced mobility of the particles at low temperatures. Received 21 July 2001     

Hypersensitivity to small signals in a stochastic system with multiplicative colored noise

Recently we discovered the phenomenon of hypersensitivity to small time-dependent signals in a simple stochastic system, the Kramers oscillator with multiplicative white noise. In the present work we study, theoretically and experimentally with analog simulations, an influence of noise correlation time on hypersensitivity in a nonlinear oscillator with piecewise-linear current-voltage characteristic and multiplicative colored dichotomous noise. We found that the region of hypersensitive behavior is defined by universal scaling index, whereas the specifics of a particular system reveals itself only in the dependence of the above index on system parameters. The dependence of gain factor on noise correlation time is of bell-shaped (resonant) type. Received 27 April 2000 and Received in final form 2 November 2000     

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes N are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions all attractors of the system become identical in the thermodynamical limit up to variations of order 1/, and thus replica symmetry is recovered for N→∞. In contrast to this, when fluctuating-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit. Received 9 July 2001     

Controlling global stochasticity of Hamiltonian systems by nonlinear perturbation

A method of controlling global stochasticity in Hamiltonian systems by applying nonlinear perturbation is proposed. With the well-known standard map we demonstrate that this control method can convert global stochasticity into regular motion in a wide chaotic region for arbitrary initial condition, in which the control signal remains very weak after a few kicks. The system in which chaos has been controlled approximates to the original Hamiltonian system, and this approach appears robust against small external noise. The mechanism underlying this high control efficiency is intuitively explained. Received 15 January 2002 Published online 6 June 2002     

Dynamics of a single ion in a perturbed Penning trap. Sextupolar perturbation

In the frame work of classical mechanics, we study the nonlinear dynamics of a single ion trapped in a Penning trap perturbed by an electrostatic sextupolar perturbation. The perturbation is caused by a deformation in the configuration of the electrodes. By using a Hamiltonian formulation, we obtain that the system is governed by three parameters: the z-component of the canonical angular momentum P _φ - which is a constant of the motion because the perturbation we assume is axial-symmetric -, the parameter δ that determines the ratio between the axial and the cyclotron frequencies, and the parameter a which indicates how far from the ideal design the electrodes are. We study the case P _φ = 0. By means of surfaces of section, we show that the phase space structure is made of three fundamental families of orbits: arch, loop and box orbits. The coexistence of these kinds of orbits depends on the parameter δ. The escape is also explained on the basis of the shape of the potential energy surface as well as of the phase space structure. Received 6 September 2001 / Received in final form 19 March 2002 Published online 28 June 2002     

Nonlinear dynamics of dimers on periodic substrates

We study the dynamics of a dimer moving on a periodic one-dimensional substrate as a function of the initial kinetic energy at zero temperature. The aim is to describe, in a simplified picture, the microscopic dynamics of diatomic molecules on periodic surfaces, which is of importance for thin film formation and crystal growth. We find a complex behaviour, characterized by a variety of dynamical regimes, namely oscillatory, “quasi-diffusive” (chaotic) and drift motion. Parametrically resonant excitations of internal vibrations can be induced both by oscillatory and drift motion of the centre of mass. For weakly bound dimers a chaotic regime is found for a whole range of velocities between two non-chaotic phases at low and high kinetic energy. The chaotic features have been monitored by studying the Lyapunov exponents and the power spectra. Moreover, for a short-range interaction, the dimer can dissociate due to the parametric excitation of the internal motion. Received 8 July 2002 / Received in final form 15 November 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: fusco@sci.kun.nl.     

Multi-affinity and multi-fractality in systems of chaotic elements with long-wave forcing

Multi-scaling properties in quasi-continuous arrays of chaotic maps driven by long-wave random force are studied. The spatial pattern of the amplitude X(x,t) is characterized by multi-affinity, while the field defined by its coarse-grained spatial derivative exhibits multi-fractality. The strong behavioral similarity of the X- and Y-fields respectively to the velocity and energy dissipation fields in fully-developed fluid turbulence is remarkable, still our system is unique in that the scaling exponents are parameter-dependent and exhibit nontrivial q-phase transitions. A theory based on a random multiplicative process is developed to explain the multi-affinity of the X-field, and some attempts are made towards the understanding of the multi-fractality of the Y-field. Received 16 November 1998     

Quasiperiodically driven Josephson junctions: strange nonchaotic attractors, symmetries and transport

We consider the dynamics of the overdamped Josephson junction under the influence of an external quasiperiodic driving field. In dependence on parameter values either a quasiperiodic motion or a strange nochaotic attractor (SNA) can be observed. The latter corresponds to a resistive state in the current-voltage characteristics while for quasiperiodic motion a finite superconducting current exists for zero voltage. It is shown that in the case of SNA a nonzero mean voltage across the junction can appear due to symmetry breakings. Based on this observation a detailed symmetry consideration of the generalized equation of motion is performed and symmetry conditions ensuring zero mean voltage across the junction are found. Received 16 August 2001 and Received in final form 22 January 2002     

Power injected in dissipative systems and the fluctuation theorem

We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the “fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of “temperature" for a dissipative system out of equilibrium. We consider how this “temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration. Received 26 June 2000 and Received in final form 24 October 2000