Nonlinear Periodic Systems Research Articles

Spontaneous symmetry breaking in nonlinear binary periodic systems

Spontaneous symmetry breaking in nonlinear binary periodic systems

Spontaneous Symmetry Breaking in Diffraction.

We demonstrate spontaneous symmetry breaking in the diffraction of a laser-driven grating with memory in its nonlinear response. We observe, experimentally and theoretically, asymmetric diffraction even when the grating and illumination are symmetric. Our analysis reveals how diffracted waves can spontaneously acquire momentum parallel to the lattice vector in quantities unconstrained by the grating period. Our findings point to numerous opportunities for imaging, sensing, and information processing with nonlinear periodic systems, which can leverage a much richer diffractive response than their linear counterparts.

Some Non-Linear Periodic Systems of Difference Equations

Some Non-Linear Periodic Systems of Difference Equations

Gap solitons with quadratic and quartic dispersion competition in one-dimensional nonlinear periodic systems

Gap solitons with quadratic and quartic dispersion competition in one-dimensional nonlinear periodic systems

The response solution for a class of nonlinear periodic systems under small perturbations

This paper considers periodic perturbations of a class of nonlinear degenerate systems. By the technique of introducing external parameters, the implicit function theorem and the theory of Brouwer topological degree, it is proved that the perturbed systems have response solutions if the perturbations are sufficiently small.

Asymmetric localized states at a nonlinear interface of fractional systems with optical lattices

We investigate the existence and stability of localized gap states at a non-linear interface of non-linear fractional systems in a one-dimensional photonic lattice. By using the direct numerical simulations and linear stability analysis, we obtain the stability of the asymmetric localized gap states in the first and second finite gaps. Our theoretical results show that the power of the localized gap states decrease gradually as the increase of propagation constant and the non-linear landscape (non-linear coefficient ratio between the left and right interface), providing insights into soliton physics in non-linear periodic systems with fractional-order diffraction.

Computational prediction of structure-borne noise in a two-stage double-helical gearbox using harmonic balance method

The acoustics of gearboxes contribute as a quality feature to the success of a gear design by increasing consumer satisfaction. Particularly in special machine engineering, ensuring quiet operation is a challenging task because the production and testing of a prototype is usually not a feasible option. A suitable calculation methodology is thus crucial for finding the optimal design. Unfortunately, an increase in accuracy most often results in high simulation times and thus complicates a continuous design support. A promising approach regarding computational efficiency for nonlinear periodic systems is the harmonic balance method (HBM). However, its application to gearboxes is still quite rare or even non-existent in terms of this acoustic motivation. Convergence behavior and memory requirements are ambitious aspects for the HBM. Consequently, tracking the vibration propagation from the gear mesh to the housing in one simulation run requires special treatment. This article examines appropriate implementation recommendations and further investigates the applicability of the HBM for gear acoustics from a practical point of view. Experimental data serves as the basis for validation purposes.

Adaptive neural control of nonlinear periodic time-varying parameterized mixed-order multi-agent systems with unknown control coefficients

Adaptive neural control of nonlinear periodic time-varying parameterized mixed-order multi-agent systems with unknown control coefficients

Adaptive neural tracking control for stochastic nonlinear multi-agent periodic time-varying systems

This paper investigates the design of adaptive tracking neural controllers for first- and second-order nonlinear multi-agent periodic time-varying systems described based on random differential equations, respectively. First, the Fourier series expansion and neural networks are used as the main tools to describe the nonlinear periodic disturbance dynamics of the follower, which transforms the completely nonlinear problem into a partially linearizable problem. Then, the fully distributed adaptive neural controllers are developed based on adaptive estimation, in which new estimation methods are designed to address periodic time-varying perturbations. Furthermore, based on the stochastic Lyapunov stability theory, it is proved that the tracking errors converge asymptotically to a small neighborhood of zero in the sense of mean square. Finally, the efficicency of the proposed controllers is verified using simulations.

Dark matter-wave gap solitons of Bose-Einstein condensates trapped in optical lattices with competing cubic-quintic nonlinearities

• The existence, property and stability of one-dimensional gap solitons and soliton clusters in periodic nonlinear media with competing cubic-quintic nonlinearities are theoretical studied. • The linear-stability analysis and direct perturbed calculations are adopted to identify the stability and instability of the corresponding dark gap modes. • The model can find its realization in periodic optical media with cubic-quintic nonlinear nonlinearities and in Bose-Einstein condensates trapped by an optical lattice. Solitons are nonlinear self-sustained wave excitations and probably among the most interesting and exciting emergent nonlinear phenomenon in the corresponding theoretical settings. Bright solitons with sharp peak and dark solitons with central notch have been well known and observed in various nonlinear systems . The interplay of periodic potentials, like photonic crystals and lattices in optics and optical lattices in ultracold atoms, with the dispersion has brought about gap solitons within the finite band gaps of the underlying linear Bloch-wave spectrum and, particularly, the bright gap solitons have been experimentally observed in these nonlinear periodic systems, while little is known about the underlying physics of dark gap solitons. Here, we theoretically and numerically investigate the existence, property and stability of one-dimensional matter-wave gap solitons and soliton clusters of Bose-Einstein condensates trapped in optical lattices with competing cubic-quintic nonlinearity, the higher-order of which is self-defocusing and the lower-order (cubic) one is chosen as self-defocusing or focusing nonlinearities. By means of the conventional linear-stability analysis and direct numerical calculations with initial perturbations, we identify the stability and instability areas of the corresponding dark gap solitons and clusters ones.

Harmonic and shock wave propagation in bistable periodic structure: Regularity, randomness, and tunability

Nonlinear periodic structures can present abundant nonlinear wave physics. The model consisting of periodic bistable oscillators (i.e., the bistable periodic structure) is essentially different from those nonlinear periodic systems consisting of monostable oscillators due to multiple equilibria in bistable periodic structure. Despite the extensive attention received, properties of harmonic and shock wave propagation in bistable periodic structure, especially the randomness and tunability behind regularity, have not been fully understood. This article reports the answers based on numerical method. We consider the varying trends of the band gap, vibration center, wave amplitude, and transmission and show their effects on energy transport. We find that the snap-through behavior always presents local intrinsic randomness with the regularity in whole, that is, it does not happen in sequence. For both harmonic and shock wave, most energy is localized inside the snap-through oscillators that changes the regularity for energy transport and is meaningful for shock wave protection. Bistable periodic structure can present very low-frequency and broadband wave attenuation by shifting the initial frequency of the band gap to nearly zero through tuning the wave amplitude to a critical value, which offers dynamic tunability. The damping and intensity of the shock pulse have significant effects on the shock wave propagation. This work provides guidance for the design and application of bistable periodic structure for elastic wave attenuation and shock wave protection.

Application of Lyapunov–Floquet transformation to the nonlinear spacecraft relative motion with periodic-coefficients

The Lyapunov–Floquet (L–F) theory, a powerful result of Floquet theory, is one of the main tools for the stability analysis and study of nonlinear periodic systems. In this paper, Lyapunov–Floquet and modal transformations of the nonlinear approximation model of spacecraft relative motion, containing periodic state dependent coefficients (SDC), are developed. The transformations, advantageously, retain the stability characteristics of the original nonlinear relative motion dynamics. To provide a form to which averaging method can be applied, the nonlinear approximated equation of motion is scaled using the chief’s eccentricity. Through the application of L–F transformation, the linear part with periodic coefficients is reduced to time-invariant form and the transformed periodic nonlinear part has coefficients with true anomaly as the independent variable. The modal transformation reduced LF transformed equation into a form containing Jordan canonical form. Afterward, the averaging method is applied to the L–F transformed equation to describe the evolution of the motion in terms of the average values of the dynamical variables. Both the LF and modal transformed relative motion equations and the averaged equations, with chief in elliptical orbit, present forms that are useful and applicable for spacecraft guidance, navigation and control, formation flying, rendezvous and proximity operations.

Elastic Wave Propagation in Strongly Nonlinear Lattices and Its Active Control

Abstract This work studies elastic wave propagation in strongly nonlinear periodic systems and its active control with specific attention to an infinite mass-in-mass lattice. Piezoelectric materials are applied to it to provide active control loads to manipulate band structures of the lattice. Governing equations of the active mass-in-mass lattice with cubic nonlinearities are established. The control loads are modeled by using linear piezoelectric springs. Due to phase differences among vibrations of different cells during wave propagation, a series of delay functions with different delays are used to represent the steady-state of a traveling wave. The incremental harmonic balance method for delay dynamic systems is employed in this case to calculate periodic solutions of the lattice. The fast Fourier transform is employed to construct the Jacobian matrix of the Newton–Raphson iteration to avoid a large number of Galerkin integrations, and thus, the efficiency is significantly improved. Amplitude-dependent dispersion curves are calculated using results of the linearized system as an initial guess for the iteration. The results are compared with existing results in the literature, which demonstrates that the present method is efficient for wave propagation analysis of strongly nonlinear structures. Effects of nonlinearities, the mass ratio, and different control actions on band structures of the mass-in-mass lattice are investigated through a comprehensive parametric study. Numerical results show that the band structures can be influenced by nonlinearities of the lattice. New stopbands and critical wave numbers can be created by the control actions.

Dispersive fractalisation in linear and nonlinear Fermi–Pasta–Ulam–Tsingou lattices

We investigate, both analytically and numerically, dispersive fractalisation and quantisation of solutions to periodic linear and nonlinear Fermi–Pasta–Ulam–Tsingou systems. When subject to periodic boundary conditions and discontinuous initial conditions, e.g., a step function, both the linearised and nonlinear continuum models for FPUT exhibit fractal solution profiles at irrational times (as determined by the coefficients and the length of the interval) and quantised profiles (piecewise constant or perturbations thereof) at rational times. We observe a similar effect in the linearised FPUT chain at timestwhere these models have validity, namelyt= O(h−2), wherehis proportional to the intermass spacing or, equivalently, the reciprocal of the number of masses. For nonlinear periodic FPUT systems, our numerical results suggest a somewhat similar behaviour in the presence of small nonlinearities, which disappears as the nonlinear force increases in magnitude. However, these phenomena are manifested on very long time intervals, posing a severe challenge for numerical integration as the number of masses increases. Even with the high-order splitting methods used here, our numerical investigations are limited to nonlinear FPUT chains with a smaller number of masses than would be needed to resolve this question unambiguously.

Existence and subharmonicity of solutions for nonsmooth $ p $-Laplacian systems

<abstract><p>In this paper we study nonlinear periodic systems driven by the vectorial $ p $-Laplacian with a nonsmooth locally Lipschitz potential function. Using variational methods based on nonsmooth critical point theory, some existence of periodic and subharmonic results are obtained, which improve and extend related works.</p></abstract>

Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian

Abstract In the present paper, we consider the nonlinear periodic systems involving variable exponent driven by p(t)-Laplacian with a locally Lipschitz nonlinearity. Our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue-Sobolev space. Applying the non-smooth critical point theory, we establish some new existence results.

Normalization of a Periodic Hamiltonian System

First, the normal form in the vicinity of the stationary solution of the autonomous Hamiltonian system is recalled. Next, linear periodic Hamiltonian systems are considered. For them, normal forms of the Hamiltonians in the complex and real cases are found. A feature of the real case in the situation of parametric resonance is discovered. Next, normal forms of Hamiltonians of nonlinear periodic systems are found. Using an additional canonical transformation, such a normal form can be always reduced to an autonomous Hamiltonian system that preserves all small parameters and symmetries of the original system. The local families of fixed points of this autonomous system are associated with families of periodic solutions to the original system. A similar theory is constructed in the neighborhood of the periodic solution to the autonomous system. All transformations are algorithmic and can be implemented in a computer algebra system.

In Memoriam: Subhash C. Sinha

In Memoriam: Subhash C. Sinha

Method of Guiding Functions for Existence Problems for Periodic Solutions of Differential Equations

We provide a review and systematic explanation of various generalizations of the guiding function method. The current state of the said method and its applications to various kinds of problems for nonlinear periodic systems described by differential and functional differential equations are considered.

Nonreciprocal phenomenon in nonlinear elastic wave metamaterials with continuous properties

The nonreciprocal phenomenon appears in nonlinear acoustic metamaterial which are composed of a linear phononic crystal and a nonlinear layer. Based on the nonlinear characteristics, the acoustic waves can pass in one direction and are prohibited in reverse. Naturally, people will wonder whether the nonreciprocal phenomenon can be found in nonlinear elastic wave metamaterial. Because of the coupling of the multi-displacements, this problem becomes more complicated but interesting for waves in solid. Motivated by this enlightenment, we will discuss the band gap and transmission properties of elastic waves in both linear and nonlinear periodic systems. The fundamental and double frequency waves can be generated by the interaction between SH waves and the nonlinearity of materials. According to continuity conditions of incident SH wave, the transfer matrices are derived. The stop band and transmission coefficient are obtained by Bloch's law. This research is expected to be helpful to develop new elastic wave metamaterial devices.