Forced oscillations of the string under conditions of 'sonic vacuum'.

We present the results of analytical study of the significant regularities which are inherent to forced nonlinear oscillations of a string with uniformly distributed discrete masses, without its preliminary stretching. It was found recently that a corresponding autonomous system admits a series of nonlinear normal modes with a lot of possible intermodal resonances and that similar synchronized solutions can exist in the presence of a periodic external field also. The paper is devoted to theoretical explanation of numerical data relating to one of possible scenarios of intermodal interaction which was numerically revealed earlier. This is unidirectional energy flow from unstable nonlinear normal mode to nonlinear normal modes with higher wavenumbers under the conditions of sonic vacuum. The mechanism of such a scenario has not yet been clarified contrary to alternative mechanisms consisting in almost simultaneous energy flow to all nonlinear normal modes with breaking the above-mentioned conditions of sonic vacuum. We begin with a description of single-mode manifolds and then show that consideration of arbitrary double mode manifolds is sufficient for solution of the problem. Because of this, the two-modal equations of motion can be reduced to a linear equation which describes a perturbation of initially excited nonlinear normal mode of the forced system in the conditions of sonic vacuum. We have found analytical representation (in the parametric space) of the thresholds for all possible energy transfers corresponding to unidirectional energy flow from unstable nonlinear normal modes. The analytical results are in a good agreement with previous numerical calculations.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Nonreciprocal acoustics and dynamics in the in-plane oscillations of a geometrically nonlinear lattice.

We study the dynamics and acoustics of a nonlinear lattice with fixed boundary conditions composed of a finite number of particles coupled by linear springs, undergoing in-plane oscillations. The source of the strongly nonlinearity of this lattice is geometric effects generated by the in-plane stretching of the coupling linear springs. It has been shown that in the limit of low energy the lattice gives rise to a strongly nonlinear acoustic vacuum, which is a medium with zero speed of sound as defined in classical acoustics. The acoustic vacuum possesses strongly nonlocal coupling effects and an orthogonal set of nonlinear standing waves [or nonlinear normal modes (NNMs)] with mode shapes identical to those of the corresponding linear lattice; in contrast to the linear case, however, all NNMs except the one with the highest wavelength are unstable. In addition, the lattice supports two types of waves, namely, nearly linear sound waves (termed "L waves") corresponding to predominantly axial oscillations of the particles and strongly nonlinear localized propagating pulses (termed "NL pulses") corresponding to predominantly transverse oscillating wave packets of the particles with localized envelopes. We show the existence of nonlinear nonreciprocity phenomena in the dynamics and acoustics of the lattice. Two opposite cases are examined in the limit of low energy. The first gives rise to nonreciprocal dynamics and corresponds to collective, spatially extended transverse loading of the lattice leading to the excitation of individual, predominantly transverse NNMs, whereas the second case gives rise to nonreciprocal acoutics by considering the response of the lattice to spatially localized, transverse impulse or displacement excitations. We demonstrate intense and recurring energy exchanges between a directly excited NNM and other NNMs with higher wave numbers, so that nonreciprocal energy exchanges from small-to-large wave numbers are established. Moreover, we show the existence of nonreciprocal wave interaction phenomena in the form of irreversible targeted energy transfers from L waves to NL pulses during collisions of these two types of waves. Additional nonreciprocal acoustics are found in the form of complex "cascading processes, as well as nonreciprocal interactions between L waves and stationary discrete breathers. The computational studies confirm the theoretically predicted transition of the lattice dynamics to a low-energy state of nonlinear acoustic vacum with strong nonlocality.

Transient and chaotic low-energy transfers in a system with bistable nonlinearity.

The low-energy dynamics of a two-dof system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both cubic nonlinear and negative linear components is investigated. The mechanisms leading to intense energy exchanges between the linear oscillator, excited by a low-energy impulse, and the nonlinear attachment are addressed. For lightly damped systems, it is shown that two main mechanisms arise: Aperiodic alternating in-well and cross-well oscillations of the nonlinear attachment, and secondary nonlinear beats occurring once the dynamics evolves solely in-well. The description of the former dissipative phenomenon is provided in a two-dimensional projection of the phase space, where transitions between in-well and cross-well oscillations are associated with sequences of crossings across a pseudo-separatrix. Whereas the second mechanism is described in terms of secondary limiting phase trajectories of the nonlinear attachment under certain resonance conditions. The analytical treatment of the two aformentioned low-energy transfer mechanisms relies on the reduction of the nonlinear dynamics and consequent analysis of the reduced dynamics by asymptotic techniques. Direct numerical simulations fully validate our analytical predictions.

Front propagation in a bistable system: how the energy is released.

In this Rapid Communication we consider a front of transition between metastable and stable states in a conservative system. Due to the difference of energies between initial and finite states, such transition front can propagate only while radiating energy. A simulation of such a process in a one-dimensional nonlinear lattice shows an essential imbalance between the energy released in each act of transition, and the density of energy of oscillations behind the front. It means that the stationary front propagation must be accompanied by an essentially nonstationary radiative process. We reveal the origin of this phenomenon and show that the characteristics of the front propagation critically depend on boundary conditions. In the framework of a simple model of a bistable system we propose analytic evaluation of all important features of the transition process, such as front velocity, radiation frequency, and oscillation amplitude. All calculated values are in good agreement with numerical simulation data.

Localization of low-frequency oscillations in single-walled carbon nanotubes.

In the framework of the continuum shell theory, we analytically predict a new phenomenon: the weak localization of optical low-frequency oscillations in carbon nanotubes. We clarify the origin of the localization by means of the concept of the limiting phase trajectory and confirm the obtained analytical results by molecular dynamics simulations of simply supported carbon nanotubes. The performed analysis contributes to the new universal approach to the treatment of nonstationary resonant processes.

Simulation of melting in crystalline polyethylene.

We carry out a molecular dynamics simulation of the first stages of constrained melting in crystalline polyethylene (PE). When heated, the crystal undergoes two structural phase transitions: from the orthorhombic (O) phase to the monoclinic (M) phase, and then to the columnar (C), quasi-hexagonal, phase. The M phase represents the tendency to the parallel packing of planes of PE zigzags, and the C phase proves to be some kind of oriented melt. We follow both the transitions O→M and M→C in real time and establish that, at their beginning, the crystal tries (and fails) to pass into the partially ordered phases similar to the RI and RII phases of linear alkanes, correspondingly. We discuss the molecular mechanisms and driving forces of the observed transitions, as well as the reasons why the M and C phases in PE crystals substitute for the rotator phases in linear alkanes.

Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling.

We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

Nonstationary regimes in a Duffing oscillator subject to biharmonic forcing near a primary resonance.

An analytical investigation of nonstationary processes in a Duffing oscillator subject to biharmonic forcing, under conditions of primary resonance, is carried out. The earlier developed methodology of limiting phase trajectories (LPTs) for studying highly nonstationary regimes, characterized by intense energy exchanges between the different degrees of freedom, is successfully applied to the system under investigation. Two distinct types of LPT trajectories are described in the first part of the study. Conditions for the recurrent transitions in time from one type of LPT to another were derived in the first part of the analysis corresponding to the undamped case. An approximation of the LPT related to the higher amplitude of oscillations was derived using nonsmooth transformations. An analysis carried out in the study has revealed the necessary and sufficient conditions for excitation of relaxation oscillations exhibited by a lightly damped system. It was also demonstrated that the mechanism of relaxations may be approximated and explained by the methodology of LPTs, characterized by strong energy exchanges between the coupled oscillators or, alternatively, a single oscillator and an external source of energy. The results of analytical approximations and numerical simulations are observed to be in quite satisfactory agreement.

Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains.

We demonstrate that the modulation instability of the zone-boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone-boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of effective particles and limiting phase trajectories. The effective particles appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone-boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the limiting phase trajectories corresponding to the most intensive energy exchange between effective particles can be considered as an alternative to nonlinear normal modes, which describe the stationary process.

Discrete breathers in vibroimpact chains: analytic solutions.

We present exact analytic solutions for discrete breathers in essentially nonlinear oscillatory chains, belonging to both of the most common universality classes (Klein-Gordon and Fermi-Pasta-Ulam). The exact solutions can be obtained due to use of vibroimpact potentials, combining extreme nonlinearity with the possibility of description in terms of a forced linear model under conditions of self-consistency. A crossover between the cases of high and low energies can be studied directly. The solutions obtained may be used as a high-energy limit for models with other realistic potentials, as well as benchmarks for the testing of approximate approaches in the theory of discrete breathers.

Wandering breathers and self-trapping in weakly coupled nonlinear chains: classical counterpart of macroscopic tunneling quantum dynamics.

We present analytical and numerical studies of the phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic transverse translation (wandering) of the low-amplitude breather between the chains and the one-chain-localization of the high-amplitude breather. These two modes of coupled nonlinear excitations, which involve a large number of anharmonic oscillators, can be mapped onto two solutions of a single pendulum equation, detached by a separatrix mode. We also show that these two regimes of coupled phase-coherent breathers are similar and are described by a similar pair of equations to the two regimes in the nonlinear tunneling dynamics of two weakly linked interacting (nonideal) Bose-Einstein condensates. On the basis of this profound analogy, we predict a tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around pi/2 mod pi. We also show that the magnitude of the static displacements of the coupled chains with nonlinear localized excitation, induced by the cubic term in the intrachain anharmonic potential, scales approximately as the total vibrational energy of the excitation, either a one- or two-chain one, and does not depend on the interchain coupling. This feature is also valid for a narrow stripe of several parallel-coupled nonlinear chains. We also study two-chain breathers which can be considered as bound states of discrete breathers, with different symmetry and center locations in the coupled chains, and bifurcation of the antiphase two-chain breather into the one-chain one. Bound states of two breathers with different commensurate frequencies are found in the two-chain system. Merging of two breathers with different frequencies into one breather in two coupled chains is observed. Wandering of the low-amplitude breather in a system of several, up to five, coupled nonlinear chains is studied, and the dependence of the wandering period on the number of chains is analytically estimated and compared with numerical results. The delocalizing transition of a one-dimensional (1D) breather in the 2D system of a large number of parallel-coupled nonlinear oscillator chains is described, in which the breather, initially excited in a given chain, abruptly spreads its vibrational energy in the whole 2D system upon decreasing the breather frequency or amplitude below the threshold one. The threshold breather frequency is above the cutoff phonon frequency in the 2D system, and the threshold breather amplitude scales as the square root of the interchain coupling constant. The delocalizing transition of the discrete vibrational breather in 2D and 3D systems of parallel-coupled nonlinear oscillator chains has an analogy with the delocalizing transition for Bose-Einstein condensates in 2D and 3D optical lattices.

Nonlinear dynamics of topological solitons in DNA.

Dynamics of topological solitons describing open states in the DNA double helix are studied in the framework of a model that takes into account asymmetry of the helix. It is shown that three types of topological solitons can occur in the DNA double chain. Interaction between the solitons, their interactions with the chain inhomogeneities, and stability of the solitons with respect to thermal oscillations are investigated.

Supersonic motion of vacancies in a polyethylene crystal.

The possibility of supersonic motion of vacancies in a polyethylene crystal is revealed by means of analytical investigation and numerical simulation. It is demonstrated that in the crystalline field of immovable neighbors, a vacancy with a core size of about 70 CH2 groups and a velocity in the range of 1.02-1.05 sound velocity preserves itself for the time scale of about 1 ns. It is demonstrated that this type of structural defect is similar to coupled supersonic solitons described earlier in the one-dimensional chains with combined gradient and nongradient nonlinearity. An analytic approach is proposed for prediction of their shape and velocity. The simulation of the crystal with all degrees of freedom released demonstrates that the supersonic vacancy is still distinguishable. Its lifetime is less than 5 ps but still may be significant for physical applications.

Simple "kink" model of melt intercalation in polymer-clay nanocomposites.

We propose a simple semiphenomenological model to describe the dynamics of polymer melt intercalation in the gallery between the adjacent clay sheets in polymer-clay nanocomposites. Within this model, the intercalation process is driven by the motion of localized excitations ("kinks") which open up the tip between the clay sheets. These kinks belong to a novel type of solitonlike excitations that appear due to the interplay between the double-well potential of the clay-clay long-range interaction, bending elasticity of the sheets, and external shear force. We find that the kink solutions can exist only if applied shear is sufficiently strong, in a qualitative agreement with experimental data.

Solitons in spiral polymeric macromolecules.

The problem of the existence and stability of dynamical soliton regimes in a helix polymer is solved numerically. For the polytetrafluoroethylene macromolecule, within a model in which deformations of the valence and torsion angles and the valence bonds are taken into account, two types of soliton solutions are found. The first type describes the propagation of a solitary wave of torsional displacements of a helix chain. The twisting of the chain is a result of the compression of dihedral (torsion) angles. The second type describes the propagation of a solitary wave of longitudinal displacements of a helix chain. The longitudinal compression of the chain is a result of the compression of the valence angles and bonds. The solitons have a finite narrow spectrum of supersonic velocities: the soliton of torsion has a spectrum above the velocity of long-wavelength phonons of torsion while the spectrum of the solitons of compression lies above the velocity of long-wavelength phonons of longitudinal displacement. Numerical simulations of the soliton dynamics show their stability in the intervals of admissible velocities. The elasticity of soliton interactions under their collisions is demonstrated. The formation of solitons induced by deformation of end bonds of the helix chain has been modeled. It is shown that helicity of the macromolecule is the necessary condition for existence of torsional solitons.