Physica D Nonlinear Phenomena

Published by Elsevier
Online ISSN: 0167-2789
Publications
Article
We develop mathematical techniques for analyzing detailed Hodgkin-Huxley like models for excitatory-inhibitory neuronal networks. Our strategy for studying a given network is to first reduce it to a discrete-time dynamical system. The discrete model is considerably easier to analyze, both mathematically and computationally, and parameters in the discrete model correspond directly to parameters in the original system of differential equations. While these networks arise in many important applications, a primary focus of this paper is to better understand mechanisms that underlie temporally dynamic responses in early processing of olfactory sensory information. The models presented here exhibit several properties that have been described for olfactory codes in an insect's Antennal Lobe. These include transient patterns of synchronization and decorrelation of sensory inputs. By reducing the model to a discrete system, we are able to systematically study how properties of the dynamics, including the complex structure of the transients and attractors, depend on factors related to connectivity and the intrinsic and synaptic properties of cells within the network.
 
Article
In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.
 
Article
Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle-node bifurcation of these solutions. An example of this bifurcation in the context of magnetohydrodynamics is given. The spectral stability of the traveling waves generated in the saddle-node bifurcation is studied via an Evans function approach. It is shown that generically one real eigenvalue of the linearization of the viscous conservation law around the parametrized family of traveling waves changes its sign at the bifurcation point. Hence this bifurcation describes the basic mechanism of a stable traveling wave which becomes unstable in a saddle-node bifurcation.
 
Article
This paper focuses on the study of certain classes of Boolean functions that have appeared in several different contexts. Nested canalyzing functions have been studied recently in the context of Boolean network models of gene regulatory networks. In the same context, polynomial functions over finite fields have been used to develop network inference methods for gene regulatory networks. Finally, unate cascade functions have been studied in the design of logic circuits and binary decision diagrams. This paper shows that the class of nested canalyzing functions is equal to that of unate cascade functions. Furthermore, it provides a description of nested canalyzing functions as a certain type of Boolean polynomial function. Using the polynomial framework one can show that the class of nested canalyzing functions, or, equivalently, the class of unate cascade functions, forms an algebraic variety which makes their analysis amenable to the use of techniques from algebraic geometry and computational algebra. As a corollary of the functional equivalence derived here, a formula in the literature for the number of unate cascade functions provides such a formula for the number of nested canalyzing functions.
 
Article
Time series from biological system often display fluctuations in the measured variables. Much effort has been directed at determining whether this variability reflects deterministic chaos, or whether it is merely "noise". Despite this effort, it has been difficult to establish the presence of chaos in time series from biological sytems. The output from a biological system is probably the result of both its internal dynamics, and the input to the system from the surroundings. This implies that the system should be viewed as a mixed system with both stochastic and deterministic components. We present a method that appears to be useful in deciding whether determinism is present in a time series, and if this determinism has chaotic attributes, i.e., a positive characteristic exponent that leads to sensitivity to initial conditions. The method relies on fitting a nonlinear autoregressive model to the time series followed by an estimation of the characteristic exponents of the model over the observed probability distribution of states for the system. The method is tested by computer simulations, and applied to heart rate variability data.
 
Article
In cases where the same real-world system can be modeled both by an ODE system ⅅ and a Boolean system 𝔹, it is of interest to identify conditions under which the two systems will be consistent, that is, will make qualitatively equivalent predictions. In this note we introduce two broad classes of relatively simple models that provide a convenient framework for studying such questions. In contrast to the widely known class of Glass networks, the right-hand sides of our ODEs are Lipschitz-continuous. We prove that if 𝔹 has certain structures, consistency between ⅅ and 𝔹 is implied by sufficient separation of time scales in one class of our models. Namely, if the trajectories of 𝔹 are "one-stepping" then we prove a strong form of consistency and if 𝔹 has a certain monotonicity property then there is a weaker consistency between ⅅ and 𝔹. These results appear to point to more general structure properties that favor consistency between ODE and Boolean models.
 
Article
Many biological and physical systems exhibit population-density dependent transitions to synchronized oscillations in a process often termed "dynamical quorum sensing". Synchronization frequently arises through chemical communication via signaling molecules distributed through an external medium. We study a simple theoretical model for dynamical quorum sensing: a heterogenous population of limit-cycle oscillators diffusively coupled through a common medium. We show that this model exhibits a rich phase diagram with four qualitatively distinct physical mechanisms that can lead to a loss of coherent population-level oscillations, including a novel mechanism arising from effective time-delays introduced by the external medium. We derive a single pair of analytic equations that allow us to calculate phase boundaries as a function of population density and show that the model reproduces many of the qualitative features of recent experiments on BZ catalytic particles as well as synthetically engineered bacteria.
 
Article
Transcription factor proteins control the temporal and spatial expression of genes by binding specific regulatory elements, or motifs, in DNA. Mapping a transcription factor to its motif is an important step towards defining the structure of transcriptional regulatory networks and understanding their dynamics. The information to map a transcription factor to its DNA binding specificity is in principle contained in the protein sequence. Nevertheless, methods that map directly from protein sequence to target DNA sequence have been lacking, and generation of regulatory maps has required experimental data. Here we describe a purely computational method for predicting transcription factor binding. The method calculates the free energy of binding between a transcription factor and possible target DNA sequences using thermodynamic integration. Approximations of additivity (each DNA basepair contributes independently to the binding energy) and linear response (the DNA-protein and DNA-solvent couplings are linear in an effective reaction coordinate representing the basepair character at a specific position) make the computations feasible and can be verified by more detailed simulations. Results obtained for MAT-alpha2, a yeast homeodomain transcription factor, are in good agreement with known results. This method promises to provide a general, computationally feasible route from a genome sequence to a gene regulatory network.
 
Post-inhibitory rebound. The cells take turns firing. When one cell jumps down, it releases the other cell from inhibition. If, at this time, the inhibited cell lies below the left knee of C 0 , then the inhibited cell will jump up to the active phase. 
An example inhibitory network with seven cells. The left panel shows the graph of the network architecture. Cell 1, for example, sends inhibition to cells 4 and 5. Subsets or clusters of cells fire in distinct episodes. Each horizontal row in the middle panels represents the time course of a single cell. A black rectangle indicates when the cell is active. In the right panel, we keep track of which cells fire during each subsequent episode. The equations and parameters used are precisely those described in Section 7 except g I syn = .4, α x = 1, β x = 4, I = 16 for the E-cells and I = 10 for the I-cells. 
Discrete dynamics corresponding to the network shown in Fig. 4. 
Assumptions on the nullclines for the I-cells. 
Solutions of the discrete model (left) and neuronal model (right) for a network of 100 excitatory and 100 inhibitory cells. Each E-cell effectively inhibits nine other E-cells chosen at random. In this example, the refractory period is one and the length of the attractor is nineteen. The top panels show the full network and the bottom panel shows a blow-up of the rectangular regions highlighted in the top panel. In the right panels, the bright areas indicate when a neuron fires; the dark areas indicate when a neuron receives inhibition. where g L = 2.25, g N a = 37.5, g K = 45, v L = −60, v N a = 55 and v K = −80 represent the maximal conductances and reversal potentials of a leak, sodium and potassium current, respectively. Moreover, = .04, I = 0, m ∞ (v) = 1/(1 + exp(−(v + 30)/15)) and w ∞ (v) = 1/(1 + exp(−(v + 45)/3)). The function τ (v) can be written as τ (v) = τ 1 + τ 2 /(1 + exp(v/.1)) where τ 1 = 4 and τ 2 = 3 for each E-cell and τ 1 = 4.5 and τ 2 = 3.5 for each I-cell. The synaptic connections are modelled as in (3); however, in this model, we do not have the I-cells send inhibition to each other. We also assume that both the excitatory and inhibitory synapses are indirect. Here we take g syn = 0.15, g I syn = 0.2, v syn = 0 and v I syn = −100. Moreover, α x = 1.2 and β x = 4.8 for both the inhibitory and excitatory indirect synapses. To simplify the equations, we assume that the synaptic variables s i and s I i turn on and off instantaneously; that is, each s i satisfies s i = H (v i −θ ) where H is the Heaviside step-function and θ = 0.1 is some threshold. A similar equation holds for each s I i. Fig. 10 shows an example in which there are 100 excitatory and 100 inhibitory cells. Each E-cell has a refractory period of one and sends excitation to one I-cell and each I-cell sends inhibition to nine E-cells, chosen at random. We show solutions 
Article
We consider a general class of purely inhibitory and excitatory-inhibitory neuronal networks, with a general class of network architectures, and characterize the complex firing patterns that emerge. Our strategy for studying these networks is to first reduce them to a discrete model. In the discrete model, each neuron is represented as a finite number of states and there are rules for how a neuron transitions from one state to another. In this paper, we rigorously demonstrate that the continuous neuronal model can be reduced to the discrete model if the intrinsic and synaptic properties of the cells are chosen appropriately. In a companion paper [1], we analyze the discrete model.
 
Article
A future quantitative genetics theory should link genetic variation to phenotypic variation in a causally cohesive way based on how genes actually work and interact. We provide a theoretical framework for predicting and understanding the manifestation of genetic variation in haploid and diploid regulatory networks with arbitrary feedback structures and intra-locus and inter-locus functional dependencies. Using results from network and graph theory, we define propagation functions describing how genetic variation in a locus is propagated through the network, and show how their derivatives are related to the network's feedback structure. Similarly, feedback functions describe the effect of genotypic variation of a locus on itself, either directly or mediated by the network. A simple sign rule relates the sign of the derivative of the feedback function of any locus to the feedback loops involving that particular locus. We show that the sign of the phenotypically manifested interaction between alleles at a diploid locus is equal to the sign of the dominant feedback loop involving that particular locus, in accordance with recent results for a single locus system. Our results provide tools by which one can use observable equilibrium concentrations of gene products to disclose structural properties of the network architecture. Our work is a step towards a theory capable of explaining the pleiotropy and epistasis features of genetic variation in complex regulatory networks as functions of regulatory anatomy and functional location of the genetic variation.
 
Article
Recently we developed a stochastic particle system describing local interactions between cyanobacteria. We focused on the common freshwater cyanobacteria Synechocystis sp., which are coccoidal bacteria that utilize group dynamics to move toward a light source, a motion referred to as phototaxis. We were particularly interested in the local interactions between cells that were located in low to medium density areas away from the front. The simulations of our stochastic particle system in 2D replicated many experimentally observed phenomena, such as the formation of aggregations and the quasi-random motion of cells. In this paper, we seek to develop a better understanding of group dynamics produced by this model. To facilitate this study, we replace the stochastic model with a system of ordinary differential equations describing the evolution of particles in 1D. Unlike many other models, our emphasis is on particles that selectively choose one of their neighbors as the preferred direction of motion. Furthermore, we incorporate memory by allowing persistence in the motion. We conduct numerical simulations which allow us to efficiently explore the space of parameters, in order to study the stability, size, and merging of aggregations.
 
Conference Paper
This paper considers multiple mobile agents moving in the space with point mass dynamics. We introduce a set of coordination control laws that enable the group to generate the desired stable flocking motion. The control laws are a combination of attractive/repulsive and alignment forces, and the control law acting on each agent relies on the state information of its flockmates and the external reference signal. By using the control laws, all agent velocities asymptotically approach the desired velocity, collisions are avoided between the agents, and the final tight formation minimizes all agent global potentials. Moreover, we show that the velocity of the center of mass either is equal to the desired velocity or exponentially converges to it. Furthermore, when the velocity damping is taken into account, we can properly modify the control laws to generate the same stable flocking motion. Finally, for the case that not all agents know the desired common velocity, we show that the desired flocking motion can still be guaranteed. Numerical simulations are worked out to illustrate our theoretical results.
 
Conference Paper
We review current progress and challenges in modeling of nonlinear phenomena in traffic flow. We are particularly interested in the nonlinear dy- namics of traffic jams: the transition to instabil- ities of traffic flow, the formation of traffic jams and the evolution of the traffic jams. A discrete dynamic system approach is proposed. The self- organized oscillatory behavior and chaotic behavior in traffic are identified and formulated. The results can help to explain the appearance of a phantom traffic jam observed in real traffic. It is also proved that coarser resolution results in richer dynamics of the traffic flow including traffic jams. This is ex- actly what the numerical simulations showed. It is clear that the spatial and temporal resolution affects traffic variable and assumptions in a considerable way. Continuum models run into the limits of de- scribing discrete phenomena in a continuous way. Therefore a combination of continuum and discrete descriptions will be necessary to further improve and declare specific traffic phenomena.
 
Article
We present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the Koopman operator. We first compare invariant measures and discuss this in terms of a “statistical Takens” theorem proved here. We also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic–theoretic treatment of a class of spectral functionals that allow for this. The formalism is extended for a class of stochastic systems: discrete Random Dynamical Systems. The ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior. As an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system.
 
Article
The mechanism for the formation of retinal vessel patterns in the developing human eye is an unresolved question of considerable importance. The current hypothesis is based on the existence of a variable oxygen gradient across the developing photoreceptors which stimulates the release of angiogenic factors which diffuse in the plane of the retina and result in the growth of retinal vessels. This implies that the limiting step in the formation of retinal blood vessels is a diffusion process. To test this hypothesis we have performed a fractal analysis of the human retinal vessels using two different methods. Within the limited range of length scales available in the red-free fundus photographs, we find that the human retinal blood vessels have a self-similar structure with a fractal dimension D ≈ 1.7. Since this value of D is the same as the value found for a diffusion limited growth process, our result supports the hypothesis that diffusion is the fundamental process in the formation of human retinal vessel patterns.
 
Article
It has been shown that uniform as well as non-uniform cellular automata (CA) can be evolved to perform certain computational tasks. Random Boolean networks are a generalization of two-state cellular automata, where the interconnection topology and the cell’s rules are specified at random.Here we present a novel analytical approach to find the local rules of random Boolean networks (RBNs) to solve the global density classification and the synchronization task from any initial configuration. We quantitatively and qualitatively compare our results with previously published work on cellular automata and show that randomly interconnected automata are computationally more efficient in solving these two global tasks. Our approach also provides convergence and quality estimates and allows the networks to be randomly rewired during operation, without affecting the global performance. Finally, we show that RBNs outperform small-world topologies on the density classification task and that they perform equally well on the synchronization task.Our novel approach and the results may have applications in designing robust complex networks and locally interacting distributed computing systems for solving global tasks.
 
Article
This paper investigates nonlinear wave–wave interactions in a system that describes a modified decay instability and consists of three Langmuir and one ion-sound waves. As a means to establish that the underlying dynamics exists in a 3D space and that it is of the Lorenz-type, both continuous and discrete-time multivariable global models were obtained from data. These data were obtained from a 10D dynamical system that describes the modified decay instability obtained from Zakharov’s equations which characterise Langmuir turbulence. This 10D model is equivariant under a continuous rotation symmetry and a discrete order-2 rotation symmetry. When the continuous rotation symmetry is modded out, that is, when the dynamics are represented with the continuous rotation symmetry removed under a local diffeomorphism, it is shown that a 3D system may describe the underlying dynamics. For certain parameter values, the models, obtained using global modelling techniques from three time series from the 10D dynamics with the continuous rotation symmetry modded out, generate attractors which are topologically equivalent. These models can be simulated easily and, due to their simplicity, are amenable for analysis of the original dynamics after symmetries have been modded out. Moreover, it is shown that all of these attractors are topologically equivalent to an attractor generated by the well-known Lorenz system.
 
Article
In this paper we use symmetry methods to study networks of coupled cells, which are models for central pattern generators (CPGs). In these models the cells obey identical systems of differential equations and the network specifies how cells are coupled. Previously, Collins and Stewart showed that the phase relations of many of the standard gaits of quadrupeds and hexapods can be obtained naturally via Hopf bifurcation in small networks. For example, the networks they used to study quadrupeds all had four cells, with the understanding that each cell determined the phase of the motion of one leg. However, in their work it seemed necessary to employ several different four-oscillator networks to obtain all of the standard quadrupedal gaits.
 
Article
Self-reproduction in cellular automata is discussed with reference to the models of von Neumann and Codd. The conclusion is drawn that although the capacity for universal construction is a sufficient condition for self-reproduction, it is not a necessary condition. Slightly more “liberal” criteria for what constitutes genuine self-reproduction are introduced, and a simple self-reproducing structure is exhibited which satisfies these new criteria. This structure achieves its simplicity by storing its description in a dynamic “loop”, rather than on a static “tape”.
 
Article
The paper studies the behavior of the trajectories of fluid particles in a compressible generalization of the Kraichnan ensemble of turbulent velocities. We show that, depending on the degree of compressibility, the trajectories either explosively separate or implosively collapse. The two behaviors are shown to result in drastically different statistical properties of scalar quantities passively advected by the flow. At weak compressibility, the explosive separation of trajectories induces a familiar direct cascade of the energy of a scalar tracer with a short-distance intermittency and dissipative anomaly. At strong compressibility, the implosive collapse of trajectories leads to an inverse cascade of the tracer energy with suppressed intermittency and with the energy evacuated by large-scale friction. A scalar density whose advection preserves mass exhibits in the two regimes opposite cascades of the total mass squared. We expect that the explosive separation and collapse of Lagrangian trajectories occur also in more realistic high Reynolds number velocity ensembles and that the two phenomena play a crucial role in fully developed turbulence.
 
Article
The articulation process of dynamical networks is studied with a functional map, a minimal model for the dynamic change of relationships through iteration. The model is a dynamical system of a function f, not of variables, having a self-reference term f∘f, introduced by recalling that operation in a biological system is often applied to itself, as is typically seen in rules in the natural language or genes. Starting from an inarticulate network, two types of fixed points are formed as an invariant structure with iterations. The function is folded with time, until it has finite or infinite piecewise-flat segments of fixed points, regarded as articulation. For an initial logistic map, attracted functions are classified into step, folded step, fractal, and random phases, according to the degree of folding. Oscillatory dynamics are also found, where function values are mapped to several fixed points periodically. The significance of our results to prototype categorization in language is discussed.
 
Article
This paper is concerned with a class of symmetric forced oscillators modeled by nonlinear ordinary differential equations. The forced Duffing oscillator and the forced pendulum belong to this class of oscillators. Another example is found in an electric power system and is associated with a phenomenon known as ferroresonance. Solutions for this class of systems can be symmetric or non-symmetric. Sometimes, as a physical parameter is varied, solutions lose or gain the symmetry at a bifurcation point. It will be shown that bifurcations which might otherwise be called symmetry breaking, symmetry creation via collision and symmetry creation via explosion are all the result of a collision between conjugate attractors (i.e., attractors that relate to each other by the symmetry) and a symmetric limit set. The same mechanism seems plausible for at least some bifurcations involving non-attracting sets. This point is illustrated with some examples.
 
Article
After presenting some basic ideas in the theory of large deviations, this paper applies the theory to a number of problems in statistical mechanics. These include deriving the form of the Gibbs state for a discrete ideal gas; describing probabilistically the phase transition in the Curie–Weiss model of a ferromagnet; and deriving variational formulas that describe the equilibrium macrostates in models of two-dimensional turbulence. A general approach to the large deviation analysis of models in statistical mechanics is also formulated.
 
Article
We improve [1: Theorem 3.2] by showing that the number of unstable eigenvalues plus the number of purely imaginary eigenvalues with negative Krein signature can be computed from the negative signature of the energy.
 
Article
A generalized Pauli master equation is established for describing the vibrational energy flow in a 1D lattice of hydrogen bounded peptide units. A Lang–Firsov transformation is applied so that the relevant excitations are small polarons corresponding to vibrational excitons dressed by virtual phonons. A special attention is thus paid to characterize the energy transfer mediated by two polarons. At biological temperature, it is shown that the polaron–phonon coupling is sufficiently strong to prevent any coherent motion. The polaron–polaron interaction occurring in such a nonlinear lattice does not affect the long time behavior of the energy flow which results from the diffusion of two independent polarons. This diffusive motion originates from the competition between two contributions related to phonon mediated transitions (incoherent contribution) and to dephasing limited coherent motion (coherent contribution).
 
Article
The Lombardo–Imbihl–Fink (LFI) ODE model of the NO+NH3 reaction on a Pt(1 0 0) surface shows stable relaxation oscillations with very sharp transitions for temperatures T between 404 and 433 K. Here we study numerically the effect of linear diffusive coupling of these oscillators in one spatial dimension. Depending on the parameters and initial conditions we find a rich variety of spatio-temporal patterns which we group into four main regimes: bulk oscillations (BOs), standing waves (SW), phase clusters (PC), and phase waves (PW). Two key ingredients for SW and PC are identified, namely the relaxation type of the ODE oscillations and a nonlocal (and nonglobal) coupling due to relatively fast diffusion of the kinetically slaved variables NH3 and H. In particular, the latter replaces the global coupling through the gas phase used to obtain SW and PC in models of related surface reactions. The PW exist only under the assumption of (relatively) slow diffusion of NH3 and H.
 
Article
Theoretical and experimental results on the control of chaotic motion in an electronic circuit modeled by a delay-differential equation are presented. The model is derived from a nonlinear optical system with time-delay feedback. Our control method is based on Pyragas approach and applied to the infinite-dimensional system. Generalized mean Floquet multipliers and mean Lyapunov exponents related to delay-differential equations are introduced in order to compare a discrete-time approximation with the continuous-time equation. This control method is also used to find the so-called isomers related to harmonic periodic solutions predicted by Ikeda and Matsumoto.
 
Article
We replace the flux term in Burger's equation by two simple alternates that contain contributions depending globally on the solution. In one case, the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal, and in the other the term is in conservation form. In both cases, the nonanalytic is due to the presence of the Hilbert transform. The equations have a loose analogy to the motion of vortex sheets. In particular, they both form singularities in finite time in the absence of viscous effects. Our motivation then is to study the influence of viscosity. In one case, viscosity does not prevent singularity formation. In the other, we can prove solutions exist for all time, and determine the likely weak solution as viscosity vanishes. An interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time. In one case, the singularity is a pole and causes the solution to blow up when it reaches the real axis. In the other, numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time, and then propagates along it. We hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations.
 
Article
Much of the nontrivial dynamics of the one-dimensional (1D) complex Ginzburg–Landau equation (CGLE) is dominated by propagating structures that are characterized by local “twists” of the phase-field. I give a brief overview of the most important properties of these various structures, formulate a number of experimental challenges and address the question how such structures may be identified in experimental space–time data sets.
 
Article
Only Misiurewicz points strictly have a Lyapunov exponent positive measure in one-dimensional (1D) quadratic maps. Hence, we represent here the Lyapunov exponent, λ, of all the Misiurewicz points which were found and inventoried by us in a former work. As a result of this representation, an infinity of alignments of Misiurewicz points are obtained, and two types of jumps in the values of λ are found.
 
Article
We describe the occurence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns. We recall experimental results and propose a model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors, which we characterize briefly and compare to those given by the experiment.
 
Schematic diagram depicting a number of possible dispositions of Floquet multipliers with reference to the unit circle, for a real eigenfrequency p  
Schematic diagram of a pair of possible dispositions of Floquet multipliers for a complex p.
Numerical integration of (1a), (1b) to exhibit stability of N = 2, δ = π, SW solution (γ = 5.9765, slightly below the critical value) starting from the profile (as function of lattice site index n) at (a) t = 0 up to τ = 5000T (the new profile coincides with (a)); (b) corresponds to τ + T 8 , and τ + 7T 8 , (c) to τ + 3T 8 , and τ + 5T 8 , and (d) to τ + 4T 8 .  
Schematic diagram of spectrum for N = 2, δ = π, SW solution similar to fig. 2, but now with ω = −3, and three values of γ, respectively less than, equal to, and greater than 23 3 .  
Schematic diagram, similar to fig. 2 and fig. 7, of the spectrum for the N = 3, δ = π SW solution, with ω = −2.5, and three values of γ, resp. smaller than, equal to, and larger than the critical value (see text).  
Article
We construct a class of exact commensurate and incommensurate standing wave (SW) solutions in a piecewise smooth analogue of the discrete non-linear Schrödinger (DNLS) model and present their linear stability analysis. In the case of the commensurate SW solutions the analysis reduces to the eigenvalue problem of a transfer matrix depending parametrically on the eigenfrequency. The spectrum of eigenfrequencies and the corresponding eigenmodes can thereby be determined exactly. The spatial periodicity of a commensurate SW implies that the eigenmodes are of the Bloch form, characterised by an even number of Floquet multipliers. The spectrum is made up of bands that, in general, include a number of transition points corresponding to changes in the disposition of the Floquet multipliers. The latter characterise the different band segments. An alternative characterisation of the segments is in terms of the Krein signatures associated with the eigenfrequencies. When one or more parameters characterising the SW solution is made to vary, one occasionally encounters collisions between the band-edges or the intra-band transition points and, depending on the the Krein signatures of the colliding bands or segments, the spectrum may stretch out in the complex plane, leading to the onset of instability. We elucidate the correlation between the disposition of Floquet multipliers and the Krein signatures, presenting two specific examples where the SW possesses a definite window of stability, as distinct from the SW’s obtained close to the anti-continuous and linear limits of the DNLS model.
 
Article
The cubic Complex Ginzburg-Landau Equation (CGLE) has a one parameter family of traveling localized source solutions. These so called “Nozaki-Bekki holes” are (dynamically) stable in some parameter range, but always structurally unstable: A perturbation of the equation in general leads to a (positive or negative) monotonic acceleration or an oscillation of the holes. This confirms that the cubic CGLE has an inner symmetry. As a consequence small perturbations change some of the qualitative dynamics of the cubic CGLE and enhance or suppress spatio-temporal intermittency in some parameter range. An analytic stability analysis of holes in the cubic CGLE and a semianalytical treatment of the acceleration instability in the perturbed equation is performed by using matching and perturbation methods. Furthermore we treat the asymptotic hole-shock interaction. The results, which can be obtained fully analytically in the nonlinear Schro¨dinger limit, are also used for the quantitative description of modulated solutions made up of periodic arrangements of traveling holes and shocks.
 
Article
We reformulate the 1D complex Ginzburg–Landau equation as a fourth-order ordinary differential equation in order to find stationary spatially periodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.
 
Article
The stability properties of certain simple periodic solutions (nonlinear modes) of the equations of motion of one-dimensional, N-particle FPU lattices, are obtained analytically by uncoupling the N linear variational equations. The energy per particle Ec/N at which these modes first become unstable is calculated and its asymptotic behavior as N→∞ is determined. We find that for lattices which experience strong energy sharing Ec/N →0 as N →∞, while for a lattice where little energy sharing is observed Ec/N →const > 0, as N increases. Certain possible connections between our local stability results and some global chaotic properties of FPU lattices are discussed.
 
Article
We investigate the exact bright and dark solitary wave solutions of an effective one dimensional (1D) Bose-Einstein condensate (BEC) by assuming that the interaction energy is much less than the kinetic energy in the transverse direction. In particular, following the earlier works in the literature P\'erez-Garc\'ia et al. [Physica D 221 (2006) 31], Serkin et al. [Phys. Rev. Lett. 98 (2007) 074102], G\"urses [arXiv:0704.2435] and Kundu [Phys. Rev. E 79 (2009) 015601], we point out that the effective 1D equation res ulting from the Gross-Pitaevskii (GP) equation can be transformed into the stand ard soliton (bright/dark) possessing, completely integrable 1D nonlinear Schr\"o dinger (NLS) equation by effecting a change of variables of the coordinates and the wave function. We consider both confining and expulsive harmonic trap potentials separately and treat the atomic scattering length, gain/loss term and trap frequency as the experimental control parameters by modulating them as a function of time. In the case when the trap frequency is kept constant, we show the existence of different kinds of soliton solutions, such as the periodic oscillating solitons, collapse and revival of condensate, snake-like solitons, stable solitons, soliton growth and decay and formation of two-soliton like bound state, as the atomic scattering length and gain/loss term are varied. However when the trap frequency is also modulated, we show the phenomena of collapse and revival of two-soliton like bound state formation of the condensate for double modulated periodic potential and bright and dark solitons for step-wise modulated potentials. Comment: 33 pages, 20 figures, 4 tables
 
Article
Mechanisms for initiating rotating waves in 1D and 2D excitable media were compared and parameters affecting wavefront formation were analyzed. The time delay between two sequentially initiated wavefronts (a conditioning wave followed by a test wave) was varied in order to induce rotating waves, a protocol similar to that utilized in cardiac muscle experiments to reveal vulnerability to rotating wave initiation.We define the vulnerability region, VR, as the range of time delays between conditioning and test waves where the test waves evolves into a rotating wave. The smaller the VR, the more resistant the heart is against origination of dangerous cardiac arrhythmias. Heterogeneity of cardiac muscle is widely recognized as the prerequisite for rotating wave initiation. We have identified the VR in homogeneous 2D excitable media. In the Belousov-Zhabotinsky (BZ) reaction with immobilized catalyst and in the Oregonator model of this reaction, a properly timed test wave gives rise to rotating waves. The VR was increased when the size of the perturbation used for test wave creation was increased or when the threshold for propagation was decreased. Increasing the dimensionality of the medium for 1D to 2D results in diminishing of VR.
 
Article
A scattering problem (or more precisely, a transmission-reflection problem) of linearized excitations in the presence of a dark soliton is considered in a one-dimensional nonlinear Schr\"odinger system with a general nonlinearity: $ \mathrm{i}\partial_t \phi = -\partial_x^2 \phi + F(|\phi|^2)\phi $. If the system is interpreted as a Bose-Einstein condensate, the linearized excitation is a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation. We exactly prove that the perfect transmission of the zero-energy phonon is suppressed at a critical state determined by Barashenkov's stability criterion [Phys. Rev. Lett. 77, (1996) 1193.], and near the critical state, the energy-dependence of the reflection coefficient shows a saddle-node type scaling law. The analytical results are well supported by numerical calculation for cubic-quintic nonlinearity. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. As a by-product of the proof, we also give all exact zero-energy solutions of the Bogoliubov equation and their finite energy extension.
 
Article
We study the dynamics of the one-dimensional complex Ginzburg–Landau equation (CGLE) in the regime where holes and defects organize themselves into composite superstructures which we call zigzags. Extensive numerical simulations of the CGLE reveal a wide range of dynamical zigzag behavior which we summarize in a “phase diagram”. We have performed a numerical linear stability and bifurcation analysis of regular zigzag structures which reveals that traveling zigzags bifurcate from stationary zigzags via a pitchfork bifurcation. This bifurcation changes from supercritical (forward) to subcritical (backward) as a function of the CGLE coefficients, and we show the relevance of this for the “phase diagram”. Our findings indicate that in the zigzag parameter regime of the CGLE, the transition between defect-rich and defect-poor states is governed by bifurcations of the zigzag structures.
 
Article
A linear system $\dot x = Ax$, $A \in \mathbb{R}^{n \times n}$, $x \in \mathbb{R}^n$, with $\mathrm{rk} A = n-1$, has a one-dimensional center manifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. If a differential equation $\dot x = f(x)$ has a one-dimensional center manifold $W^c$ at an equilibrium $x^*$ then $E^c$ is tangential to $W^c$ with $A = Df(x^*)$ and for stability of $W^c$ it is necessary that $A$ has no spectrum in $\mathbb{C}^+$, i.e.\ if $A$ is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to $A$, we formulate meso-scale conditions with certain principal minors of $A$ which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.
 
Article
We present experimental results on hydrothermal traveling waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean wavenumber of the waves, we produce modulated wave patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern breaks by traveling holes and spatio-temporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg–Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [N. Garnier, A. Chiffaudel, F. Daviaud, Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions, Physica D (2003), this issue] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.
 
Article
In this work, we analyze the linear stability of singular homoclinic stationary solutions and spatially-periodic stationary solutions in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of selfreplicating pulses. For each solution constructed in [5], we analytically find a large open region in the space of the two scaled parameters in which it is stable. Specifically, for each value of the scaled inhibitor feed rate, there exists an interval, whose length and location depend on the solution type, of values of the activator (autocatalyst) decay rate for which the solution is stable. The upper boundary of each interval corresponds to a subcritical Hopf bifurcation point, and the lower boundary is explicitly determined by finding the parameter value where the solution `disappears,' i.e., below which it no longer exists as a solution of the steady state system.
 
Article
We begin with a brief review of the ionization of 3D hydrogen atoms with large principal quantum number n0, first by a static electric field Fs and then by linearly polarized (LP) electric field. Near its onset, LP ionization can be understood with (1D + time) theory. Various kinds of resonant phenomena are important. We continue with a brief review of the polarization dependence. When the dynamics is dominated by the main pendulum-like resonance zone, a separation of timescales leads to ionization near onset that is independent of polarization. In other parameter ranges, polarization-dependent effects occur that can be understood only in higher dimensions, minimally (2D + time). Finally, we present preliminary results from new experiments using collinear LP and Fs fields, both of which can be strong. The data show the importance of (a) multiphoton resonances driven between Stark substates of the initial n0 manifold and (b) striking, regular oscillations recorded for fixed microwave parameters as a function of Fs. The mechanisms responsible for (a) are understood. Those responsible for (b) are not, but the oscillations exhibit empirical scaling behavior that will help to unravel the wave packet dynamics.
 
Article
The transfer matrix method is applied to finite quasi-1D disordered samples attached to perfect leads. The model is described by structured band matrices with random and regular entries. We investigate numerically the level-spacing distribution for finite-length Lyapunov exponents as well as the conductance and its fluctuations for different channel numbers and sample sizes. A comparison is made with theoretical predictions and with numerical results recently obtained with the scattering matrix approach. The role of the coupling and finite size effects is also discussed.
 
Article
One model of randomness observed in physical systems is that low-dimensional deterministic chaotic attractors underly the observations. A phenomenological theory of chaotic dynamics requires an accounting of the information flow from the observed system to the observer, the amount of information available in observations, and just how this information affects predictions of the system's future behavior. In an effort to develop such a description, we discuss the information theory of highly discretized observations of random behavior. Metric entropy and topological entropy are well-defined invariant measures of such an attractor's “level of chaos”, and are computable using symbolic dynamics. Real physical systems that display low dimensional dynamics are, however, inevitably coupled to high-dimensional randomless, e.g. thermal noise. We investigate the effects of such fluctuations coupled to deterministic chaotic systems, in particular, the metric entropy's response to the fluctuations. We find that the entropy increases with a power law in the noise level, and that the convergence of the entropy and the effect of fluctuations can be cast as a scaling theory. We also argue that in addition to the metric entropy, there is a second scaling invariant quantity that characterizes a deterministic system with added fluctuations: I0, the maximum average information obtainable about the initial condition that produces a particular sequence of measurements (or symbols).
 
Article
The nature of high-dimensional chaos exhibited by a class of delay-differential equation is investigated by various methods. This delay-differential equation models systems with time delayed feedback such as nonlinear optical resonators. We first describe briefly the bifurcation phenomena exhibited by the system: With an increase in a control parameter representing the energy flow rate, periodic states bi(multi)furcate themselves successively, forming a hierarchy of multistable periodic states with increasing complexities. Finally each branch of multistable periodic states makes transition to chaos. The possibility of applying such a multistability as a memory device for complicated information is discussed. In the chaotic regime each of the bifurcated states, in turn, merge successively into fewer sets of states with larger attractor dimensions, and finally a single developed chaos with a very large attractor dimension is formed. Lyapunov analysis is introduced to study the high-dimensional chaotic states. The Lyapunov vectors as well as Lyapunov spectrum are shown to be very useful to understand the underlying mechanism of the successive merging process mentioned above. Characteristics of developed chaos are investigated by high-pass filtered time series (HFTS). The intermittency characteristics of the HFTS changes markedly at a certain frequency. The Lyapunov analysis reveals that this frequency corresponds to a characteristic Lyapunov mode number. This characteristic number can be looked upon as the dimension of subspace in which active chaotic information is generated and is different from the attractor dimension in the customary sense.
 
Article
In this paper we consider the influence of static and dynamic random perturbations of the properties of reactive media on the formation of structures and propagation of fronts. First we study the front and nuclei dynamics in uniform media. For systems with two order parameters the existence of stable nuclei is shown. Then we describe several effects on propagating fronts and nucleation dynamics resulting from random perturbations.
 
Article
The self-focusing singularity of the attractive 2D cubic Schrödinger equation arises in nonlinear optics and many other situations, including certain models of Bose–Einstein condensates. This 2D case is very sensitive to perturbations of the equation and so solutions can be regularized in a number of ways. Here the effect of linear potentials is considered, such as could arise in models of optical fibres with narrow cores of different refractive index, wave-guides induced in a nonlinear medium by another beam, and as part of the Gross–Pitaevskii model of Bose–Einstein condensates. It is observed that in critical dimension only, one can have inhibition of collapse by attractive linear potentials, without dissipation, and that this can lead to a stable oscillating beam, as opposed to the dispersion or dissipation seen with previously studied regularizing mechanisms.
 
Article
We adapt the formalism of the statistical theory of 2D turbulence to the case where the Casimir constraints are replaced by the specification of a prior vorticity distribution. A phenomenological relaxation equation is obtained for the evolution of the coarse-grained vorticity. This equation monotonically increases a generalized entropic functional (determined by the prior) while conserving circulation and energy. It can be used as a thermodynamical parametrization of forced 2D turbulence, or as a numerical algorithm for constructing (i) arbitrary statistical equilibrium states in the sense of Ellis, Haven and Turkington, (ii) particular statistical equilibrium states in the sense of Miller, Robert and Sommeria, (iii) arbitrary stationary solutions of the 2D Euler equation that are formally nonlinearly dynamically stable according to the Ellis–Haven–Turkington stability criterion refining the Arnold theorems.
 
Article
We consider two particular applications of a multispecies, multispeed lattice gas with energy levels. In the first one, we study the mass diffusion properties of a model where the collisions preserved the partial masses. In the second one, we are looking at heat diffusion for a model where the total mass, momentum and energy are the only conserved quantities. The diffusion equations are derived by the Chapman-Enskog method and some numerical simulations are presented.
 
Top-cited authors
Leonid Rudin
  • Cognitech
Stanley Osher
  • University of California, Los Angeles
Harry Swinney
  • University of Texas at Austin
Alan Wolf
  • The Cooper Union for the Advancement of Science and Art
Peter Grassberger
  • The University of Calgary