Symmetry breaking, coupling management, and localized modes in dual-core discrete nonlinear-Schr\"{o}dinger lattices

We introduce a system of two linearly coupled discrete nonlinear Schr\"{o}dinger equations (DNLSEs), with the coupling constant subject to a rapid temporal modulation. The model can be realized in bimodal Bose-Einstein condensates (BEC). Using an averaging procedure based on the multiscale method, we derive a system of averaged (autonomous) equations, which take the form of coupled DNLSEs with additional nonlinear coupling terms of the four-wave-mixing type. We identify stability regions for fundamental onsite discrete symmetric solitons (single-site modes with equal norms in both components), as well as for two-site in-phase and twisted modes, the in-phase ones being completely unstable. The symmetry-breaking bifurcation, which destabilizes the fundamental symmetric solitons and gives rise to their asymmetric counterparts, is investigated too. It is demonstrated that the averaged equations provide a good approximation in all the cases. In particular, the symmetry-breaking bifurcation, which is of the pitchfork type in the framework of the averaged equations, corresponds to a Hopf bifurcation in terms of the original system.Comment: 6 pages, 3 figure

A delay-dependent LMI approach to dynamics analysis of discrete-time recurrent neural networks with time-varying delays

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Elsevier Ltd.In this Letter, the analysis problem for the existence and stability of periodic solutions is investigated for a class of general discrete-time recurrent neural networks with time-varying delays. For the neural networks under study, a generalized activation function is considered, and the traditional assumptions on the boundedness, monotony and differentiability of the activation functions are removed. By employing the latest free-weighting matrix method, an appropriate Lyapunov–Krasovskii functional is constructed and several sufficient conditions are established to ensure the existence, uniqueness, and globally exponential stability of the periodic solution for the addressed neural network. The conditions are dependent on both the lower bound and upper bound of the time-varying time delays. Furthermore, the conditions are expressed in terms of the linear matrix inequalities (LMIs), which can be checked numerically using the effective LMI toolbox in MATLAB. Two simulation examples are given to show the effectiveness and less conservatism of the proposed criteria.This work was supported in part by the National Natural Science Foundation of China under Grant 50608072, an International Joint Project sponsored by the Royal Society of the UK and the National Natural Science Foundation of China, and the Alexander von Humboldt Foundation of Germany

Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation

We demonstrate how time-integration of stochastic differential equations (i.e. Brownian dynamics simulations) can be combined with continuum numerical bifurcation analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) coarse macroscopic equations, estimating the necessary quantities through appropriately initialized, short bursts of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their coarse stability is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated

Return-Map Cryptanalysis Revisited

As a powerful cryptanalysis tool, the method of return-map attacks can be used to extract secret messages masked by chaos in secure communication schemes. Recently, a simple defensive mechanism was presented to enhance the security of chaotic parameter modulation schemes against return-map attacks. Two techniques are combined in the proposed defensive mechanism: multistep parameter modulation and alternative driving of two different transmitter variables. This paper re-studies the security of this proposed defensive mechanism against return-map attacks, and points out that the security was much over-estimated in the original publication for both ciphertext-only attack and known/chosen-plaintext attacks. It is found that a deterministic relationship exists between the shape of the return map and the modulated parameter, and that such a relationship can be used to dramatically enhance return-map attacks thereby making them quite easy to break the defensive mechanism.Comment: 11 pages, 7 figure

Discrete breathers in dissipative lattices

We study the properties of discrete breathers, also known as intrinsic localized modes, in the one-dimensional Frenkel-Kontorova lattice of oscillators subject to damping and external force. The system is studied in the whole range of values of the coupling parameter, from C=0 (uncoupled limit) up to values close to the continuum limit (forced and damped sine-Gordon model). As this parameter is varied, the existence of different bifurcations is investigated numerically. Using Floquet spectral analysis, we give a complete characterization of the most relevant bifurcations, and we find (spatial) symmetry-breaking bifurcations which are linked to breather mobility, just as it was found in Hamiltonian systems by other authors. In this way moving breathers are shown to exist even at remarkably high levels of discreteness. We study mobile breathers and characterize them in terms of the phonon radiation they emit, which explains successfully the way in which they interact. For instance, it is possible to form ``bound states'' of moving breathers, through the interaction of their phonon tails. Over all, both stationary and moving breathers are found to be generic localized states over large values of $C$, and they are shown to be robust against low temperature fluctuations.Comment: To be published in Physical Review

Criticality and Bifurcation in the Gravitational Collapse of a Self-Coupled Scalar Field

We examine the gravitational collapse of a non-linear sigma model in spherical symmetry. There exists a family of continuously self-similar solutions parameterized by the coupling constant of the theory. These solutions are calculated together with the critical exponents for black hole formation of these collapse models. We also find that the sequence of solutions exhibits a Hopf-type bifurcation as the continuously self-similar solutions become unstable to perturbations away from self-similarity.Comment: 18 pages; one figure, uuencoded postscript; figure is also available at http://www.physics.ucsb.edu/people/eric_hirschman