Heteroclinic dynamics in a model of Faraday waves in a square container

We study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross-section and single-frequency forcing. These periodic orbits correspond to quasiperiodic surface waves in the physical system. The heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable Hamiltonian limit, some of which persist in the presence of small damping. The dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate Poincaré maps, with predictions that agree with numerical computations. The results suggest a great richness of possible dynamics of Faraday waves even in simple geometries and with single-frequency forcing.     

Cardiac arrhythmias and circle maps-A classical problem

The periodic forcing of nonlinear oscillations can often be cast as a problem involving self-maps of the circle. Consideration of the effects of changes in the frequency and amplitude of the periodic forcing leads to a problem involving the bifurcations of circle maps in a two-dimensional parameter space. The global bifurcations in this two-dimensional parameter space is described for periodic forcing of several simple theoretical models of nonlinear oscillations. As was originally recognized by Arnold, one motivation for the formulation of these models is their connection with theoretical models of cardiac arrhythmias originating from the competition and interaction between two pacemakers for the control of the heart.     

Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999     

耦合电路中的复杂振荡行为分析

讨论了两个非线性电路适当连接后的耦合系统随耦合强度变化的演化过程.给出了两子系统各自的分岔行为及通向混沌的过程,指出原子系统均为周期运动时,耦合系统依然会由倍周期分岔进入混沌,同时在混沌区域中存在有周期急剧增加及周期增加分岔等现象.而当周期运动和混沌振荡相互作用时,在弱耦合条件下,受混沌子系统的影响,原周期子系统会在其原先的轨道邻域内作微幅振荡,其振荡幅值随耦合强度的增加而增大,混沌的特征越加明显,相反,周期子系统不仅可以导致混沌子系统的失稳,也会引起混沌吸引子结构的变化. 关键词： 非线性电路 耦合强度 分岔 混沌     

From multi-layered resonance tori to period-doubled ergodic tori

The Letter presents a number of new bifurcation structures that can be observed when a multi-dimensional period-doubling system is subjected to a periodic forcing. We show how multi-layered tori arise through transverse period-doubling bifurcations of the resonant saddle and node cycles, and how these multi-layered tori transform into period-doubled ergodic tori through sets of saddle-node bifurcations.     

Painleve analysis of the perturbed sine-Gordon equation

We are reporting on numerical investigations of a seven-variable model corresponding to a class of chemical reactions which exhibit, as a function of the control parameter, a sequence of periodic and chaotic states strikingly similar to that observed in bench experiments. This scenario involves period-doubling cascades, tangent bifurcations and intermittency, in good agreement with a dynamical evolution predicted by a multi-humped one-dimensional map. This strongly suggests an interpretation of the strange-attractor-like behavior observed along such paths, in terms of the chaotic behavior which occurs nearby homoclinic conditions.     

Forcing and control of localized states in optical single feedback systems

Under feedback extended nonlinear optical systems spontaneously show a variety of periodic patterns and structures. Control gives new insight into these phenomena and it can open the way for potential application of nonlinear optical structures. We briefly review methods to control localized states in single feedback experiments. Application of a Fourier control method allows to modify interaction behavior of the localized states. As a further approach we study a forcing method, using externally created light fields as additional input to the system. Recent experiments show that the forcing method enables to favor addressing positions for localized structures. We demonstrate static addressing and favoring of addressing positions. We extend the forcing method to a dynamic forcing scheme, which allows to move and reposition localized states. Additionally forcing is used to balance experimental imperfections. PACS 05.45.Gg; 42.60.Jf; 42.65.Tg     

Homoclinic bifurcation in Chua’s circuit

We report our experimental observations of the Shil’nikov-type homoclinic chaos in asymmetry-induced Chua’s oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcations. The asymmetry is introduced in the circuit by forcing a DC voltage. For a selected asymmetry, when a system parameter is controlled, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of periodic mixed-mode oscillations interspersed by chaotic states. Moreover, we observed two intermediate bursting regimes. Experimental evidences of homoclinic chaos are verified with PSPICE simulations.     

The dynamics of a symmetric coupling of three modified quadratic maps

We investigate the dynamical behavior of a symmetric linear coupling of three quadratic maps with exponential terms, and identify various interesting features as a function of two control parameters. In particular, we investigate the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable period-l, period-2, and period-3 orbits. We also investigate the multistability in the same coupling. Lyapunov exponents, parameter planes, phase space portraits, and bifurcation diagrams are used to investigate transitions from periodic to quasiperiodic states, from quasiperiodic to mode-locked states and to chaotic states, and from chaotic to hyperchaotic states.     

Optimal subsystem approach to multi-qubit quantum state discrimination and experimental investigation

Quantum computing is a significant computing capability which is superior to classical computing because of its superposition feature. Distinguishing several quantum states from quantum algorithm outputs is often a vital computational task. In most cases, the quantum states tend to be non-orthogonal due to superposition; quantum mechanics has proved that perfect outcomes could not be achieved by measurements, forcing repetitive measurement. Hence, it is important to determine the optimum measuring method which requires fewer repetitions and a lower error rate. However, extending current measurement approaches mainly aiming at quantum cryptography to multi-qubit situations for quantum computing confronts challenges, such as conducting global operations which has considerable costs in the experimental realm. Therefore, in this study, we have proposed an optimum subsystem method to avoid these difficulties. We have provided an analysis of the comparison between the reduced subsystem method and the global minimum error method for two-qubit problems; the conclusions have been verified experimentally. The results showed that the subsystem method could effectively discriminate non-orthogonal two-qubit states, such as separable states, entangled pure states, and mixed states; the cost of the experimental process had been significantly reduced, in most circumstances, with acceptable error rate. We believe the optimal subsystem method is the most valuable and promising approach for multi-qubit quantum computing applications.     

Coherence and stochastic resonance in threshold crossing detectors with delayed feedback

We consider the dynamical behavior of threshold systems driven by external periodic and stochastic signals and internal delayed feedback. Specifically, the effect of positive delayed feedback on the sensitivity of a threshold crossing detector (TCD) to periodic forcing embedded in noise is investigated. The system has an intrinsic ability to oscillate in the presence of positive feedback. We first show conditions under which such reverberatory behavior is enhanced by noise, which is a form of coherence resonance (CR) for this system. Further, for input signals that are subthreshold in the absence of feedback, the open-loop stochastic resonance (SR) characteristic can be sharply enhanced by positive delayed feedback. This enhancement is shown to depend on the stimulus period, and is maximal when this period is matched to an integer multiple of the delay. Reverberatory oscillations, which are particularly prominent after the offset of periodic forcing, are shown to be eliminated by a summing network of such TCDs with local delayed feedback. Theoretical analysis of the crossing rate dynamics qualitatively accounts for the existence of CR and the resonant behavior of the SR effect as a function of delay and forcing frequency.     

Three types of chaos in the forced nonlinear Schrödinger equation

Three different types of chaotic behavior and instabilities (homoclinic chaos, hyperbolic resonance, and parabolic resonance) in Hamiltonian perturbations of the nonlinear Schr?dinger (NLS) equation are described. The analysis is performed on a truncated model using a novel framework in which a hierarchy of bifurcations is constructed. It is demonstrated numerically that the forced NLS equation exhibits analogous types of chaotic phenomena. Thus, by adjusting the forcing frequency, the behavior near the plane wave solution may be set to any one of the three different types of chaos for any periodic box length.     

Spatial Unfolding of Elementary Bifurcations

We consider solutions of a partial differential equation which are homogeneous in space and stationary or periodic in time. We study the stability with respect to large wavelength perturbations and the weakly nonlinear behavior around these solutions, especially when they are close to bifurcations for the ordinary differential equation governing the homogeneous solutions of the PDE. We distinguish cases where a spatial parity symmetry holds. All bifurcations occurring generically for two-dimensional ODES are treated. Our main result is that for almost homoclinic periodic solutions instability is generic.     

Characterizing bifurcations and chaos in multiwavelength lasers with intensity-dependent loss and saturable homogeneous gain

We consider the nonlinear dynamics of multiwavelength laser cavities with saturable transmitter and saturating homogeneous gain using a simple and general discrete model. Saturable transmitter is an intensity dependent loss in which the transmittance decreases when the incident optical power increases. We determine the condition under which the saturable transmitter will generate behaviors such as stable steady-state lasing states, periodic lasing states, and chaotic lasing states. Indeed, for sufficiently large power, steady-state operation is first destabilized through a Hopf bifurcation which generates periodic lasing solutions. This is followed by a sequence of period doubling bifurcations to chaotic lasing. The bifurcation structure leading to chaos is characterized by three key methods of dynamical systems: a Feigenbaum series, the calculation of Lyapunov exponents and the computation of the correlation dimension of the system. We found that even single wavelength operation can exhibit complex nonlinear dynamics if the loss element is a saturable transmitter.     

Strange Attractors in Periodically-Kicked Degenerate Hopf Bifurcations

We prove that spiral sinks (stable foci of vector fields) can be transformed into strange attractors exhibiting sustained, observable chaos if subjected to periodic pulsatile forcing. We show that this phenomenon occurs in the context of periodically-kicked degenerate supercritical Hopf bifurcations. The results and their proofs make use of a k-parameter version of the theory of rank one maps.     

Collections of heteroclinic cycles in the Kuramoto-Sivashinsky equation

We study the Kuramoto-Sivashinky equation with periodic boundary conditions in the case of low-dimensional behavior. We analyze the bifurcations that occur in a six-dimensional (6D) approximation of its inertial manifold. We mainly focus on the attracting and structurally stable heteroclinic connections that arise for these parameter values. We reanalyze the ones that were previously described via a 4D reduction to the center-unstable manifold (Ambruster et al., 1988, 1989). We also find a parameter region for which a manifold of structurally stable heteroclinic cycles exist. The existence of such a manifold is responsible for an intermittent behavior which has some features of unpredictability.     

Clustering of arrays of chaotic chemical oscillators by feedback and forcing

Feedback and external forcing are applied to an array of chaotic electrochemical oscillators through variations in the applied potential. We see transitions from intermittent clusters to stable chaotic clusters to stable periodic clusters to synchronized states as the feedback gain and forcing amplitude, respectively, are varied. With forcing up to four clusters are observed in stable states. The transition to synchronization with feedback occurs by the increase in the size of one cluster at the expense of the others.     

Chaotic Motion in a Harmonically Excited Soliton System

The influence of a soliton system under an external harmonic excitation is considered. We take the compound KdV-Burgers equation as an example, and investigate numerically the chaotic behavior of the system with a periodic forcing. Different routes to chaos such as period doubling, quasi-periodic routes, and the shapes of strange maps.