In this work, we consider a generalized form of the classical (2+1)-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results.

In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order β of the temporal derivative to that of a Caputo fractional type and found that, for 1<β<2, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α=2, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.

The aim of this article is to introduce the distributed-order hyperchaotic detuned (DOHD) laser model. Its dissipative dynamics, invariance, and fixed points (FPs) and their stability are investigated. Numerical solutions of the DOHD laser model are computed using the modified Predictor–Corrector approach. Its viscoelasticity is described by the so-called DO derivative, allowing for the study of different technical systems and materials, and the model is found to have a whole circle of FPs as a hyperchaotic attractor. We discuss the coexistence of more attractors under various initial conditions and the same sets of parameters for our model (multistability). We also introduce the notion of dual combination synchronization (DCS), using four integer-order drive models and two DO response models. A theorem is stated and proved to obtain an analytical control function that ensures DCS for our models. Numerical simulations are presented to support these analytical results. Regarding the use of the well–known Caputo derivative, the results are very similar to those of DO, except when the Caputo order, 0<σ≤1, is very close to 1, where the dynamics shows a “spiralling behavior” towards a fixed point. In all other cases, both Caputo and DO exhibit a very similar behavior.

In honor of the great Russian mathematician A. N. Kolmogorov, we would like to draw attention in the present paper to a curious mathematical observation concerning fractional differential equations describing physical systems, whose time evolution for integer derivatives has a time-honored conservative form. This observation, although known to the general mathematical community, (Achar et al., 2001; Stanislavsky, 2004; Diethelm and Ford, 2010; Chung and Jung, 2014; Olivar-Romero and Rosas-Ortiz, 2017; Baleanu et al., 2020) has not, in our view, been satisfactorily addressed. More specifically, we follow the recent exploration of Caputo–Riesz time–space-fractional nonlinear wave equation of Macias Diaz (2022), in which two of the present authors introduced an energy-type functional and proposed a finite-difference scheme to approximate the solutions of the continuous model. The relevant Klein–Gordon equation considered here has the form: (0.1)∂βϕ(x,t)∂tβ−Δαϕ(x,t)+F′(ϕ(x,t))=0,∀(x,t)∈(−∞,∞)where we explore the sine-Gordon nonlinearity F(ϕ)=1−cos(ϕ) with smooth initial data. For α=β=2, we naturally retrieve the exact, analytical form of breather waves expected from the literature. Focusing on the Caputo temporal derivative variation within 1<β<2 values for α=2, however, we observe artificial dissipative effects, which lead to complete breather disappearance, over a time scale depending on the value of β. We compare such findings to single degree-of-freedom linear and nonlinear oscillators in the presence of Caputo temporal derivatives and also consider anti-damping mechanisms to counter the relevant effect. These findings also motivate some interesting directions for further study, e.g., regarding the consideration of topological solitary waves, such as kinks/antikinks and their dynamical evolution in this model.

We investigated a class of one-dimensional (1D) Hamiltonian N-particle lattices whose binary interactions are quadratic and/or quartic in the potential. We also included on-site potential terms, frequently considered in connection with localization phenomena, in this class. Applying a sinusoidal perturbation at one end of the lattice and an absorbing boundary on the other, we studied the phenomenon of supratransmission and its dependence on two ranges of interactions, 0<α<∞ and 0<β<∞, as the effect of the on-site potential terms of the Hamiltonian varied. In previous works, we studied the critical amplitude As(α,Ω) at which supratransmission occurs, for one range parameter α, and showed that there was a sharp threshold above which energy was transmitted in the form of large-amplitude nonlinear modes, as long as the driving frequency Ω lay in the forbidden band-gap of the system. In the absence of on-site potentials, it is known that As(α,Ω) increases monotonically the longer the range of interactions is (i.e., as α⟶0). However, when on-site potential terms are taken into account, As(α,Ω) reaches a maximum at a low value of α that depends on Ω, below which supratransmission thresholds decrease sharply to lower values. In this work, we studied this phenomenon further, as the contribution of the on-site potential terms varied, and we explored in detail their effect on the supratransmission thresholds.

Ферромагнитные диссипативные системы, описывающиеся изотропным уравнением Ландау-Лифшица-Гильберта, изучаются с точки зрения их пространственно локализованных динамических возбуждений. В частности, строятся диссипативные солитонные решения нелокального нелинейного уравнения Шредингера, в которое преобразуется уравнение Ландау-Лифшица-Гильберта. Для доказательства существования этих решений при достаточно малом рассеянии используется теория Мельникова. Для проверки достоверности полученных аналитических результатов применяются псевдоспектральные численные методы и физически-информированные нейронные сети в схеме машинного обучения. Такие локализованные структуры были обнаружены экспериментально в магнитных системах и наблюдались в наноосцилляторах, а магнитные капли, описывающиеся диссипативными солитонами, были исследованы теоретически и наблюдались в эксперименте.

We study ferromagnetic dissipative systems described by the isotropic LLG equation, from the standpoint of their spatially localized dynamical excitations. In particular, we focus on dissipative soliton solutions of a nonlocal NLS equation to which the LLG equation is transformed and use Melnikov’s theory to prove the existence of these solutions for sufficiently small dissipation. Next, we employ pseudospectral and PINN (physics-informed neural network) numerical techniques of machine learning to demonstrate the validity of our analytic results. Such localized structures have been detected experimentally in magnetic systems and observed in nano-oscillators, while dissipative magnetic droplet solitons have also been found theoretically and experimentally.

The important phenomenon of "stickiness" of chaotic orbits in low dimensional dynamical systems has been investigated for several decades, in view of its applications to various areas of physics, such as classical and statistical mechanics, celestial mechanics and accelerator dynamics. Most of the work to date has focused on two-degree of freedom Hamiltonian models often represented by two-dimensional (2D) area preserving maps. In this paper, we extend earlier results using a 4-dimensional extension of the 2D McMillan map, and show that a symplectic model of two coupled McMillan maps also exhibits stickiness phenomena in limited regions of phase space. To this end, we employ probability distributions in the sense of the Central Limit Theorem to demonstrate that, as in the 2D case, sticky regions near the origin are also characterized by "weak" chaos and Tsallis entropy, in sharp contrast to the "strong" chaos that extends over much wider domains and is described by Boltzmann Gibbs statistics. Remarkably, similar stickiness phenomena have been observed in higher dimensional Hamiltonian systems around unstable simple periodic orbits at various values of the total energy of the system.

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

The quasipolynomial (QP) formalism and the Painlevé property constitute two distinct approaches for studying the integrability of systems of ordinary differential equations with polynomial nonlinearities. The former relies on a set of quasimonomial variable transformations, which explore the existence of hidden quasipolynomial invariants, while the latter requires that all solutions be meromorphic, expressed in the form of Laurent series in the complex time domain. In this paper, we compare the effectiveness of these approaches as independent methods for identifying integrals of motion, in many examples of polynomial dynamical systems of physical interest.

Energy transport in 1-dimensional oscillator arrays has been extensively studied to date in the conservative case, as well as under weak viscous damping. When driven at one end by a sinusoidal force, such arrays are known to exhibit the phenomenon of supratransmission, i.e. a sudden energy surge above a critical driving amplitude. In this paper, we study 1-dimensional oscillator chains in the presence of hysteretic damping, and include nonlinear stiffness forces that are important for many materials at high energies. We first employ Reid's model of local hysteretic damping, and then study a new model of nearest neighbor dependent hysteretic damping to compare their supratransmission and wave packet spreading properties in a deterministic as well as stochastic setting. The results have important quantitative differences, which should be helpful when comparing the merits of the two models in specific engineering applications.

Квазиполиномиальный формализм и свойство Пенлеве составляют два различных подхода к изучению интегрируемости систем обыкновенных дифференциальных уравнений с полиномиальными нелинейностями. Первый подход основан на наборе квазимономиальных преобразований переменных, с помощью которых исследуется существование скрытых квазиполиномиальных инвариантов, а во втором подходе требуется, чтобы все решения были мероморфными и представлялись в виде рядов Лорана в комплексной временно́й области. На многих примерах полиномиальных динамических систем, представляющих физический интерес, проведено сравнение эффективности этих подходов как независимых методов определения интегралов движения.

In a recent paper [1], completely integrable cases were discovered of the Lotka Volterra Hamiltonian (LVH) system without linear terms, dx(i)/dt = (x)over dot(i) = Sigma(n)(j=1) a(ij)x(i)x(j) = 1, ..., n, a(ij) = -a(ji), satisfying the condition H = Sigma(n)(i=1) x(i) = h = const. In this paper, we first generalize this system to one that includes an arbitrary set of linear terms that preserve the Hamiltonian integral. We thus discover a wide class of LVH systems which we claim to be integrable, since their equations possess the Painleve property, i.e. their solutions have only poles as movable singularities in the complex t-plane. Next, we focus on the case n = 3 and vary some of the parameters, including additional nonlinearities to look for nonintegrable extensions with interesting dynamical properties. Our results suggest that, in this class of systems, non integrability generally yields simple dynamics far removed from the type of complexity one expects from non - integrable 3 - dimensional nonlinear systems. (C) 2021 Published by Elsevier B.V.

We study the solitary wave solutions of a non-integrable generalized KdV equation proposed by Fokas [A. S. Fokas, Physica D 87, 145 (1995)], aiming to describe unidirectional waves in shallow water with greater accuracy than the standard KdV equation.This generalized equation includes higher-order terms in the small parameters α and β, representing respectively the height and inverse width of the wave compared to the thickness of the water sheet.The solitary waves we find have a smaller height and a larger width than the corresponding KdV soliton at the same propagation velocity.Extrapolating these results we conjecture that -in the limit of arbitrarily high order in α and β -the solitary waves will attain a specific, finite height and width as the wave speed c increases.

We investigate a family of one-dimensional (1D) Hamiltonian semi-infinite particle lattices whose interactions involve exclusively terms of fourth order in the potential. Our aim is to examine their distinct role in the dynamics, in the absence of quadratic (harmonic) interactions, which are typically included in most studies, as they are known to play an important role in many physical phenomena. We also include in our potentials on-site terms of the sine-Gordon type, which are also considered in many studies in connection with localization effects. Our 1D lattices are subjected to sinusoidal perturbation on one end and an absorbing boundary on the other. To simulate a semi-infinite chain, we will consider a relatively long chain with string coupling. Using reliable finite difference discretization schemes, we establish the existence of nonlinear supratransmission for both short-range and long-range interactions, and demonstrate that the presence of quadratic interactions is not necessary for a system to show nonlinear supratransmission. Additionally, we provide diagrams depicting novel relations between the critical amplitude at which supratransmission is triggered versus driving frequency and a parameter measuring the length of the interactions. Our investigation also shows that the presence of on-site potentials is also not crucial for the system to present supratransmission.

We investigate the local and global dynamics of two 1-Dimensional (1D) Hamiltonian lattices whose inter-particle forces are derived from nonanalytic potentials. In particular, we study the dynamics of a model governed by a "graphene-type" force law and one inspired by Hollomon's law describing "work-hardening" effects in certain elastic materials. Our main aim is to show that, although similarities with the analytic case exist, some of the local and global stability properties of nonanalytic potentials are very different than those encountered in systems with polynomial interactions, as in the case of 1D Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion in the neighborhood of simple periodic orbits representing continuations of normal modes of the corresponding linear system, as the number of particles N and the total energy E are increased. We find that the graphene-type model is remarkably stable up to escape energy levels where breakdown is expected, while the Hollomon lattice never breaks, yet is unstable at low energies and only attains stability at energies where the harmonic force becomes dominant. We suggest that, since our results hold for large N, it would be interesting to study analogous phenomena in the continuum limit where 1D lattices become strings.

Arrays of coupled semiconductor lasers are systems possessing radically complex dynamics that makes them useful for numerous applications in beam forming and beam shaping. In this work, we investigate the spatial controllability of oscillation amplitudes in an array of coupled photonic dimers, each consisting of two semiconductor lasers driven by differential pumping rates. We consider parameter values for which each dimer's stable phase-locked state has become unstable through a Hopf bifurcation and we show that, by assigning appropriate pumping rate values to each dimer, high-amplitude oscillations coexist with negligibly low-amplitude oscillations. The spatial profile of the amplitude of oscillations across the array can be dynamically controlled by appropriate pumping rate values in each dimer. This feature is shown to be quite robust, even for random detuning between the lasers, and suggests a mechanism for dynamically reconfigurable production of a large diversity of spatial profiles of laser amplitude oscillations.

•Cantilever beam.•Swelling pressure.•Lumped mass model.•Periodic solution and Resonances.

We consider the integrable (zero perturbation) two–dimensional standard map, in light of current developments on ergodic sums of irrational rotations, and recent numerical evidence that it might possess non-trivial q-Gaussian statistics. Using both classical and recent results, we show that the phase average of the sum of centered positions of an orbit, for long times and after normalization, obeys the Cauchy distribution (a q-Gaussian with q=2), while for almost all individual orbits such a sum does not obey any distribution at all. We discuss the question of existence of distributions for KAM tori.