Chain Of Nonlinear Oscillators Research Articles

A design methodology for nonlinear oscillator chains enabling energy localization tuning and soliton stability enhancement with optimal damping

In this paper, the vibration energy localization in coupled nonlinear oscillators is investigated, based on the creation of standing solitons. The main objective is to establish a design methodology for mechanical lattices using the Nonlinear Schrödinger Equation (NLSE) as a guide strategy, even in the presence of damping. A three-dimensional diagram is used to illustrate stable parameter regions for damped stationary solitons. Moreover, an analysis of the influence of the number of oscillators in the system, and a numerical investigation regarding the stability of solitonic behavior is done. Through numerical analyses, it is observed that the developed algorithm not only has the capability to locate the highest amplitudes in the chain of oscillators, but also to control the intensity at which these amplitudes are located according to design requirements. The outcomes of the proposed methodology elucidate the impact that the coupling stiffness has on the stabilization of the NLSE, as well as the influence of the number of oscillators on the continuity hypothesis. The developed algorithm holds potential for practical applications in mechanical engineering since the NLSE is used as a design line rather than as a consequence of the phenomenon description.

Near-integrable dynamics of the Fermi-Pasta-Ulam-Tsingou problem.

It is well known that the classic Fermi-Pasta-Ulam-Tsingou (FPUT) study of a chain of nonlinear oscillators is closely related to a number of completely integrable systems, including the Toda lattice. Here, we present a method that captures the departure of nonintegrable FPUT dynamics from those of a nearby integrable Toda lattice. Using initial long-wave data, we find that the former depart rather sharply from the latter near the predicted shock time of an asymptotic partial differential equation approximation, at which point energy cascades into higher lattice modes. Our method provides an appropriate frame of reference for one to distinguish the short-term dynamics of the two systems, whose macroscopic trajectories diverge noticeably only on a much longer timescale, when the FPUT dynamics thermalize.

An artificial intelligence approach to design periodic nonlinear oscillator chains under external excitation with stable damped solitons

In this paper, an artificial intelligence approach is developed to create a metamodel of damped solitons in periodic nonlinear oscillator chains under external excitation. Unlike resonators subjected to parametric excitation, chains subjected exclusively to external excitation do not have known analytical solutions for Intrinsic Localized Modes (ILMs) when damping is pondered. The proposed metamodel is a quick design tool of nonlinear damped periodic structures describing effectively the behavior of solitonic solutions, demonstrating a computational cost considerably lower than that of the Newton method. The results of this work highlight the potential of ANN regression models in the description of physical phenomena represented by the Nonlinear Schrödinger Equation when damping is considered.

Behavior of circular chains of nonlinear oscillators with Kuramoto-like local coupling

The conditions under which synchronization is achieved for a one-dimensional ring of identical phase oscillators with Kuramoto-like local coupling are studied. The system is approached in the weakly coupled approximation as phase units. Instead of global couplings, the nearest-neighbor interaction is assumed. Units are pairwise coupled by a Kuramoto term driven by their phase differences. The system exhibits a rich set of behaviors depending on the balance between the natural frequency of isolated units and the self-feedback. The case of two oscillators is solved analytically, while a numerical approach is used for N > 2. Building from Kuramoto, the approach to synchronization, when possible, is studied through a local complex order parameter. The system can eventually evolve as a set of coupled local communities toward a given phase value. However, the approach to the stationary state shows a non-monotonous non-trivial dynamic.

Neimark Sacker bifurcations and non-linear energy exchange in chains of non-linear oscillators

We treat a chain of oscillators with linear stiffness and internal and external cubic non-linearities. The method of harmonic balance is used to determine the non-linear modes of the Hamiltonian system. For each equilibrium point, a stability analysis is performed by means of the associated monodromy matrix. The numerical results shows that branch points, limit points and Neimark–Sacker bifurcations exist in the system. The aim of the paper is to study how they drive the energy in the system: branch points make energy transfer between non-linear modes possible, while Neimark–Sacker bifurcations can lead to chaotic behaviour. As the number of oscillators increases, the energy required to reach the first Neimark–Sacker bifurcation follows a remarkable regularity.

A quasi-periodic route to chaos in a parametrically driven nonlinear medium

Small-sized systems exhibit a finite number of routes to chaos. However, in extended systems, not all routes to complex spatiotemporal behavior have been fully explored. Starting from the sine-Gordon model of parametrically driven chain of damped nonlinear oscillators, we investigate a route to spatiotemporal chaos emerging from standing waves. The route from the stationary to the chaotic state proceeds through quasiperiodic dynamics. The standing wave undergoes the onset of oscillatory instability, which subsequently exhibits a different critical frequency, from which the complexity originates. A suitable amplitude equation, valid close to the parametric resonance, makes it possible to produce universe results. The respective phase-space structure and bifurcation diagrams are produced in a numerical form. We characterize the relevant dynamical regimes by means of the largest Lyapunov exponent, the power spectrum, and the evolution of the total intensity of the wave field.

Two decades of INO research on laser arrays

We summarize in this work the salient points of the studies carried out at the National Institute of Optics (INO) in Florence in the vast field of coherent laser arrays. The research began in the early 90s under the push of Tito Arecchi with the study of coupled laser oscillators. These systems stimulated Tito's curiosity by representing an accessible sample of nonlinear oscillator chains. In the following years at INO we continued our research in this field, obtaining numerical and experimental results both with the realization of laser arrays and in the field of their possible application. Of particular importance for obtaining a fruitful application was the development of optical phase filtering techniques aimed at increasing the optical quality of the beam emitted from these sources, which is actually far from having a traditional geometry.

Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices.

The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi etal. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955 (unpublished)]. We introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi etal. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.

Mobile Dissipative Breathers in a Chain of Nonlinear Oscillators

In a numerical experiment, it is found that at least three localized stationary breather type modes can exist in a chain of Rayleigh oscillators with a nonlinear coupling between them: immobile, “slow,” and “fast” dissipative breathers. The dynamics of synchronous motions of chain elements depends on the relation of characteristic time scales of the system related to the frequency of the oscillators and rigidity of the coupling between them; it is multistable.

Heat flux in one-dimensional systems.

Understanding heat transport in one-dimensional systems remains a major challenge in theoretical physics, both from the quantum as well as from the classical point of view. In fact, steady states of one-dimensional systems are commonly characterized by macroscopic inhomogeneities, and by long-range correlations, as well as large fluctuations that are typically absent in standard three-dimensional thermodynamic systems. These effects violate locality-material properties in the bulk may be strongly affected by the boundaries, leading to anomalous energy transport-and they make more problematic the interpretation of mechanical microscopic quantities in terms of thermodynamic observables. Here, we revisit the problem of heat conduction in chains of classical nonlinear oscillators, following a Lagrangian and a Eulerian approach. The Eulerian definition of the flux is composed of a convective and a conductive component. The former component tends to prevail at large temperatures where the system behavior is increasingly gaslike. Finally, we find that the convective component tends to be negative in the presence of a negative pressure.

Beats in the system of three non-linear oscillators and their quantum analog

The classical beats phenomenon is usually demonstrated for the system of two coupled oscillators. The only known realization of similar process in the short homogenous chain with more then two elements refers to the three-well quantum system with the same coupling between the wells providing the special condition on the equations describing the system. However, this condition does not appear to hold in classical homogenous coupled systems, that makes the full periodic energy exchange in the systems of three classical oscillators questionable. Here we prove that fully reciprocal periodic energy transport between the ends of a short homogenous chain of nonlinear oscillators is possible in the conservative classical system with the ‘soft’ nonlinearity. The analogy with the quantum system of the coherent (harmonic) quantum Rabi oscillations in a superposition of three quantum states helps to reveal the special condition on the effective on-site potentials, which does not exist in the classical linear system or system of more than two oscillators with ‘hard’ nonlinearity. We study the periodic energy transport and its localization with use of the regular asymptotic analysis in the reduced phase space. The reported effects can be significant for many fundamental and applied areas of sciences where the coherent energy transport is important.

N-break states in a chain of nonlinear oscillators.

In the present work we explore a prestretched oscillator chain where the nodes interact via a pairwise Lennard-Jones potential. In addition to a homogeneous solution, we identify solutions with one or more (so-called) "breaks," i.e., jumps. As a function of the canonical parameter of the system, namely, the precompression strain d, we find that the most fundamental one-break solution changes stability when the monotonicity of the Hamiltonian changes with d. We provide a proof for this (motivated by numerical computations) observation. This critical point separates stable and unstable segments of the one-break branch of solutions. We find similar branches for two- through five-break branches of solutions. Each of these higher "excited state" solutions possesses an additional unstable pair of eigenvalues. We thus conjecture that k-break solutions will possess at least k-1 (and at most k) pairs of unstable eigenvalues. Our stability analysis is corroborated by direct numerical computations of the evolutionary dynamics.

Experimental results on the vibratory energy exchanges between a linear system and a chain of nonlinear oscillators

Experimental results on a nonlinear chain coupled to a main system are presented. The chain is composed of eight moving masses, each one possesses local nonlinear restoring forcing function and global linear springs for coupling to other masses. The main system is coupled to the first mass of the chain via a linear spring. The main system is under external sinusoidal excitation with sweeping frequency around its targeted mode. Experimental results show that according to the amplitude of the excitation, the system can reach periodic or modulated regimes. The goal of using the nonlinear chain is to examine experimentally the possibility of localization of the vibratory energy of the main system into the chain for the aim of passive control.

Complex spatiotemporal behavior and coherent excitations in critically-coupled chains of neural circuits.

We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the "synaptic matrix" embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a "critical" synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period , thus given by a unitary matrix . It is now explicit that such critical couplings are volume-preserving in the sense of Liouville's theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.

Glassy dynamics in disordered oscillator chains.

The escape of energy injected into one site in a disordered chain of nonlinear oscillators is examined numerically. When the disorder has a "fractal" pattern, the decay of the residual energy at the injection site can be fit to a stretched exponential with an exponent that varies continuously with the control parameter. At low temperature, we see evidence that energy can be trapped for an infinite time at the original site, i.e., classical many body localization.

Dynamics of a linear system coupled to a chain of light nonlinear oscillators analyzed through a continuous approximation

The continuous approximation is used in this work to describe the dynamics of a nonlinear chain of light oscillators coupled to a linear main system. A general methodology is applied to an example where the chain has local nonlinear restoring forces. The slow invariant manifold is detected at fast time scale. At slow time scale, equilibrium and singular points are sought around this manifold in order to predict periodic regimes and strongly modulated responses of the system. Analytical predictions are in good accordance with numerical results and represent a potent tool for designing nonlinear chains for passive control purposes.

Hamiltonian formulation of systems with balanced loss–gain and exactly solvable models

A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occurs in a pairwise fashion. It is also shown that with the choice of a suitable co-ordinate, the Hamiltonian can always be reformulated in the background of a pseudo-Euclidean metric. If the equations of motion of some of the well-known many-body systems like Calogero models are generalized to include balanced loss and gain, it appears that the same may not be amenable to a Hamiltonian formulation. A few exactly solvable systems with balanced loss and gain, along with a set of integrals of motion are constructed. The examples include a coupled chain of nonlinear oscillators and a many-particle Calogero-type model with four-body inverse square plus two-body pair-wise harmonic interactions. For the case of nonlinear oscillators, stable solution exists even if the loss and gain parameter has unbounded upper range. Further, the range of the parameter for which the stable solutions are obtained is independent of the total number of the oscillators. The set of coupled nonlinear equations are solved exactly for the case when the values of all the constants of motions except the Hamiltonian are equal to zero. Exact, analytical classical solutions are presented for all the examples considered.

Complex behavior in chains of nonlinear oscillators.

This article outlines sufficient conditions under which a one-dimensional chain of identical nonlinear oscillators can display complex spatio-temporal behavior. The units are described by phase equations and consist of excitable oscillators. The interactions are local and the network is poised to a critical state by balancing excitation and inhibition locally. The results presented here suggest that in networks composed of many oscillatory units with local interactions, excitability together with balanced interactions is sufficient to give rise to complex emergent features. For values of the parameters where complex behavior occurs, the system also displays a high-dimensional bifurcation where an exponentially large number of equilibria are borne in pairs out of multiple saddle-node bifurcations.

Vibratory control of a linear system by addition of a chain of nonlinear oscillators

An N-degree-of-freedom model consisting of a single-degree-of-freedom linear system coupled to a chain of $$(N-1)$$ light nonlinear oscillators is studied. The connection between the chain and the single-degree-of-freedom system is supposed to be linear. Time multi-scale system behaviors at fast and slow time scales are investigated and lead to the detection of the slow invariant manifold and equilibrium and singular points. These points correspond to periodic regimes and strongly modulated responses, respectively. These analytical developments are used to provide evidence of transfer of vibratory energy of the main system to the chain in the form of localized modes during periodic regimes and extreme energy exchanges between modes when the overall structure faces singularities. Furthermore, analytical predictions at slow time scale and nonlinear normal modes of the system are compared with numerical results obtained from direct time integration of the system equations, showing a good agreement between them. Finally, we present a procedure showing how these analytical developments can be used to study a system where the main structure is replaced by a multi-degree-of-freedom linear system, by projecting its dynamics on one of its modes.

Landau-Zener tunneling of solitons.

We consider Landau-Zener tunneling of solitons in a weakly coupled two-channel system, for this purpose we construct a simple mechanical system using two weakly coupled chains of nonlinear oscillators with gradually decreasing (first chain) and increasing (second chain) masses. The model allows us to consider soliton propagation and Landau-Zener tunneling between the chains. It is shown that soliton tunneling characteristics become drastically dependent on its amplitude in nonlinear regime. The validity of the developed tunneling theory is justified via comparison with direct numerical simulations on oscillator ladder system.