Nonlinear Dissipative Systems Research Articles

Optical bullets and double bullet complexes in dissipative systems

We show that optical light bullets can coexist with double bullet complexes in nonlinear dissipative systems. Coexistence occurs for a relatively large range of the system parameters, and is associated with either marginal stability or bistable existence of the two dissipative soliton species. In the case of marginal stability, spontaneous transformations of single bullets into double bullet complexes are observed. Among the bistable cases, we show how both clockwise and anticlockwise rotating double bullet complexes can be formed out of the phase-controlled interaction of two single bullets. The internal dynamics of pulsating double bullet complexes, with oscillations in both the spatial separation between the two bullets and the bullet shape in time domain is also detailed.

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift–Hohenberg equation and reaction–diffusion systems of activator–inhibitor type.

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples.

Optical bullets and “rockets” in nonlinear dissipative systems and their transformations and interactions

We demonstrate the existence of stable optical light bullets in nonlinear dissipative media for both cases of normal and anomalous chromatic dispersion. The prediction is based on direct numerical simulations of the (3+1)-dimensional complex cubic-quintic Ginzburg-Landau equation. We do not impose conditions of spherical or cylindrical symmetry. Regions of existence of stable bullets are determined in the parameter space. Beyond the domain of parameters where stable bullets are found, unstable bullets can be transformed into "rockets" i.e. bullets elongated in the temporal domain. A few examples of the interaction between two optical bullets are considered using spatial and temporal interaction planes.

Analysis of prebreakdown current in a gas discharge system with a semiconducting cathode

Results are presented of an analysis of the prebreakdown current of a Townsend discharge in a planar gas discharge system with a GaAs semiconductor cathode for a gap thickness of 45 µm and gas pressures of 6.9, 8.9, 34.2 and 36.9 kPa. IR radiation excites the cathode in the system, thus controlling the current density in the gas discharge layer. It is shown that the prebreakdown current is caused by electron emission from the semiconductor surface. The present experimental investigations show that the prebreakdown current decreases anomalously with increase of the feeding voltage and that it also decreases under the illumination of a semiconductor cathode. Both these anomalies depend on gas pressure. The current density change through the cross-section of the discharge gap, i.e. the appearance of the spatio-temporal self-organization of non-linear dissipative systems, causes these observed effects. These characteristics of the system can be applied for fast IR imaging and a high speed UV light source.

Large time profile of solutions for a dissipative nonlinear evolution system with conservational form

We examine the Cauchy problem for a nonlinear dissipative evolution system with conservation form with initial condition where ?, ?, ? and ? are positive constants such that ? < ? and ? < 1. We establish the global existence and decay rate of the solution subject to the parameter restriction . The optimal decay rate is obtained if (?0, ?0) L1(R, R2) furthermore. The global existence of the solution to the same problem has been studied in Jian and Chen (1998 Acta Math. Sin. 14 17?34) without giving the decay rate and optimal decay order of the solution, in which (?0, ?0) H1(R, R2)?L1(R, R2).

Light bullets and dynamic pattern formation in nonlinear dissipative systems

In the search for suitable new media for the propagation of (3+1) D optical light bullets, we show that nonlinear dissipation provides interesting possibilities. Using the complex cubic-quintic Ginzburg-Landau equation model with localized initial conditions, we are able to observe stable light bullet propagation or higher-order transverse pattern formation. The type of evolution depends on the model parameters.

Two-dimensional laser soliton complexes with weak, strong, and mixed coupling

A review is given of features and motion of two-dimensional dissipative solitons in lasers and laser amplifiers with saturable absorption. We present a rich variety of stable complexes with weak, strong, and mixed coupling of individual laser solitons. The type of coupling is determined by the topology of the distribution of energy flows within the complex. We reveal the existence of stable dissipative soliton complexes with curvilinear motion of their centre of mass. This type of motion results from the field distribution asymmetry and is well pronounced for complexes of laser solitons with strong and mixed types of coupling. Similar complexes are expected to exist in different spatially distributed nonlinear dissipative systems, including schemes with discrete dissipative solitons.

Analysis of input-to-state stability for discrete time nonlinear systems via dynamic programming

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L 2 -gain analysis and nonlinear H ∞ theory). As such, the results presented address an obvious gap in the literature.

Curvilinear Motion of Multivortex Laser-Soliton Complexes with Strong and Weak Coupling

We reveal the existence of stable dissipative soliton complexes with curvilinear motion of their center of mass. This type of motion results from the field distribution asymmetry and is well pronounced for asymmetric complexes of laser solitons with strong coupling. We present results of numerical simulations of such complexes in a model of wide-aperture lasers or laser amplifiers with saturable gain and absorption. The complex consists of a pair of strongly coupled vortex solitons weakly coupled with a number of other vortex solitons. Similar complexes are expected to exist in different spatially distributed nonlinear dissipative systems, including schemes with discrete dissipative solitons.

Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours

From viewpoint of nonlinear dynamics, the model reduction and its influence on the long-term behaviours of a class of nonlinear dissipative autonomous dynamical system with higher dimension are investigated theoretically under some assumptions. The system is analyzed in the state space with an introduction of a distance definition which can be used to describe the distance between the full system and the reduced system, and the solution of the full system is then projected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. As a result, the influence of mode series truncation on the long-term behaviours and the error estimate are derived, showing that the error is dependent on the first products of frequencies and damping ratios in the subspace spanned by the eigenvectors with higher modal damping. Furthermore, the fundamental understanding for the topological change of the solution due to the application of different model reduction is interpreted in a mathematically precise way, using the qualitative theory of nonlinear dynamics.

DISSIPATIVE SOLITON PULSATIONS WITH PERIODS BEYOND THE LASER CAVITY ROUND TRIP TIME

We review recent results on periodic pulsations of the soliton parameters in a passively mode-locked fiber laser. Solitons change their shape, amplitude, width and velocity periodically in time. These pulsations are limit cycles of a dissipative nonlinear system in an infinite-dimensional phase space. Pulsation periods can vary from a few to hundreds of round trips. We present a continuous model of a laser as well as a model with parameter management. The results of the modeling are supported with experimental results obtained using a fiber laser.

Additive mixing modulation for public key encryption based on distributed dynamics

We introduce a public key encryption scheme that is based on additive mixing of a message with chaotic nonlinear dynamics. A high-dimensional dissipative nonlinear dynamical system is distributed between transmitter and receiver. The transmitter dynamics is public (known to all) and the receiver dynamics is private (known only to the authorized receiver). Bidirectional signals that couple transmitter and receiver are transmitted over a public channel. Once the chaotic dynamics which is initialized with a random state converges to the attractor, a message is mixed with the chaotic dynamics at the transmitter. The authorized receiver who knows the entire dynamics can use a simple algorithm to decode the message. An unauthorized receiver does not know the receiver dynamics and needs to use computationally unfeasible algorithms in order to decode the message. Security is maintained by altering the private receiver dynamics during transmission. We show that using additive mixing modulation is more efficient than the attractor position modulation distributed dynamics encryption scheme. We demonstrate the concept of this new scheme by simulating a simple coupled map lattice.

Curvilinear Motion of Laser Soliton Complexes

Conditions under which the center of inertia of stable complexes of dissipative optical solitons moves curvilinearly have been determined. Such a character of the motion of dissipative structures is caused by asymmetry in the distribution of the intensity and energy fluxes, and it is pronounced for laser solitons with strong interaction. The results of the numerical simulation of these complexes in the model of surface emitting lasers or laser amplifiers with saturated amplification and absorption are presented. Such complexes may be observed in various spatially distributed nonlinear dissipative systems, in particular, in the form of discrete solitons.

Conditions for the existence of continuous storage functions for nonlinear dissipative systems

The problem of existence of continuous storage functions for dissipative nonlinear systems is considered. It is shown that, if a nonlinear system is dissipative in the state x * , then, under certain assumptions, a continuous storage function can be constructed on a set of points accessible from x * by concatenation of a finite number of forward and backward motions of the system. Most of these assumptions are weaker than certain controllability-type properties and can be checked using similar tests.

QUENCHING LORENZIAN CHAOS

How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on "nonlinear quenching" [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) [Formula: see text], ẏ=-xzq+bx-y and ż=xyq-cz (where (x, y, z) are time(t)-dependent dimensionless variables and [Formula: see text], etc.) in representative cases when q, the "quenching function", satisfies q=1-e+ey with 0≤e≤1. Control parameter space based on a,b and c can be divided into two "regions", an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The "Hopf boundary" between the two regions is located where b=bH(a, c; e) (say), with the much studied point (a, b, c)=(10, 28, 8/3), where the persistent "Lorenzian" chaos that arises in the case when e=0 was first found lying close to b=bH(a, c; 0). As e increases from zero the S-region expands in total "volume" at the expense of F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the entire parameter space.

Dendrite-like texture growth in the nematic liquid crystal phase of 4-n-heptyl- and 4-n-octyl-oxybenzoic acids aligned by a polyimide coating

An oriented dendrite-like texture is reported, appearing at a definite temperature in the nematic phase range of 4-n-heptyl- and 4-n-octyl-oxybenzoic acids (HOBA and OOBA), aligned by rubbed polyimide and preceding the smectic C phase, on cooling. Two preferred directions with respect to the ‘easy’ axis are indicated in the dendrites grown of HOBA and OOBA. We discuss a possible mechanism, at molecular and supramolecular levels, for this dendrite growth; and assume that the building ‘blocks' of the dendrites are oligomers, or mixture of oligomers with ‘free’ closed and open dimers, constituting a detached crystalline layered state (named by us SmX, a smectic state intermediate between the N1 ordinary nematic and SmC phases). The study of the dynamics of the dendrite growth demonstrates a scaling relationship typical for non-equilibrium systems. The observed dendrites can be considered as patterns formed in complex non-linear dissipative systems, driven outside of equilibrium.

Pattern formation at the bicritical point of the Faraday instability.

We present measurements on parametrically driven surface waves (Faraday waves) performed in the vicinity of a bicritical point in parameter space, where modes with harmonic and subharmonic time dependence interact. The primary patterns are squares in the subharmonic and hexagons in the harmonic regime. If the primary instability is harmonic we observe a hysteretic secondary transition from hexagons to squares without a perceptible variation of the fundamental wavelength. The transition is understood in terms of a set of coupled Landau equations and related to other canonical examples of phase transitions in nonlinear dissipative systems. Moreover, the subharmonic-harmonic mode competition gives rise to a variety of new superlattice states. These structures are interpreted as mediator modes involved in the transition between patterns of fourfold and sixfold rotational symmetry.

Dissipative systems: from physics to robust nonlinear control

AbstractIn this paper, we explore connections between the underlying physics of dissipative systems and nonlinear robust control. In particular, we concentrate on the problem of stabilizing stationary solutions of nonlinear dissipative systems with states distributed in space. Dissipative systems are equipped with an entropy function which we employ to relate dissipation with the Hamilton–Jacobi–Bellman equation. This relation allows us to establish formal links between the dynamic properties of dissipative systems, passivity and optimal stabilizing control, as it is understood in systems theory. Robustness issues in controller design, are also discussed in the context of front or pulse spatial pattern stabilization. Copyright © 2004 John Wiley &amp; Sons, Ltd.

A general formula for decay rates of nonlinear dissipative systems

This work is concerned with stabilization of hyperbolic systems by a nonlinear feedback which can be localized on part of the boundary or locally distributed. We present here a general formula which gives the energy decay rates in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We give also two other significant examples of nonpolynomial growth at the origin. We also show that we either obtain or improve significantly the decay rates of Lasiecka and Tataru (Differential Integral Equations 8 (1993) 507–533) and Martinez (Rev. Mat. Comput. 12 (1999) 251–283). To cite this article: F. Alabau-Boussouira, C. R. Acad. Sci. Paris, Ser. I 338 (2004).