Solitons in coupled nonlinear Schrödinger equations with variable coefficients

We study the coupled nonlinear Schrodinger equation with variable coefficients (VCNLS), which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. By constructing an explicit transformation, which maps VCNLS to the classical coupled nonlinear Schrödinger equations (CNLS), we obtain Bright–Dark and Bright–Bright solitons for VCNLS. Furthermore, the optical super-lattice potentials (or periodic potentials) and hyperbolic cosine potentials with parameters are designed, which are two kinds of important potentials in physics. This method can be used to design a large variety of external potentials in VCNLS, which could be meaningful for manipulating solitons experimentally.     

Identification of Stochastically Perturbed Autonomous Systems from Temporal Sequences of Probability Density Functions

The paper introduces a method for reconstructing one-dimensional iterated maps that are driven by an external control input and subjected to an additive stochastic perturbation, from sequences of probability density functions that are generated by the stochastic dynamical systems and observed experimentally.     

Dynamics of maps with a global multiplicative coupling

The dynamics of coupled logistic maps with a multiplicative coupling is analyzed. We determine the transition to chaos and the multifractal properties of some of the attractors studied in the particular case of only two coupled maps. This transition cannot be deduced from the subharmonic cascade typical of a single map. The results are generalized to an ensemble of globally coupled maps with a similar multiplicative coupling. The global quantities have different attractors depending on the coupling strength and the number of elements in the ensemble.     

“Coherence–incoherence” transition in ensembles of nonlocally coupled chaotic oscillators with nonhyperbolic and hyperbolic attractors

We consider in detail similarities and differences of the “coherence–incoherence” transition in ensembles of nonlocally coupled chaotic discrete-time systems with nonhyperbolic and hyperbolic attractors. As basic models we employ the Hénon map and the Lozi map. We show that phase and amplitude chimera states appear in a ring of coupled Hénon maps, while no chimeras are observed in an ensemble of coupled Lozi maps. In the latter, the transition to spatio-temporal chaos occurs via solitary states. We present numerical results for the coupling function which describes the impact of neighboring oscillators on each partial element of an ensemble with nonlocal coupling. Varying the coupling strength we analyze the evolution of the coupling function and discuss in detail its role in the “coherence–incoherence” transition in the ensembles of Hénon and Lozi maps.     

Generalized synchronization in discrete maps. New point of view on weak and strong synchronization

In the present Letter we show that the concept of the generalized synchronization regime in discrete maps needs refining in the same way as it has been done for the flow systems Koronovskii et al. [Koronovskii AA, Moskalenko OI, Hramov AE. Nearest neighbors, phase tubes, and generalized synchronization. Phys Rev E 2011;84:037201]. We have shown that, in the general case, when the relationship between state vectors of the interacting chaotic maps are considered, the prehistory must be taken into account. We extend the phase tube approach to the systems with a discrete time coupled both unidirectionally and mutually and analyze the essence of the generalized synchronization by means of this technique. Obtained results show that the division of the generalized synchronization into the weak and the strong ones also must be reconsidered. Unidirectionally coupled logistic maps and Hénon maps coupled mutually are used as sample systems.     

How useful randomness for cryptography can emerge from multicore-implemented complex networks of chaotic maps

This article is devoted to the study of several topologies of complex networks of chaotic maps, in order to design new Chaotic Pseudo Random Number Generators for cryptographic purpose in a bottom up approach of doing mathematics. We examine topologies of special 2 -D coupled maps which are built combining piecewise linear and logistic maps. We present also a family of p -Dimensional networks whose we study numerically a particular realization, up to one hundred trillion iterates using multicores computers. From those topologies emerges useful randomness for cryptographic purpose.     

Topological Persistence for Circle-Valued Maps

We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.     

Rate of afferent stimulus dependent synchronization and coding in coupled neurons system

Sequences of intervals between firing times (interspike interval (ISI)) from a pair of locus ceruleus (LC) neurons coupled by axon–dendrite synapse with stimulus of constant and chaos are investigated in this paper. We analyze how the dynamical properties of chaotic input determine those of the output ISI sequences, and assess how various strength of stimulus and coupling affects the input–output relationship. The attractors constructed from delay embeddings of ISIs and of chaotic input are compared from the points of view of geometry and nonlinear dynamics characteristics, i.e., Lyapunov exponent spectrum (LES), Kaplan–York fractal dimension (KY_D) and unstable periodic orbit (UPO). For the coupled LC neurons system investigated, with the moderate strength of stimulus and coupling, the synchronous oscillation of the two neurons is well preserved even if the external stimulus is chaotic; the similarity between these attractors is high only when the afferent stimulus strength is smaller and rate is lower. When these conditions are satisfied, the output two ISI sequences are reciprocally related to input signals, and their oscillation wave shape in time course can be derived from that of the input signals variation, furthermore, the similar input sequence of order of UPOs, distribution of LES and value of KY_D remain in attractors reconstructed from ISI sequences. But these phenomena will disappear in higher rate of stimulus activity or in changing of the strength of stimulus and coupling, for this situation, the ISIs shows bifurcate behavior. These results may be of vital importance for any kind of information processing based on the neurons and temporal coding.     

Vibrational resonance in biological nonlinear maps

We investigate vibrational resonance in two different nonlinear maps driven by a biharmonic force: the Bellows and the Rulkov map. These two maps possess dynamical features of particular interest for the study of these phenomena. In both maps, the resonance occurs at the low-frequency of the biharmonic signal as the amplitude of the high-frequency signal is varied. We also consider an array of unidirectionally coupled maps with the forcing signal applied to the first unit. In this case, a signal propagation with several interesting features above a critical value of the coupling strength is found, while the response amplitude of the ith unit is greater than the first one. This response evolves in a sigmoidal fashion with the system number i, meaning that at some point the amplitudes saturate. The unidirectional coupling acts as a low-pass filter for distant units. Moreover, the analysis of the mean residence time of the trajectory in a given region of the phase space unveils a multiresonance mechanism in the coupled map system. These results point at the relevance of the discrete-time models for the study of resonance phenomena, since analyses and simulations are much easier than for continuous-time models.     

Dynamics of spikes in delay coupled semiconductor lasers

We derive the discrete maps to describe the dynamics of coupled laser diodes. The maps allow us to find analytically regions of parameters and initial conditions in the functional phase space that correspond to spiking with stable (or nearly stable) phase shift. The method developed is promising for further discussion of controlled switching between periodic states by an impulse injection signal.     

Dynamics of a coupled atmosphere–ocean model

We consider a coupled atmosphere–ocean model, which involves hydrodynamics, thermodynamics and nonautonomous interaction at the air–sea interface. First, we show that the coupled atmosphere–ocean system is stable under the external fluctuation in the atmospheric energy balance relation. Then, we estimate the atmospheric temperature feedback in terms of the freshwater flux, heat flux and the external fluctuation at the air–sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Finally, we prove that the coupled atmosphere–ocean system has time-periodic, quasiperiodic and almost periodic motions, whenever the external fluctuation in the atmospheric energy balance relation is time-periodic, quasiperiodic and almost periodic, respectively.     

On–off intermittency and coherent bursting in stochastically-driven coupled maps

On–off intermittency is a phase space mechanism for bursting in dynamical systems. Here we recall how the simple example of a logistic map with a time-dependent control parameter, considered as a dynamical variable of the system, gives rise to bursting or on–off behavior. We show that, for a given realization of the driver, a stochastically driven logistic map in the on–off intermittent regime always converges to the same temporal dynamics, independently of initial conditions. In that sense, the map is not chaotic. We then explore the behavior of two coupled on–off logistic maps, each driven by a separate random process, and show that, for a wide range of coupling strengths, bursting becomes at least partially coherent. The bursting coherence has a smooth dependence on the coupling parameter and no sharp transition from coherence to incoherence is detected. In the system of two coupled on–off maps studied here, coherent bursting is rooted in the behavior during off phases when the mapped coordinates take on extremely small values.     

Maps serving the combined coupling for use in environmental models and their behaviour in the presence of dynamical noise

Many physical, biological as well as the environmental problems, can be described by the dynamics of driven coupled oscillators. In order to study their behaviour as a function of coupling strength and nonlinearity, we considered dynamics of two maps serving the combined coupling (diffusive and linear) in the above fields. Firstly, we have considered a logistic difference equation on extended domain that is a part of the maps, that is discussed using its bifurcation diagram, Lyapunov exponent, sample as well as the permutation entropy. Secondly we have performed the dynamical analysis of the coupled maps using Lyapunov exponent and cross sample entropy in dependence on two coupling parameters. Further, we investigated how dynamical noise can affects the structure of their bifurcation diagrams. It was done (i) by the noise entering in two specific ways, that disturbs either the logistic parameter on extended domain or (ii) by an additive “shock” to the state variables. Finally, we demonstrated the effect of forcing by parametric noise, introduced in all maps’ parameter, on Lyapunov exponent of coupled maps.     

Dynamics of Inhomogeneous Chains of Coupled Quadratic Maps

A new effective local analysis method is elaborated for coupled map dynamics. In contrast to the previously suggested methods, it allows visually investigating the evolution of synchronization and complex-behavior domains for a distributed medium described by a set of maps. The efficiency of the method is demonstrated with examples of ring and flow models of diffusively coupled quadratic maps. An analysis of a ring chain in the presence of space defects reveals some new global-behavior phenomena.     

Stability analysis of mathematical model of competition in a chain of chemostats in series with delay

We study a nonlinear system of differential delay equations describing a model of a chain of two chemostats, where one contains two microbial species in competition for a single limiting nutrient and receives an external input of the less advantaged competitor, which is cultivated in an external chemostat. We obtain sufficient conditions ensuring coexistence of all the species in competition which consist in upper delay bounds.     

Stability and Asymptotic Analysis of a Fluid-Particle Interaction Model

We are interested in coupled microscopic/macroscopic models describing the evolution of particles dispersed in a fluid. The system consists in a Vlasov–Fokker–Planck equation to describe the microscopic motion of the particles coupled to the Euler equations for a compressible fluid. We investigate dissipative quantities, equilibria and their stability properties and the role of external forces. We also study some asymptotic problems, their equilibria and stability and the derivation of macroscopic two-phase models.     

Čech Type Approach to Computing Homology of Maps

A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations, this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of necessary input. We show that the approach is particularly advantageous in the case of homology map algorithms.     

Application of the GALI method to localization dynamics in nonlinear systems

We investigate localization phenomena and stability properties of quasiperiodic oscillations in N$N$ degree of freedom Hamiltonian systems and N$N$ coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of N$N$ coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date.     

Micromechanics of Electro- and Magneto-active Soft Composites

We study the coupled behavior in soft active microstructured materials undergoing large deformations in the presence of an external electric or magnetic field. We focus on the role of the microstructures on the coupled behavior, and examine the phenomenon in the composites with (a) periodic composites with rectangular and hexagonal periodic unit cells, and (b) in composites with the random distributions of active particles embedded in a soft matrix. We show that for these similar microstructures exhibit very different responses in terms of the actuation, and the coupling phenomenon. Next, we consider the macroscopic and microscopic instabilities in the active composites. We show that the external field has a significant influence of the instability phenomena, and can stabilize or destabilize the composites depending on the direction relative to composite geometry. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)