On the effect of the noise in dynamical systems

Summary We discuss the complexity in dynamical systems with a random perturbation by considering the rateK of divergence of nearby orbits evolving under two different noise realization. This quantity has a clear meaning in the context of the information theory and it has physical relevance for the analysis of experimental data. Our definition of complexity becomes crucial for strongly intermittent systems whereK is very different from the Lyapunov exponent. Paper presented at the International Workshop ?Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena?, Elba, 5–10 June 1994.     

Wave chaotic behaviour generated by linear systems

It is shown that regimes with dynamical chaos are inherent not only to nonlinear system but they can be generated by initially linear systems and the requirements for chaotic dynamics and characteristics need further elaboration. Three simplest physical models are considered as examples. In the first, dynamic chaos in the interaction of three linear oscillators is investigated. Analogous process is shown in the second model of electromagnetic wave scattering in a double periodical inhomogeneous medium occupying half-space. The third model is a linear parametric problem for the electromagnetic field in homogeneous dielectric medium which permittivity is modulated in time.     

Equivalent linearization finds nonzero frequency corrections beyond first order

We show that the equivalent linearization technique, when used properly, enables us to calculate frequency corrections of weakly nonlinear oscillators beyond the first order in nonlinearity. We illustrate the method by applying it to the conservative anharmonic oscillators and the nonconservative van der Pol oscillator that are respectively paradigmatic systems for modeling center-type oscillatory states and limit cycle type oscillatory states. The choice of these systems is also prompted by the fact that first order frequency corrections may vanish for both these types of oscillators, thereby rendering the calculation of the higher order corrections rather important. The method presented herein is very general in nature and, hence, in principle applicable to any arbitrary periodic oscillator.     

Finite-time stabilization of three-dimensional chaotic systems based on CLF

This Letter investigates the stabilization of three-dimensional chaotic systems in a finite time. Based on the finite-time stability theory, a control law is proposed to realize finite-time chaos stabilization of three-dimensional chaotic systems. Several illustrative examples with numerical simulations are studied by using the results obtained in this Letter. Study of examples shows that our control methods work very well in stabilizing a class of chaotic systems in a finite time.     

Experiments on node-to-node pinning control of Chua’s circuits

In this paper, we study the global intermittent pinning controllability of networks of coupled chaotic oscillators. We explore the feasibility of the recently presented node-to-node pinning control strategy through experiments on Chua’s circuits. We focus on the case of two peer-to-peer coupled Chua’s circuits and we build a novel test-bed platform comprised of three inductorless Chua’s oscillators. We investigate the effect of a variety of design parameters on synchronization performance, including the coupling strength between the oscillators, the control gains, and the switching frequency of node-to-node pinning control. Experimental results demonstrate the effectiveness of this novel pinning control strategy in rapidly taming chaotic oscillator dynamics onto desired reference trajectories while minimizing the overall control effort and the number of pinned network sites. From an analytical standpoint, we present sufficient conditions for global node-to-node pinning controllability and we estimate the maximum switching period for network controllability by adapting and integrating available results on Lyapunov stability theory and partial averaging techniques.     

Persistent spin oscillations in a spin-orbit-coupled superconductor

Quasi-two-dimensional superconductors with tunable spin-orbit coupling are very interesting systems with properties that are also potentially useful for applications. In this Letter we demonstrate that these systems exhibit undamped collective spin oscillations that can be excited by the application of a supercurrent. We propose to use these collective excitations to realize persistent spin oscillators operating in the frequency range of 10 GHz-1 THz.     

Automatic control of phase synchronization in coupled complex oscillators

We present an automatic control method for phase locking of regular and chaotic nonidentical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic Rössler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic Rössler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Step Density Profiles in Localized Chains

We consider two types of strongly disordered one-dimensional Hamiltonian systems coupled to baths (energy or particle reservoirs) at the boundaries: strongly disordered quantum spin chains and disordered classical harmonic oscillators. These systems are believed to exhibit localization, implying in particular that the conductivity decays exponentially in the chain length L. We ask however for the profile of the (very slowly) transported quantity in the steady state. We find that this profile is a step-function, jumping in the middle of the chain from the value set by the left bath to the value set by the right bath. This is confirmed by numerics on a disordered quantum spin chain of 9 spins and on much longer chains of harmonic oscillators. From theoretical arguments, we find that the width of the step grows not faster than \(\sqrt{L}\), and we confirm this numerically for harmonic oscillators. In this case, we also observe a drastic breakdown of local equilibrium at the step, resulting in a heavily oscillating temperature profile.     

Nonuniform energy level broadening in open quantum dots: the influence of the closed dot eigenstates on transport

We present simulations of a realistically modeled quantum dot with soft boundaries and open leads supporting several modes. The wavefunctions of this open dot can be decomposed in terms of the eigenstates of a corresponding closed dot my means of projection. At particular resonances, this decomposition reveals that transport in the open dot can be mediated through a single eigenstate. Thus, the quantization of closed-dot energy levels can be preserved even as the dot is opened, but we find that there is a selection of particular eigenstates that depends strongly on the positions and nature of the contacts.     

耦合振子系统的多稳态同步分析

本文讨论了一维闭合环上Kuramoto相振子在非对称耦合作用下同步区域出现的多定态现象. 研究发现在振子数N≤3情形下系统不会出现多态现象, 而N≥4多振子系统则呈现规律的多同步定态. 我们进一步对耦合振子系统中出现的多定态规律及定态稳定性进行了理论分析, 得到了定态渐近稳定解. 数值模拟多体系统发现同步区特征和理论描述相一致. 研究结果显示在绝热条件下随着耦合强度的减小, 系统从不同分支的同步态出发最终会回到同一非同步态. 这说明, 耦合振子系统在非同步区由于运动的遍历性而只具有单一的非同步态, 在发生同步时由于遍历性破缺会产生多个同步定态的共存现象.     

Coherence resonance in coupled chaotic oscillators

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.     

Synchronization of oscillators coupled through an environment

We study synchronization of oscillators that are indirectly coupled through their interaction with an environment. We give criteria for the stability or instability of a synchronized oscillation. Using these criteria we investigate synchronization of systems of oscillators which are weakly coupled, in the sense that the influence of the oscillators on the environment is weak. We prove that arbitrarily weak coupling will synchronize the oscillators, provided that this coupling is of the ‘right’ sign. We illustrate our general results by applications to a model of coupled GnRH neuron oscillators proposed by Khadra and Li [A. Khadra, Y.X. Li, A model for the pulsatile secretion of gonadotropin-releasing hormone from synchronized hypothalamic neurons, Biophys. J. 91 (2006) 74-83.], and to indirectly weakly-coupled λ-ω oscillators.     

Self-synchronization of coupled oscillators with hysteretic responses

We analyze a large system of nonlinear phase oscillators with sinusoidal nonlinearity, uniformly distributed natural frequencies and global all-to-all coupling, which is an extension of Kuramoto's model to second-order systems. For small coupling, the system evolves to an incoherent state with the phases of all the oscillators distributed uniformly. As the coupling is increased, the system exhibits a discontinuous transition to the coherently synchronized state at a pinning threshold.of the coupling strength, or to a partially synchronized oscillation coherent state at a certain threshold below the pinning threshold. If the coupling is decreased from a strong coupling with all the oscillators synchronized coherently, this coherence can persist until the depinning threshold which is less than the pinning threshold, resulting in hysteretic synchrony depending on the initial configuration of the oscillators. We obtain analytically both the pinning and depinning threshold and also expalin the discontinuous transition at the thresholds for the underdamped case in the large system size limit. Numerical exploration shows the oscillatory partially coherent state bifurcates at the depinning threshold and also suggests that this state persists independent of the system size. The system studied here provides a simple model for collective behaviour in damped driven high-dimensional Hamiltonian systems which can explain the synchronous firing of certain fireflies or neural oscillators with frequency adaptation and may also be applicable to interconnected power systems.     

Entanglement production in quantized chaotic systems

Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems.     

A W-Band Frequency Tracing Transceiver System

In millimeter-wave, phase-locked oscillators are often used because the frequency drift of oscillators is very large. However, they are very complex. In this paper, a W-band frequency tracing transceiver system is presented and tested, which is used for digital communication. Although the communication distance is less than that of phase-locked systems, the scheme is simple and the cost is low.     

Integrability lost: Chaotic dynamics of classical strings on a confining holographic background

It is known that classical string dynamics on pure AdS₅×S⁵${\mathit{AdS}}_5\times S^5$ is integrable and plays an important role in solvability. This is a deep and central issue in holography. Here we investigate similar classical integrability for a more realistic confining background and provide a negative answer. The dynamics of a class of simple string configurations on AdS soliton background can be mapped to the dynamics of a set of non-linearly coupled oscillators. In a suitable limit of small fluctuations we discuss a quasi-periodic analytic solution of the system. Numerics indicates chaotic behavior as the fluctuations are not small. Integrability implies the existence of a regular foliation of the phase space by invariant manifolds. Our numerics shows how this nice foliation structure is eventually lost due to chaotic motion. We also verify a positive Lyapunov index for chaotic orbits. Our dynamics is roughly similar to other known non-integrable coupled oscillator systems like Hénon–Heiles equations.     

Controlling chaotic oscillators by altering their energy

We developed the control technique for nonlinear oscillators when simple feedback alters the oscillation energy. The control does not require any computation nor knowledge of the system equations. Depending on the control perturbation parameters, two scenarios take place: (i) changing (linearly or nonlinearly) the oscillator damping, and (ii) suppressing (enhancing) the driving force. In the case of large deviations between phases of the chaotic oscillator and the driving force, only first scenario holds. Generalization of the technique to the case of oscillator networks is considered.     

Coulomb blockade in single tunnel-junctions: Quantum mechanical effects of the electromagnetic environment

We discuss the interaction of a tunneling electron with its equilibrium electromagnetic environment. The environment of an isolated tunnel junction is modeled by a set of harmonic oscillators that are suddenly displaced when an electron tunnels across the junction. We treat these displaced oscillators quantum mechanically, predicting behavior that is very different than that predicted by a semiclassical treatment. In particular, the shape of the zero-bias anomaly caused by the Coulomb blockade (a single-electron charging effect), is found to be strongly dependent on the impedance,Z (), of the leads connected to the junction. Comparison with three recent experiments demonstrates that the quantum mechanical treatment of this model correctly describes the essential physics in these systems.     

Switching phenomenon induced by breakdown of chaotic phase synchronization

We investigate chaotic phase synchronization (CPS) in three-coupled chaotic oscillator systems. According to the coupling strength and mismatches in the frequencies of these oscillators, we can observe complete CPS where all three oscillators exhibit CPS, and partial CPS where only two oscillators exhibit CPS. When the coupling strength is weakened, we observe a phenomenon that complete CPS among the three oscillators is suddenly disrupted without going through partial CPS. In this case oscillators exhibit quasi-CPS where two oscillators appear to exhibit CPS transiently, and the combination of the two oscillators changes with time. We call this phenomenon CPS switching D. It is revealed that phase fluctuation plays an important role in CPS switching D. It is also shown that the amplitude with a specific structure strengthens the degree of CPS switching. In the present paper, we characterize this CPS switching and discuss its mechanism.