Periodic Solutions Research Articles

A Resonant Lyapunov Centre Theorem with an Application to Doubly Periodic Travelling Hydroelastic Waves

We present a Lyapunov centre theorem for an antisymplectically reversible Hamiltonian system exhibiting a nondegenerate 1 : 1 or 1:-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1:-1$$\\end{document} semisimple resonance as a detuning parameter is varied. The system can be finite- or infinite-dimensional (and quasilinear) and have a non-constant symplectic structure. We allow the origin to be a ‘trivial’ eigenvalue arising from a translational symmetry or, in an infinite-dimensional setting, to lie in the continuous spectrum of the linearised Hamiltonian vector field provided a compatibility condition on its range is satisfied. As an application, we show how Kirchgässner’s spatial dynamics approach can be used to construct doubly periodic travelling waves on the surface of a three-dimensional body of water (of finite or infinite depth) beneath a thin ice sheet (‘hydroelastic waves’). The hydrodynamic problem is formulated as a reversible Hamiltonian system in which an arbitrary horizontal spatial direction is the time-like variable, and the infinite-dimensional phase space consists of wave profiles which are periodic (with fixed period) in a second, different horizontal direction. Applying our Lyapunov centre theorem at a point in parameter space associated with a 1 : 1 or 1:-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1:-1$$\\end{document} semisimple resonance yields a periodic solution of the spatial Hamiltonian system corresponding to a doubly periodic hydroelastic wave.

Systematic conservation prioritization with the prioritizr R package.

Plans for expanding protected area systems (prioritizations) need to fulfill conservation objectives. They also need to account for other factors, such as economic feasibility and anthropogenic land-use requirements. Although prioritizations are often generated with decision support tools, most tools have limitations that hinder their use for decision-making. We outlined how the prioritizr R package (https://prioritizr.net) can be used for systematic conservation prioritization. This decision support tool provides a flexible interface to build conservation planning problems. It can leverage a variety of commercial (e.g., Gurobi) and open-source (e.g., CBC and SYMPHONY) exact algorithm solvers to identify optimal solutions in a short period. It is also compatible with a variety of spatially explicit (e.g., ESRI Shapefile, GeoTIFF) and nonspatial tabular (e.g., Microsoft Excel Spreadsheet) data formats. Additionally, it provides functionality for evaluating prioritizations, such as assessing the relative importance of different places selected by a prioritization. To showcase the prioritizr R package, we applied it to a case study based in Washington state (United States) for which we developed a prioritization to improve protected area coverage of native avifauna. We accounted for land acquisition costs, existing protected areas, places that might not be suitable for protected area establishment, and spatial fragmentation. We also conducted a benchmark analysis to examine the performance of different solvers. The prioritization identified 12,400km2 of priority areas for increasing the percentage of species' distributions covered by protected areas. Although open source and commercial solvers were able to quickly solve large-scale conservation planning problems, commercial solvers were required for complex, large-scale problems.. The prioritizr R package is available on the Comprehensive R Archive Network (CRAN). In addition to reserve selection, it can inform habitat restoration, connectivity enhancement, and ecosystem service provisioning. It has been used in numerous conservation planning exercises to inform best practices and aid real-world decision-making.

Steady-State Rotary Periodic Solutions of Rigid and Flexible Mechanisms With Large Spatial Rotations Using the Incremental Harmonic Balance Method for Differential-Algebraic Equations

Abstract Steady-state rotary periodic responses of mechanisms lead to stress cycling in flexible structures or connecting joints, which in turn can result in structural fatigue. A general approach is developed to study rotary periodic solutions of rigid and flexible mechanisms with large spatial rotations based on the incremental harmonic balance (IHB) method. The challenge in analyzing such dynamic systems emanates from the noncommutativity of the spatial rotation and the nonsuperposition nature of the rotational coordinates. The generally used rotational coordinates, such as Euler angles, cannot be expanded into Fourier series, which prevents direct usage of the IHB method. To overcome the problem, the natural coordinates method and absolute nodal coordinate formulation (ANCF) are used herein for the dynamic modeling of the rigid and flexible bodies, respectively. The absolute positions and gradients are used as generalized coordinates, and rotational coordinates are naturally avoided. Equations of motions of the system are differential-algebraic equations (DAEs), and they are solved by the IHB method to obtain the steady-state rotary periodic solutions. The effectiveness of the proposed approach is verified by the simulation of rigid and flexible examples with spatial rotations. The approach is general and robust, and it has the potential to be further extended for other extensive multibody dynamic systems.

Stability and Hopf Bifurcation Analysis of a Predator–Prey Model with Weak Allee Effect Delay and Competition Delay

The purpose of this paper is to study a predator–prey model with Allee effect and double time delays. This research examines the dynamics of the model, with a focus on positivity, existence, stability and Hopf bifurcations. The stability of the periodic solution and the direction of the Hopf bifurcation are elucidated by applying the normal form theory and the center manifold theorem. To validate the correctness of the theoretical analysis, numerical simulations were conducted. The results suggest that a weak Allee effect delay can promote stability within the model, transitioning it from instability to stability. Nevertheless, the competition delay induces periodic oscillations and chaotic dynamics, ultimately resulting in the population’s collapse.

Bistability in a predator–prey model characterized by the Crowley–Martin functional response: Effects of fear, hunting cooperation, additional foods and nonlinear harvesting

This study focuses on unraveling key factors influencing predator–prey interactions with Crowley–Martin functional response. Specifically, it explores the roles of additional food sources, harvesting practices, hunting cooperation, fear and its carry-over effects. We analyze equilibrium points and their stability properties through rigorous mathematical methods. Numerical illustrations showcase a diverse range of bifurcations including Hopf, saddle–node, and transcritical, providing a comprehensive understanding of the system’s dynamics. We find that the collaboration among predators during hunting induces instability in the system, leading to the emergence of population cycles from a stable state. Further, we place emphasis on investigating the impact of seasonal forcing by introducing time-varying parameters into our model. We reveal the emergence of periodic solutions, higher periodic solutions and chaotic dynamics due to the seasonal variations of the prey’s birth rate and the degree of hunting cooperation. We also emphasize the significance of incorporating different periodicity of seasonally forced parameters, leading to a more precise understanding of predator–prey dynamics.

Comparison of feed-forward control strategies for simplified vertical hopping model with intrinsic muscle properties

To analyse walking, running or hopping motions, models with high degrees of freedom are usually used. However simple reductionist models are advantageous within certain limits. In a simple manner, the hopping motion is generally modelled by a spring-mass system, resulting in piecewise smooth dynamics with marginally stable periodic solutions. For a more realistic behaviour, the spring is replaced by a variety of muscle models due to which asymptotically stable periodic motions may occur. The intrinsic properties of the muscle model, i.e. preflexes, are usually taken into account in three complexities-constant, linear and Hill-type. In this paper, we propose a semi-closed form feed-forward control which represents the muscle activation and results in symmetrical hopping motion. The research question is whether hopping motions with symmetric force-time history have advantages over asymmetric ones in two aspects. The first aspect is its applicability for describing human motion. The second aspect is related to robotics where the efficiency is expressed in term of performance measures. The symmetric systems are compared with each other and with those from the literature using performance measures such as hopping height, energetic efficiency, stability of the periodic orbit, and dynamical robustness estimated by the local integrity measure (LIM). The paper also demonstrates that the DynIn MatLab Toolbox that has been developed for the estimation of the LIM of equilibrium points is applicable for periodic orbits.

Exploring the dynamics of the heavy magnetized dusty plasma in laboratory experiments and solar wind interaction with lunar terminator: Theoretical predictions into the role of polarization force

This study is motivated by the lack of research that connects the dynamics of space dusty plasma (DP) with the existing experimental findings and the role of the magnetic field in the case of massive DP. For that purpose, we presented a theoretical interpretation of the dynamics of strongly (experimental) and weakly (space) magnetized DP, including the influence of the polarization force (PF). We used the multifluid model to describe the dynamics of dust particles while the hot plasma of electrons and ions is described by the Boltzmann distribution. We derived the dispersion relation to investigate the behavior of the compressive dust-acoustic waves (DAWs). To get a full picture of the mechanism of dust particles, we used the reductive perturbation technique to obtain the periodic solution for the Zakharov–Kuznetsov equation (ZK), which is coincidental with the observed wave profiles. In the case of the experimental application, we observed that the non-propagating instability starts at a small value of the polarization parameter (R), which is 0.02. Therefore, for values of R less than 0.02, the linear analysis revealed that increasing the value of the polarization parameter is not significant in distinguishing the growth rates. Moreover, we noted that both R and the applied magnetic field enhance the strength of the wave electric field. Interestingly, calculating the value of the electric field required to levitate the dust particles that are subject to a high magnetic field agrees with our theoretical findings. The current model has also been applied to DP observed in the lunar terminator. Our results reveal no role for the PF in the lunar terminator. The analysis conducted using the New Hampshire Dispersion Relation Solver (NHDS) confirms the dominance of electrostatic modes in response to the solar wind interaction with the lunar terminator. Moreover, at certain values of plasma parameters such as the temperature of solar wind electrons and ions and the density of dust particles and solar wind electrons, there is a cutoff in the fluid dispersion curve corresponding to certain values of the normalized wavenumber. Such electrostatic profiles could contribute to the observed glow in the horizontal direction of the lunar terminator.

Bright and dark optical solitons in optical metamaterials using a variety of distinct schemes for a generalized Schrodinger equation

Purpose The purpose of this study is to investigate the nonlinear Schrödinger equation (NLS) incorporating spatiotemporal dispersion and other dispersive effects. The goal is to derive various soliton solutions, including bright, dark, singular, periodic and exponential solitons, to enhance the understanding of soliton propagation dynamics in nonlinear metamaterials (MMs) and contribute new findings to the field of nonlinear optics. Design/methodology/approach The research uses a range of powerful mathematical approaches to solve the NLS. The proposed methodologies are applied systematically to derive a variety of optical soliton solutions, each demonstrating unique optical behaviors and characteristics. The approach ensures that both the theoretical framework and practical implications of the solutions are thoroughly explored. Findings The study successfully derives several types of soliton solutions using the aforementioned mathematical approaches. Key findings include bright optical envelope solitons, dark optical envelope solitons, periodic solutions, singular solutions and exponential solutions. These results offer new insights into the behavior of ultrashort solitons in nonlinear MMs, potentially aiding further research and applications in nonlinear wave studies. Originality/value This study makes an original contribution to nonlinear optics by deriving new soliton solutions for the NLS with spatiotemporal dispersion. The diversity of solutions, including bright, dark, periodic, singular and exponential solitons, adds substantial value to the existing body of knowledge. The use of distinct and reliable methodologies to obtain these solutions underscores the novelty and potential applications of the research in advancing optical technologies. The originality lies in the novel approaches used to obtain these diverse soliton solutions and their potential impact on the study and application of nonlinear waves in MMs.

Periodic Solutions of Wave Propagation in a Strongly Nonlinear Monatomic Chain and Their Novel Stability and Bifurcation Analyses

Abstract A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill’s method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature.

The localized excitation on the Weierstrass elliptic function periodic background for the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem associated with the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (gvcKP) equation with the Weierstrass function as the external potential is investigated based on the Lamé function, from which some new localized nonlinear wave solutions on the Weierstrass elliptic ℘-function periodic background are obtained by the Darboux transformation. The degenerate solutions on the ℘-function periodic background for the gvcKP equation can be derived by taking the limits of the half-periods ω 1, ω 2 of ℘(x), whose evolution and corresponding dynamics are also discussed. The findings show that nonlinear waves on the ℘-function periodic background behave as different types of nonlinear waves in different spaces, including periodic waves, vortex solitons and interaction solutions, aiding in elucidating some physical phenomena in the related fields, such as the physical ocean and nonlinear optics.

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Periodic solution for neutral-type differential equation with piecewise impulses on time scales

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg–de Vries (KdV) of the mean-field analysis. Published by the American Physical Society 2024

Lax integrability and nonlinear dispersive wave phenomenon for the (3 + 1) dimensional Kudryashov–Sinelshchikov equation

A (3 + 1) dimensional Kudryashov–Sinelshchikov equation is investigated in this paper, which describes bubbles in the liquid fluctuations. By virtue of the binary Bell polynomials, the bilinear representation, bilinear Bäcklund transformation with associated Lax pair are obtained, respectively. Moreover, utilizing Hirota’s bilinear representation, four new lump solutions are constructed and the interaction phenomenon between lump and periodic solution is thoroughly examined. The work also illustrates the intriguing dynamical behavior with the aid of Maple software, which plots the three-dimensional surface, two-dimensional density, and contour profiles of the solutions constructed in this work in various planes.

Soliton-based modeling of nano-ionic currents in transmission line

Many nonlinear evolution equations, such as the nano-ionic currents (NIC) equation, are used extensively in many scientific and technological domains particularly in nanoelectronics and bioelectronics. The mathematical modeling of NIC phenomena is vital for understanding their behavior and optimizing device performance. Our research leverages an array of mathematical methods, including multi-wave analysis, periodic wave solutions, lump soliton dynamics, breather wave phenomena, homoclinic breathers, M-shaped waveforms, and rogue wave analysis. Additionally, our investigation encompasses the exploration of single kink and double kink configurations, interactions between periodic and kink waves, interaction between M shaped with kink and rogue, interaction between M shaped with one kink, interaction between M shaped with kink and periodic, interaction between M shaped with two kinks as well as periodic wave interactions with lump waves. To further emphasize the structure of solutions derived from particular parameter choices, we include three-dimensional, two-dimensional, streamplot, and contour graphs.

Long-time emergent dynamics of liquid films undergoing thermocapillary instability.

The study of viscous thin film flow has led to the development of highly nonlinear partial differential equationsthat model how the evolution of the film height is affected by different forces. We investigate a model of interaction between surface tension and the thermocapillary Marangoni effect, with a particular focus on the long-time limit. In this limit, the model predicts the creation of an infinite cascade of successively smaller satellite droplets near points where the film thickness vanishes. Motivated by recent progress on the analysis of discrete self-similarity in thin film equations, we compute solutions in a space- and time-rescaled coordinate system. Using this rescaled system we observe the dynamics much further in time than has previously been achieved. The observed behavior is close to, but distinct from, previous observations of discretely self-similar thin film flows, in that the rescaled system does not settle down to a periodic solution, but instead has aspects that continue to evolve monotonically in scaled time. This discovery suggests there are as-yet unexplored ways in which discrete self-similarity may be exhibited.

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Bifurcation analysis of a two-neuron central pattern generator model for both oscillatory and convergent neuronal activities.

The neural oscillator model proposed by Matsuoka is a piecewise affine system that exhibits distinctive periodic solutions. Although such typical oscillation patterns have been widely studied, little is understood about the dynamics of convergence to certain fixed points and bifurcations between the periodic orbits and fixed points in this model. We performed fixed point analysis on a two-neuron version of the Matsuoka oscillator model, the result of which explains the mechanism of oscillation and the discontinuity-induced bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like and grazing bifurcations. Furthermore, it provided theoretical predictions concerning a logarithmic oscillation-period scaling law and noise-induced oscillations observed around those bifurcations. These results are expected to underpin further investigations into oscillatory and transient neuronal activities concerning central pattern generators.

Analytical solution for strongly nonlinear vibration of a stringer shell

In this paper, a new approach called the global residue harmonic balance technique is applied to construct analytical solutions for nonlinear oscillators that appear in cylindrical shells. Periodic solution is analytically proved, and consequently, the relation between amplitude and frequency is obtained in an analytical form. Numerical and analytical comparisons with the Runge–Kutta method and previously existing homotopy perturbation method (HPM) are given, to show the stability, quality, and efficiency of the present approach. The GRHBM provides an outstanding agreement with the numerical results for different amplitude values, leading to sufficiently accurate solutions for nonlinear oscillators.