Experimental study of steady-state localization in coupled beams with active nonlinearities

Summary Steady-state nonlinear motion confinement is experimentally studied in a system of weakly coupled cantilever beams with active stiffness nonlinearities. Quasistatic swept-sine tests are performed by periodically forcing one of the beams at frequencies close to the first two closely spaced modes of the system, and experimental nonlinear frequency response curves for certain nonlinearity levels are generated. Of particular interest is the detection of strongly localized steady-state motions, wherein vibrational energy becomes spatially confined mainly to the directly excited beam. Such motions exist in neighborhoods of strongly localized antiphase nonlinear normal modes (NNMs) which bifurcate from a spatially extended NNM of the system. Steady-state nonlinear motion confinement is an essentially nonlinear phenomenon with no counterpart in linear theory, and can be implemented in vibration and shock isolation designs of mechanical systems.Presently Assistant Professor of Aerospace and Mechanical Engineering, Boston University (from January 1995).     

Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearities modulated in space and time

This work deals with soliton solutions of the nonlinear Schrödinger equation with a diversity of nonlinearities. We solve the equation in a potential which oscillates in time between attractive and expulsive behavior, in the presence of nonlinearities which are modulated in space and time. Despite the presence of the periodically expulsive behavior of the potential, the results show that the nonlinear equation can support a diversity of localized excitations of the bright and dark types.     

Multi-instability of gap solitons and dynamics of nonlinear excitations in the array of optical fibers

The nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method. The perturbative analysis in nonlinear discrete systems leads to a system of four coupled nonlinear oscillated equations without second members. The method of multiple scales is used to show the different possibilities of instability. The averaging method is considered to show how the angular frequency received the perturbation with the evolution of time.     

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part I: Experimental study

This paper presents an experimental investigation on nonlinear low frequency gravity water waves in a partially filled cylindrical shell subjected to high frequency horizontal excitations. The characteristics of natural frequencies and mode shapes of the water–shell coupled system are discussed. The boundaries for onset of gravity waves are measured and plotted by curves of critical excitation force magnitude with respect to excitation frequency. For nonlinear water waves, the time history signals and their spectrums of motion on both water surface and shell are recorded. The shapes of water surface are also measured using scanning laser vibrometer. In particular, the phenomenon of transitions between different gravity wave patterns is observed and expressed by the waterfall graphs. These results exhibit pronounced nonlinear properties of shell–fluid coupled system.     

Nonlinear optical media in photonic crystal waveguides: Intrinsic localized modes and device applications

A review is given of some of the properties of photonic crystal waveguides containing or interacting with nonlinear optical media including discussions of potential device applications. An introduction is given to photonic crystals, photonic crystal waveguide technologies, the properties of Kerr nonlinear dielectric media, the properties of new types of excitations in these systems known as intrinsic localized modes, and optical bistability. Device designs for switches, waveguide couplers, interferometers, transistors, photonic crystal receivers and transmitters, and photonic crystal sensors are reviewed. © 2007 Wiley Periodicals, Inc. Complexity 12: 18–32, 2007     

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.     

Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.     

Exchange of stability and finite-dimensional dynamics in a bifurcation problem with marginally stable continuous spectrum

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a real eigenvalue crosses the imaginary axis. For a model we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a pitchfork bifurcation of equilibria and the nonlinear stability of the bifurcating equilibria, again with respect to spatially localized perturbations.     

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field. Oblatum: 6-XI-1998 & 12-VI-1998 / Published online: 14 January 1999     

Weakly Nonlinear Analysis of a Localized Disturbance in Poiseuille Flow

In this article, we investigate, via a perturbation analysis, some important nonlinear features related to the process of transition to turbulence in a wall-bounded flow subject to a spatially localized disturbance that is harmonic in time. We show that the perturbation expansion, truncated at second order, is able to capture the generation of streamwise vorticity as a weakly nonlinear effect. The results of the perturbation approach are discussed in comparison with direct numerical simulation data for a sample case by extracting the contribution of the different orders. The main aim is to provide a tool to select the most effective nonlinear interactions to enlighten the essential features of the transitional process.     

Domain effect on the behavior of solutions of a distributed kinetic system

A distributed kinetic system that is in homogeneous equilibrium in a flat circular reactor is considered. Its behavior under deformations of the circular domain is studied. It is shown that a domain deformation may lead to the formation of stable spatially inhomogeneous oscillatory solutions in the neighborhood of the homogeneous equilibrium. The possibility of developing chaotic oscillations is discussed. This mechanism of creating spatially inhomogeneous nonlinear oscillations in the distributed kinetic system is called the domain effect.     

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.     

Classical Skyrmions—static solutions and dynamics

Atomic nuclei can be modelled by Skyrmions with quantised spin and isospin. Skyrmions with a realistic value of the pion mass parameter should be quite compact structures, but few examples beyond baryon number 8 are known. The largest of these is the cubic Skyrmion with baryon number 32, which is a truncated piece of the Skyrme crystal. Here, it is argued that many Skyrmions are pieces of the Skyrme crystal. Particular attention is given to cubic crystal chunks and ways to reduce the baryon number by chopping corners off. It is also argued that quantised Skyrmion states can be approximated by classically spinning Skyrmions, and the orientations of spinning Skyrmions corresponding to polarised protons, neutrons and deuterons are identified. Nuclear collisions and the nuclear spin–orbit force are discussed in terms of classically spinning Skyrmions. Going beyond the rigid collective motions of Skyrmions, there are spatially modulated collective motions, oscillatory in time, which model vibrational excitations of nuclei and also giant resonance states. A speculative proposal for identifying quarks inside Skyrmions is briefly discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

The variational approximation for 2‐dimension solitons at cubic‐quintic nonlinearity in quasiperiodic potentials

We apply the variational approximation to study the dynamics of solitary waves of the nonlinear Schrödinger equation with compensative cubic‐quintic nonlinearity for asymmetric 2‐dimension setup. Such an approach allows to study the behavior of the solitons trapped in quasisymmetric potentials without an axial symmetry. Our analytical consideration allows finding the soliton profiles that are stable in a quasisymmetric geometry. We show that small perturbations of such states lead to generation of the oscillatory‐bounded solutions having 2 independent eigenfrequencies relating to the quintic nonlinear parameter. The behavior of solutions with large amplitudes is studied numerically. The resonant case when the frequency of the time variations (time managed) potential is near of the eigenfrequencies is studied too. In a resonant situation, the solitons acquire a weak time decay.     

The Einstein-Vlasov System with a Scalar Field

We study the Einstein-Vlasov system coupled to a nonlinear scalar field with a nonnegative potential in locally spatially homogeneous space-time, as an expanding cosmological model. It is shown that solutions of this system exist globally in time. When the potential of the scalar field is of an exponential form, the cosmological model corresponds to accelerated expansion. The Einstein-Vlasov system coupled to a nonlinear scalar field whose potential is of an exponential form shows the causal geodesic completeness of the space-time towards the future. The asymptotic behavior of solutions of this system in the future time is analyzed in various aspects, which shows power-law expansion.Communicated by Sergiu Klainermansubmitted 01/04/04, accepted 14/06/04     

Impurity modes in a curved Fermi–Pasta–Ulam chain

The spatial localization properties of nonlinear excitations/modes supported by a curved Fermi–Pasta–Ulam (FPU) lattice chain in presence of an isolated impurity of mass lighter or heavier than the host mass, is investigated. The impurity modes oscillate locally at and around the impurity site. It is examined that a light-mass impurity mode fulfills non-resonance with the linear (or phonon) spectrum because its frequency is located above the phonon band whereas frequency of a heavy-mass impurity mode drops into the phonon band. The phenomenon of resonance of impurities with plane waves explains the lifetimes of localized impurity modes in the nonlinear system.     

Oscillatory and asymptotic criteria of third order nonlinear delay dynamic equations with damping term on time scales

In this paper, we are concerned with oscillatory and asymptotic behavior of third order nonlinear delay dynamic equations with damping term on time scales. By using a generalized Riccati function and inequality technique, we establish some new oscillatory and asymptotic criteria. The established results on one hand extend some known results in the literature, on the other hand unify continuous and discrete analysis as two special cases of an arbitrary time scale. We also present some applications for the established results.     

Modelling of nonlinear modal interactions in the transient dynamics of an elastic rod with an essentially nonlinear attachment

We perform system identification and modelling of the strongly nonlinear modal interactions in a system composed of a linear elastic rod with an essentially nonlinear attachment at its end. Our method is based on slow/fast decomposition of the transient dynamics of the system, combined with empirical mode decomposition (EMD) and Hilbert transforms. The derived reduced order models (ROMs) are in the form of sets of uncoupled linear oscillators (termed intrinsic modal oscillators – IMOs), each corresponding to a basic frequency of the dynamical interaction and forced by transient excitations that represent the nonlinear modal interactions between the rod and the attachment at each of these basic frequencies. A main advantage of our proposed technique is that it is nonparametric and multi-scale, so it is applicable to a broad range of linear as well as nonlinear dynamical systems. Moreover, it is computationally tractable and conceptually meaningful, and it leads to reduced order models of rather simple form that fully capture the basic strongly nonlinear resonant interactions between the subsystems of the problem.     

Dynamics of an impact-progressive system

An impact oscillator with a frictional slider is considered. The basic function of the investigated system is to overcome the frictional force and move downwards. Based on the analysis of the oscillatory and progressive motions of the system, we introduce an impact Poincaré map with dynamical variables defined at the impact instants. The nonlinear dynamics of the impact system with a frictional slider is analyzed by using the impact Poincaré map. The stability and bifurcations of single-impact periodic motions are analyzed, and some information about the existence of other types of periodic-impact motions is provided. Since the system equilibrium is moving downwards, one way to monitor the progression rate is to calculate its progression in a finite time. The simulation results show that in a finite time, the largest progression of the system is found to occur for period-1 multi-impact motions existing in the regions of low forcing frequencies. Secondly, the progression of the period-1 single-impact motion with peak-impact velocity is also distinct enough. However, it is important to note, that the largest progression for period-1 multi-impact motion existing at a low forcing frequency is not an optimal choice for practical engineering applications. The greater the number of the impacts in an excitation period, the more distinct the adverse effects such as high noise levels and wear and tear caused by impacts. As a result, the progression of the period-1 single-impact motion with the peak-impact velocity is still optimal for practical applications. The influence of parameter variations on the oscillatory and progressive motions of the impact-progressive system are elucidated accordingly, and feasible parameter regions are provided.     

Vibrations of linear structures with spatial local nonlinearities

Classical Modal Analysis can be applied to linear systems if the corresponding damping matrix is proportional to the mass or/and stiffness matrices. Otherwise, e.g., in case of structures with single external damping devices, an alternative or approximate solution procedure for determining the dynamic response has to be chosen, compare [1]–[3]. Vibration problems of linear structures with spatially localized nonlinearities are related to those non-classically damped systems. Such systems are characterized by the fact that their nonlinear behavior is largely restricted to a limited number of single points in the structure. The objective of this paper is to present an approximate semi-analytical procedure for analyzing the steady-state harmonic response of those locally nonlinear structures, where special emphasis is laid on beams with single nonlinear devices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)