Chaotic Breathers Research Articles

Chaotic discrete breathers in bcc lattice: Effect of the first- and second-neighbor interactions

Numerical simulations are performed to investigate the formation of chaotic discrete breathers (DBs) in a bcc lattice using four zone-boundary modes with frequencies exceeding the crystal’s phonon spectrum. The study analyzes the impact of the stiffness of elastic bonds between first and second neighbors and identifies a specific range where the formation of chaotic DBs due to modulational instability of four vibrational modes is possible. The time evolution of the energy localization parameter and the maximum energy of all particles were monitored to control the formation of chaotic DBs. Their spontaneous nucleation was observed in a wide range of stiffness of elastic bonds and depends on the mode’s symmetry. Considering DBs in bcc metals, the paper focuses on the scenario where first-neighbor bonds are stiffer than second-neighbor bonds, as bond stiffness typically decreases with interatomic distance. In all four zone-boundary modes, formation of chaotic DB is observed under this condition.

Equilibria and dynamics of two coupled chains of interacting dipoles.

We explore the energy transfer dynamics in an array of two chains of identical rigid interacting dipoles. Varying the distance b between the two chains of the array, a crossover between two different ground-state (GS) equilibrium configurations is observed. Linearizing around the GS configurations, we verify that interactions up to third nearest neighbors should be accounted to accurately describe the resulting dynamics. Starting with one of the GS, we excite the system by supplying it with an excess energy ΔK located initially on one of the dipoles. We study the time evolution of the array for different values of the system parameters b and ΔK. Our focus is hereby on two features of the energy propagation: the redistribution of the excess energy ΔK among the two chains and the energy localization along each chain. For typical parameter values, the array of dipoles reaches both the equipartition between the chains and the thermal equilibrium from the early stages of the time evolution. Nevertheless, there is a region in parameter space (b,ΔK) where even up to the long computation time of this study, the array does neither reach energy equipartition nor thermalization between chains. This fact is due to the existence of persistent chaotic breathers.

Influence of the relative stiffness of second-neighbor interactions on chaotic discrete breathers in a square lattice

It is known that the modulational instability of a delocalized nonlinear vibrational mode (DNVM) with frequency outside the phonon band can lead to spontaneous energy localization in a nonlinear lattice on chaotic discrete breathers (CDBs). Considering a β-FPUT square lattice with nearest and next-nearest interactions, the appearance of CDBs is analyzed for different stiffnesses of the first- and second-nearest interactions, k1 and k2, keeping the density of the lattice unchanged. There are two DNVMs in the square lattice with frequencies above the phonon spectrum and both are studied. The appearance of CDBs in the lattice is monitored by calculating the time evolution of the energy localization parameter L and the maximum energy of the particles emax. For solid state physics and materials science, the important range of stiffness parameters is k2<k1, since the stiffness of chemical bonds typically decreases with the distance between atoms. It is found that CDBs form in the square lattice when k2>k1/4. This means that they can form in crystals if the stiffness of the second-neighbor bonds is smaller than that of the first-neighbor bonds, but not too small. Depending on the relation between the anharmonicity parameters of the first- and second-neighbor bonds CDBs can have different polarization.

Chaotic discrete breathers in bcc lattice

It is well known that a modulationally unstable short-wavelength delocalized nonlinear vibrational mode (DNVM) can create chaotic discrete breathers (DBs) in a nonlinear lattice. A necessary condition for this is that the DNVM must have a frequency outside the phonon spectrum of the lattice. This phenomenon has been repeatedly analyzed for one- and two-dimensional lattices, and here it is studied for a bcc lattice with β-Fermi–Pasta–Ulam–Tsingou potential. Using the group-theoretical approach developed by Chechin and Sakhnenko, four DNVMs are found with a wave vector at the boundary of the first Brillouin zone and frequencies above the phonon spectrum. It is shown that the development of the modulational instability of all four DNVMs with amplitudes above a certain value leads to the formation of chaotic DBs, which is justified by calculating the energy localization parameter and the maximum particle energy. Chaotic DBs in the three-dimensional bcc lattice radiate their energy faster than in previously studied two-dimensional lattices. The results obtained describe one of the possible mechanisms of energy dissipation by a crystal lattice in a far-from-equilibrium system.

Modulational Instability of Delocalized Modes in fcc Copper

Delocalized nonlinear vibrational modes (DNVMs) are exact solutions of the equations of motion, and therefore, DNVMs exist at any vibration amplitude and do not depend on interaction potentials. For the first time, modulation instability of four one-component three-dimensional DNVMs is studied in a single crystal of fcc copper with the use of methods of molecular dynamics. DNVMs frequencies, evolution of stresses, kinetic and potential energies, and heat capacity depending on the oscillation amplitudes are analyzed. It is found that all four DNVMs are characterized by a hard-type anharmonicity. Modulation instability of DNVMs results in a formation of chaotic discrete breathers (DBs) with frequency above the upper edge of the phonon spectrum of the crystal. The lifetime of chaotic DBs is found to be in the range of 30–100 ps. At low-oscillation frequencies, longer-lived DBs are formed. The growth of modulation instability leads to an increase in mechanical stresses and a decrease in the heat capacity of the crystal. The results obtained in this work enrich our understanding of the influence of the modulation instability of DNVMs on the properties of metals.

Chaos and thermalization in a classical chain of dipoles.

We explore the connection between chaos, thermalization, and ergodicity in a linear chain of N interacting dipoles. Starting from the ground state, and considering chains of different numbers of dipoles, we introduce single site excitations with excess energy ΔK. The time evolution of the chaoticity of the system and the energy localization along the chain is analyzed by computing, up to a very long time, the statistical average of the finite-time Lyapunov exponent λ(t) and the participation ratio Π(t). For small ΔK, the evolution of λ(t) and Π(t) indicates that the system becomes chaotic at approximately the same time as Π(t) reaches a steady state. For the largest considered values of ΔK the system becomes chaotic at an extremely early stage in comparison with the energy relaxation times. We find that this fact is due to the presence of chaotic breathers that keep the system far from equipartition and ergodicity. Finally, we show numerically and analytically that the asymptotic values attained by the participation ratio Π(t) fairly correspond to thermal equilibrium.

Chaotic discrete breathers and their effect on macroscopic properties of triangular lattice

The localization of energy on chaotic discrete breathers (DBs) arising in a two- dimensional triangular lattice due to the modulation instability of delocalized nonlinear vibrational modes (DNVMs) is analyzed. Three DNVMs with frequencies above the phonon band and demonstrating hard-type anharmonicity (an increase in the vibration frequency with amplitude) are considered. Chaotic DBs have long lifetime, slowly radiate their energy and eventually disappear. The evolution of the macroscopic characteristics of the lattice is observed during the transition from the regime with chaotic DBs to thermal equilibrium. It is established that chaotic DBs with a hard type of anharmonicity reduce the ratio of the total energy to the kinetic energy (and, consequently, reduce the heat capacity). They also reduce lattice pressure at constant area (and therefore reduce thermal expansion). The tensile rigidity of the lattice also decreases due to DBs with a hard type of anharmonicity. The most sensitive to the presence of DBs is the pressure, which in the presence of DBs is approximately 30% less than in thermal equilibrium. The ratio of the total energy to the kinetic energy in the regime of chaotic DBs decreases by about 3%, and the tensile rigidity by only 0.1%.

Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system

Investigation is made on a vector nonlinear Schrödinger equation in the anomalous dispersion regime describing the optical pulses in a birefringent optical fiber or in a wavelength division multiplexed system. We derive the analytical breather solutions. Fission and fusion of the breathers are investigated via the relation between the phase velocity and group velocity of the plane wave. Stability of the numerical breather is studied via the pseudospectral method: The breather with the white noise propagates stably. Breathers in the chaotic wave fields are derived via the split-step Fourier method: The breathers in the positive x axis are observed more distinctly than in the negative, where x is the spatial coordinate. Complex eigenvalue is gotten to study those phenomena. Effects of the modulation instability on the breathers in the chaotic wave fields are investigated.

Energy transfer mechanisms in a dipole chain: From energy equipartition to the formation of breathers.

We study the energy transfer in a classical dipole chain of N interacting rigid rotating dipoles. The underlying high-dimensional potential energy landscape is analyzed in particular by determining the equilibrium points and their stability in the common plane of rotation. Starting from the minimal energy configuration, the response of the chain to excitation of a single dipole is investigated. Using both the linearized and the exact Hamiltonian of the dipole chain, we detect an approximate excitation energy threshold between a weakly and a strongly nonlinear dynamics. In the weakly nonlinear regime, the chain approaches in the course of time the expected energy equipartition among the dipoles. For excitations of higher energy, strongly localized excitations appear whose trajectories in time are either periodic or irregular, relating to the well-known discrete or chaotic breathers, respectively. The phenomenon of spontaneous formation of domains of opposite polarization and phase locking is found to commonly accompany the time evolution of the chaotic breathers. Finally, the sensitivity of the dipole chain dynamics to the initial conditions is studied as a function of the initial excitation energy by computing a fast chaos indicator. The results of this study confirm the aforementioned approximate threshold value for the initial excitation energy, below which the dynamics of the dipole chain is regular and above which it is chaotic.

Interaction between Liénard and Ikeda dynamics in a nonlinear electro-optical oscillator with delayed bandpass feedback.

We report on experimental and theoretical analysis of the complex dynamics generated by a nonlinear time-delayed electro-optic bandpass oscillator. We investigate the interaction between the slow- and fast-scale dynamics of autonomous oscillations in the breather regime. We analyze in detail the coupling between the fast-scale behavior associated to a characteristic low-pass Ikeda behavior and the slow-scale dynamics associated to a Liénard limit-cycle. Finally, we show that when projected onto a two-dimensional phase space, the attractors corresponding to periodic and chaotic breathers display a spiral-like pattern, which strongly depends on the shape of the nonlinear function.

Broadband Chaotic Signals and Breather Oscillations in an Optoelectronic Oscillator Incorporating a Microwave Photonic Filter

We propose a technique to generate broadband chaotic and breather signals employing an optoelectronic oscillator (OEO) comprising a phase modulator (PM) and a linearly chirped fiber Bragg grating (LCFBG). The joint operation of the PM and the LCFBG forms a broadband microwave photonic filter (MPF), which allows the OEO to generate chaotic signals and breathers taking advantage of the interplay between the broadband MPF and the time-delayed feedback loop provided by a long optical fiber delay line. The breather excitations are characterized by nanosecond chaotic oscillations breathing periodically at a significantly lower time-scale determined by the OEO large delay time. A theoretical analysis based on a modified Ikeda time-delayed model to include the effect of the broadband filtering process is provided. The analysis is verified by an experiment. The proposed LCFBG-based OEO and the possibility to control in the optical domain its broadband bandpass characteristics considering the flexibility, accuracy, and precision in FBG fabrication can find applications in chaos-based communications and in fast optical processing systems, such as random number generation, or optical processing in reservoir computing taking advantage of the intrinsic multiple time scales of the LCFBG-based OEO.

Periodic, quasiperiodic, and chaotic breathers in two-dimensional discrete β-Fermi—Pasta—Ulam lattice

Using numerical method, we investigate whether periodic, quasiperiodic, and chaotic breathers are supported by the two-dimensional discrete Fermi—Pasta—Ulam (FPU) lattice with linear dispersion term. The spatial profile and time evolution of the two-dimensional discrete β-FPU lattice are segregated by the method of separation of variables, and the numerical simulations suggest that the discrete breathers (DBs) are supported by the system. By introducing a periodic interaction into the linear interaction between the atoms, we achieve the coupling of two incommensurate frequencies for a single DB, and the numerical simulations suggest that the quasiperiodic and chaotic breathers are supported by the system, too.

Discrete breathers in integrate-and-fire oscillator networks

A number of novel dynamical patterns are found in an integrate-and-fire network, including periodic breathers with single, double, and triple wavefronts, and chaotic breathers. It is found that the periodic breathers always propagate whereas the chaotic ones always remain stationary. A novel route to chaotic breathers through intermittency is identified, as opposed to the paths through quasi-periodicity and period-doubling mostly discussed in dissipative lattice systems. The system is found to display prominent bistability, resulting in “hysteresis” loops in single-oscillator inter-spike intervals as the parameter varies.

Spreading of energy in the Ding-Dong model

We study properties of energy spreading in a lattice of elastically colliding harmonic oscillators (Ding-Dong model). We demonstrate that in the regular lattice the spreading from a localized initial state is mediated by compactons and chaotic breathers. In a disordered lattice the compactons do not exist, and the spreading eventually stops, resulting in a finite configuration with a few chaotic spots.

Different kinds of discrete breathers in a Sine-Gordon lattice

We study a one-dimensional Sine-Gordon lattice of anharmonic oscillators with cubic and quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the one-dimensional Sine-Gordon lattice no matter whether the nonlinear interaction is cubic or quartic. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.

Periodic, quasiperiodic and chaotic discrete breathers in a parametrical driven two-dimensional discrete diatomic Klein–Gordon lattice

We study a two-dimensional (2D) diatomic lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DBs) can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the 2D discrete diatomic Klein–Gordon lattice with hard and soft on-site potentials. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers (QDBs) and chaotic discrete breathers (CDBs) by changing the amplitude of the driver. But the DBs and QDBs with symmetric and anti-symmetric profiles that are centered at a heavy atom are more stable than at a light atom, because the frequencies of the DBs and QDBs centered at a heavy atom are lower than those centered at a light atom.

Periodic, Quasiperiodic and Chaotic Discrete Breathers in a Parametrical Driven Two-Dimensional Discrete Klein-Gordon Lattice

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein–Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.

Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators

The response of a nonlinear optical oscillator subject to a delayed broadband bandpass filtering feedback is studied experimentally, numerically, and analytically. The oscillator loop is characterized by a high cutoff frequency with a response time tau approximately 10 ps and by a low cutoff frequency with a response time theta approximately 1 micros. Moreover, the optoelectronic feedback also consists of a significant delay tauD of the order of 100 ns. Depending on two key physical parameters, the loop gain beta and the nonlinearity operating point Phi, a large variety of multiple time scale regimes are reported, including slow or fast periodic oscillations with different waveforms, regular or chaotic breathers, slow time envelope dynamics, complex and irregular self-pulsing, and fully developed chaos. Many of these regimes are exhibiting new features that are absent in the classical first-order scalar nonlinear delay differential equations (DDEs), which differ in the modeling by the low cutoff only. Nearly all kinds of solutions are recovered numerically by a new class of integro-DDE (iDDE) that take into account both the high and low cutoff frequencies of the feedback loop. For moderate feedback gain, asymptotic solutions are determined analytically by taking advantage of the relative values of the time constants tau, theta, and tauD. We confirm the experimental observation of two distinct routes to oscillatory instabilities depending on the value of Phi. One route is reminiscent of the square wave oscillations of the classical first-order DDE, but the other route is quite different and allows richer wave forms. For higher feedback gain, these two distinct regimes merge leading to complex nonperiodic regimes that still need to be explored analytically and numerically. Finally, we investigate the theoretical limits of our iDDE model by experimentally exploring phenomena at extreme physical parameter setting, namely, high-frequency locking at strong feedback gain or pulse packages for very large delays. The large variety of oscillatory regimes of our broadband bandpass delay electro-optic oscillator is attractive for applications requiring rich optical pulse sources with different frequencies and/or wave forms (chaos-based communications, random number generation, chaos computing, and generation of stable multiple GHz frequency oscillations).

Modulational instability in isolated and driven Fermi–Pasta–Ulam lattices

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi--Pasta--Ulam (FPU) lattices. The growth of the instability is followed by a process of relaxation to equipartition, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, while in the original FPU problem low frequency excitations of the lattice were considered. This relaxation process leads to the formation of chaotic breathers in both one and two space dimensions. The system then relaxes to energy equipartition, on time scales that increase as the energy density is decreased. We supplement this study by considering the nonconservative case, where the FPU lattice is homogeneously driven at high frequencies. Standing and travelling nonlinear waves and solitonic patterns are detected in this case. Finally we investigate the dynamics of the FPU chain when one end is driven at a frequency located above the zone boundary. We show that this excitation stimulates nonlinear bandgap transmission effects.

Discrete breathers in two-dimensional nonlinear lattices

Properties of discrete breathers in two- and three-dimensional nonlinear lattice models are discussed with special emphasis on possible structures of localization in two-dimensional case. In contrast to the extensive studies addressing the various features of discrete breathers in one-dimensional lattices, less studies have been done in regard to the higher dimensional breathers. To begin with a brief survey of existing results for discrete breathers in higher dimensional lattices, we attempt to look into the possible structures of higher dimensional breathers. Based on the survey, we address the following topics concerning the discrete breathers in two-dimensional lattices: types of localized structures; chaotic breathers; quasi-continuum approximation; discrete breathers versus solitons in continuum.