sine-Gordon Chain Research Articles

Output Feedback Control of Sine-Gordon Chain over the Limited Capacity Digital Communication Channel

With the digitalization of mechatronic systems in the conditions of a shortage of available bandwidth of digital communication channels, the problem of ensuring the transfer of information between various components of the system can arise. This problem can be especially challenging in the observation and control of spatially distributed objects due to the complexity of their dynamics, wide frequency band, and other factors. In such cases, a useful approach is to employ smart sensors, in which the measurement results are encoded for transmission over a digital communication channel. Specifically, the article is focused on the transmission of measurement data for the control of energy for a spatially-distributed sine-Gordon chain. The procedures for binary coding of measurements by first- and full-order coder-decoder pairs are proposed and numerically investigated, for each of which the use of stationary and adaptive coding procedures is studied. The procedures for estimating the state of the circuit when measuring outputs are studied, and for each of them, the accuracy of not only estimating the state but also controlling the system by output with the help of an observer is considered. The results of comparative modeling are presented, demonstrating the dependence of the accuracy of estimation and control on the data transfer rate.

Optimal and efficient generation of sine-Gordon breathers.

We investigate analytically and numerically whether an optimally designed defect structure can significantly reduce the threshold for breather train generation by almost two orders of magnitude in a sine-Gordon chain. By optimizing the parameters of defects based on a variational approach, we show that the decoupling of in-phase and antiphase branches allows us to independently minimize the driving threshold and reach the goal of emission of high-amplitude and well-formed breathers by an ultraweak driving. These results not only provide an optimal way for highly controllable and efficient emission of breathers, but also provide some insights into the mechanism of breather excitations in such processes as the DNA-protein interaction during transcription.

Energy transmission in nonlinear chains of harmonic oscillators with long-range interactions

In this work, the phenomenon of supratransmission in nonlinear sine-Gordon chains with global interactions will be predicted analytically for the first time in the literature. The model considered is an extension of the classical sine-Gordon chain which describes the dynamics of pendula attached through strings. Using a relatively simple analytical theory, we derive the threshold that triggers the supratransmission process for relatively high degrees of interaction. The threshold for the classical model is a particular case of our predictions when the degree of interaction tends to infinity. We confirm numerically the validity of our results using a systematic computational approach. We obtain numerically bifurcation diagrams for the supratransmission threshold and compare them with the theoretical predictions, obtaining a good agreement.

Numerical evaluation of sine-Gordon chain energy control via subdomains state feedback under quantizationand time sampling

The paper is devoted to the numerical performance evaluation of the speed-gradient algorithms, recently developed in (Orlov et al., 2018; Orlov et al., 2019) for controlling the energy of the sine-Gordon spatially distributed systems with several in-domain actuators. The influence of the level quantization of the state feedback control signal (possibly coupled to the time sampling) on the steady-state energy error and the closed loop system stability is investigated in the simulation study. The following types of quantization are taken into account: sampling-in-time control signal quantization, the level quantization for control, continuous in time; control signal quantization on level jointly with time sampling; control signal transmission over the binary communication channel with time-invariant first order coder; control signal transmission over the binary communication channel with first order coder and time-based zooming; control signal transmission over the binary communication channel with adaptive first order coder. A resulting impact on the closed-loop performance in question is concluded for each type of the quantization involved.

Shape and symmetry of a fluid-supported elastic sheet

A connection between the dynamics of a sine-Gordon chain and a certain static membrane folding problem was recently found. The one-dimensional membrane profile is a cross section of the position-time sine-Gordon amplitude profile. Here we show that when one system is embedded in a higher-dimensional system in this way, obvious symmetries in the larger system can lead to nontrivial symmetries in the embedded system. In particular, a thin buckled membrane on a fluid substrate has a continuous degeneracy that interpolates between a symmetric and an antisymmetric fold. We find the Hamiltonian generator of this symmetry and the corresponding conserved momentum by interpreting the simple translational symmetries of the sine-Gordon chain in terms of the embedded coordinates. We discuss possible extensions to other embedded dynamical systems.

On the simulation of the energy transmission in the forbidden band-gap of a spatially discrete double sine-Gordon system

In this work, we present a numerical method to consistently approximate solutions of a spatially discrete, double sine-Gordon chain which considers the presence of external damping. In addition to the finite-difference scheme employed to approximate the solution of the difference-differential equations of the model under investigation, our method provides positivity-preserving schemes to approximate the local and the total energy of the system, in such a way that the discrete rate of change of the total energy with respect to time provides a consistent approximation of the corresponding continuous rate of change. Simulations are performed, first of all, to assess the validity of the computational technique against known qualitative solutions of coupled sine-Gordon and coupled double sine-Gordon chains. Secondly, the method is used in the investigation of the phenomenon of nonlinear transmission of energy in double sine-Gordon systems; the qualitative effects of the damping coefficient on the occurrence of the nonlinear process of supratransmission are briefly determined in this work, too.

Noise induced breather generation in a sine–Gordon chain

We consider a sine–Gordon chain driven sinusoidally at one end. In the absence ofnoise, there exists a well known critical value of the amplitude beyond whichbreather modes can be generated via the phenomenon of supratransmission. Weconsider values of the driving amplitude below the critical amplitude such thatno breather propagates in the medium. We show that noise induces breathergeneration with a given probability depending on the noise intensity. We alsopropose a bifurcation diagram which extends the supratransmission effect to a morerealistic signal, namely a noisy sinusoidal excitation. We finally discuss somepromising signal processing applications that can be developed by taking intoaccount the contribution of noise in the media sharing this supratransmissionphenomenon.

Internal modes in sine-Gordon chain

We address the issue of internal modes of a stable kink in a discrete sine-Gordon equation. The aim of the present study is to elucidate the effects due to the detachment of the frequency dependence of antisymmetric internal mode from the spectrum. We analyze the frequencies of the lowest modes as functions of both the number of sites and the discreteness parameter. Using a simplified approach we explain the origin of the spectrum peculiarity, which arises when the frequency dependence detaches from the quasi-continuous spectrum at some value of the intersite coupling.

Quantum Creep and Quantum-Creep Transitions in 1D Sine-Gordon Chains.

Discrete sine-Gordon (SG) chains are studied with path-integral molecular dynamics. Chains commensurate with the substrate show the transition from pinning to quantum creep at bead masses slightly larger than in the continuous SG model. Within the creep regime, a field-driven transition from creep to complete depinning is identified. The effects of disorder on the chain's dynamics depend on the potential's roughness exponent H. For example, quantum fluctuations are generally too small to depin the chain if H=1/2, while an H=0 chain can be pinned or unpinned depending on the bead masses. Thermal fluctuations always depin the chain.

Diffusion of intrinsic localized modes by attractor hopping

Propagating intrinsic localized modes exist in the damped-driven discrete sine-Gordon chain as attractors of the dynamics. The equations of motion of the system are augmented with Gaussian white noise in order to model the effects of temperature on the system. The noise induces random transitions between attracting configurations corresponding to opposite signs of the propagation velocity of the mode, which leads to a diffusive motion of the excitation. The Heun method is used to numerically generate the stochastic time-evolution of the configuration. We also present a theoretical model for the diffusion which contains two parameters, a transition probability Θ and a delay time τA. The mean value and the variance of the position of the intrinsic localized mode, obtained from simulations, can be fitted well with the predictions of our model, Θ and τA being used as parameters in the fit. After a transition period following the switching on of the noise, the variance shows a linear behaviour as a function of time and the mean value remains constant. An increase in the strength of the noise lowers the variance, leads to an increase in Θ, a decrease in τA and reduces the average distance a mode travels during the transition period.

Resonant Steps in the Discrete sine-Gordon Chain**The project supported in part by the Research Grant Council RGC, the Hong Kong Baptist University Faculty Research Grant FRG, and National Natural Science Foundation of China

Low-velocity dynamics of a dc-driven discrete sine-Gordon chain is studied in the damped case. It is shown that resonant steps of the average velocity emerge due to the resonant phase locking between the kink motion and its radiated linear modes in its wake. A mean-field description is given, which can predict precisely the numerical results. The mechanism and formula proposed in this paper can be directly applied to the discussion of fluxon dynamics of one-dimensional Josephson junction arrays and Josephson junction ladders.

Instabilities of Coupled Quasi-Homoclinic Limit Cycles

It has been recently shown that the continuous coupling of non-linear oscillators close to an Andronov homoclinic bifurcation generically leads to two types of instabilities: a self-focusing Kuramoto like instability or a finite wavelength period doubling instability. The purpose of this paper is to study simple examples where these instabilities have been observed. It includes the coupling of two pendula, the sine-Gordon chain and the coupling of dissipative non-linear oscillators.

Kink dynamics in finite discrete sine-Gordon chains

The dynamics of one-dimensional finite discrete sine-Gordon (Frenkel-Kontorova) chains is studied for small values of misfit. It is shown that nonintegrability leads to kink trapping at the system surface, stopping the system motion as a whole and localizing energy at the system boundary. An interpretation from the point of view of both dislocation dynamics and proton transport is proposed.

Numerical Simulation of Topological Solitons Dynamics in Coupled Sine-Gordon Chains and Interaction with Localized Impurities

We consider the numerical simulation of the dynamics of topological solitons in coupled Sine-Gordon chains. Oscillating bound states and mutual pinning of solitons belonging to different chains are observed. The interaction of pairs of solitons with localized impurities is also investigated. Symmetrical and antisymmetrical impurity modes are excited in the case of an attractive impurity with the same strength in both chains. We find that the critical velocity below, which resonant interactions between the solitons and impurity modes take place, increases as we move from the highly repulsive state to the highly attractive state. In the case of attractive impurities with different strengths, the pair of solitons can be either captured, separated or transmitted with the formation of a moving bound state afterwards.

Spatiotemporal dynamics of discrete sine-Gordon lattices with sinusoidal couplings

The spatiotemporal dynamics of a damped sine-Gordon chain with sinusoidal nearest-neighbor couplings driven by a constant uniform force are discussed. The velocity characteristics of the chain versus the external force is shown. Dynamics in the high- and low-velocity regimes are investigated. It is found that in the high-velocity regime, the dynamics is dominated by rotating modes, the velocity shows a branching bifurcation feature, while in the low-velocity regime, the velocity exhibits step-like dynamical transitions, broken by the destruction of strong resonances.

Long-range self-affine correlations in a random soliton gas

The random sine-Gordon chain experiences a sharp crossover to an ordered state associated with the activation of a soliton gas. The spatial coherence of the stationary regimes has three well-defined scaling behaviors. At larger scales there is no correlation, reflecting the independence between the different local structures in the chain. In this paper we introduce a different potential that produces solitons that exhibit long-range interactions. We show that a gas of such solitons can extend self-affinity to all scales.

Kink dynamics in the highly discrete and coupled sine-Gordon system.

We study the dynamics of kinks in the highly discrete and coupled sine-Gordon chains. We analyze the effects of the interchain coupling on the dressings of solitons (kinks and antikinks) and on the Peierls-Nabarro barrier. We find the binding of solitons into a bisoliton mode.

Noise-Induced Organization in a sine-Gordon chain

The appearance of thermally activated solitons in a sine-Gordon system is related to the roughening exponent ζ, defined as the scaling exponent of the length of the ensemble average of the standard deviation of the height of the spatiotemporal profile. Before the onset of the noise-induced transition to the solitonic regime, the roughening exponent is zero as it is for a white noise signal. After the activation of solitons, there is a very interesting crossover from non-KPZ behavior (ζ ∼ 0.70) to KPZ behavior (ζ ∼ 0.50); additionally, for sufficiently large scales, a crossover to a zero roughening exponent takes place. In this paper we precise the underlying dynamics of the different regimes that appear at different scales via geometric characterization as the size of the system and the friction coefficient are varied.

Statistical mechanics of nonuniform magnetization reversal.

The magnetization reversal rate via thermal creation of soliton pairs in quasi-1D ferromagnetic systems is calculated. Such a model describes e.g. the time dependent coercivity of elongated particles as used in magnetic recording media. The energy barrier that has to be overcome by thermal fluctuations corresponds to a soliton-antisoliton pair whose size depends on the external field. In contrast to other models of first order phase transitions such as the phi^4 model, an analytical expression for this energy barrier is found for all values of the external field. The magnetization reversal rate is calculated using a functional Fokker-Planck description of the stochastic magnetization dynamics. Analytical results are obtained in the limits of small fields and fields close to the anisotropy field. In the former case the hard-axis anisotropy becomes effectively strong and the magnetization reversal rate is shown to reduce to the nucleation rate of soliton-antisoliton pairs in the overdamped double sine-Gordon model. The present theory therefore includes the nucleation rate of soliton-antisoliton pairs in the double sine-Gordon chain as a special case. These results demonstrate that for elongated particles, the experimentally observed coercivity is significantly lower than the value predicted by the standard theories of N\'eel and Brown.

DNA promoters and nonlinear dynamics

We investigate the role played by DNA promoters as dynamical activators of transport processes of the RNA polymerase along DNA macromolecules, by introducing an effective potential for the kink of a discrete sine-Gordon chain.