Periodic Solutions Research Articles

Linear stability of the regular N-gon periodic solutions for the planar N-body problem with quasi-homogeneous potential

Abstract This paper considers the planar N-body problem with a quasi-homogeneous potential given by $$ \begin{align*}W = \sum_{1\leq k<j\leq N} \left[ \frac{ m_k m_j}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|} +\frac{m_k m_j C_{jk}}{\|\boldsymbol{r}_k-\boldsymbol{r}_j\|^p} \right], \end{align*} $$ where $ m_k>0 $ are the masses and $C_{jk}= C_{kj}$ are nonzero real constants, and the exponent g being $ p> $ 1. Generalizing techniques of the classical N-body problem, we first characterize the periodic solutions that form a regular polygon (relative equilibria) with equal masses ( $m_k= m$ , $k=1, \ldots , N$ ) and equal constants $C_{jk}= C$ , for all $j, k=1, \ldots , N$ (for short, N-gon solutions). Indeed, for $C>0$ we prove that there exists a unique regular N-gon solution for each fixed positive mass m. In contrast, for the case $C <0$ , we demonstrate that there can be a maximum of two distinct regular N-gon solutions for a fixed positive mass m. More precisely, there is a range of values for the mass parameter m for which no solutions of the form of an N-gon exist. Furthermore, we examine the linear stability of these solutions, with a particular focus on the special case $ N=3 $ , which is fully characterized.

Bifurcations and exact solutions of the complex Ginzburg–Landau equation with Kudryashov's law

In this paper, we investigate the complex Ginzburg–Landau equation with Kudryashov's law using the method of dynamical systems. By using travelling-wave transformation, the model can be converted into a singular integrable travelling-wave system. Then we discuss the dynamical behaviour of the associated regular system and we obtain bifurcations of the phase portraits of the travelling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, homoclinic and peakon solutions.

Investigating nonlinear dynamic properties of an inertial sensor with rotational velocity-dependent rigidity

This study investigates the nonlinear dynamics of a system with frequency-dependent stiffness using a MEMS-based capacitive inertial sensor as a case study. The sensor is positioned directly on a rotating component of a machine and consists of a microbeam clamped at both ends by fixed supports with a fixed central proof mass. The nonlinear behavior is determined by electrostatic forces, axial and bending motion coupling, and frequency-dependent stiffness. The numerical Galerkin approach is employed for discretization of the coupled differential equations in spatial coordinates. To obtain the sensor response as a function of frequency, a continuation arc-length method based on weak formulation energy balance method is used. This approach uses a physical gradient descent learning based method to obtain unknown coefficients of the considered response. The presented method computes the periodic steady-state solution of the design by considering different frequency contents within the response, including different harmonics of the response expansion. However, the primary and secondary resonances at various harmonic frequencies within the response can be predicted. In this article, the dynamic behavior of the design is investigated by testing different levels of shaft acceleration and different bias voltages. The purpose of the evaluation is to determine the frequency-dependent stiffness of the accelerometer for different levels of shaft speed, taking into account various sensor geometric parameters. The analysis demonstrates the behavior of the sensors under different conditions, particularly highlighting the nonlinearity of the sensor, which causes primary and secondary resonances in the first, second, and third harmonic responses of the accelerometer, especially at higher bias voltages. Fast Fourier analyses indicated that the transversal vibration amplitude of the structure in the second and third harmonies are so considerable that cannot be neglected. The mathematical method used in this study can be applied by researchers and engineers in the design of such sensors to effectively create these devices.

On the existence of periodic solutions under the light of semi-Fredholm operators for some evolution equations

In this work, we investigate the existence of periodic solutions for a class of evolution equations defined as w(t) = (L + C(t))w(t) + H(t). Sufficient conditions on the family of operators {C(t), t ≥ 0}, L and H are proposed to derive the periodicity of solutions from bounded ones on the positive real half-line. The results are obtained by applying the theory of perturbation of semi-Fredholm operators. We suppose that L is a generally nondensely defined operator and satifies the Hille-Yosida condition. Finally, an example of the theoretical findings is presented with numerical simulations.

Smith’s reduction for random processes and application for stochastic differential equations

Abstract The main focus of this work is a result related to multidimensional stochastic differential equations and one-dimensional SDEs (stochastic differential equations), using the Smith’s π-projection approach. This method is a commonly used technique in mathematics to simplify complex systems by projecting them onto lower-dimensional spaces, which can facilitate analysis and resolution. In the context of multidimensional stochastic processes (i.e., systems involving multiple random variables), extending the Smith’s π-projection approach we mention would likely involve defining a multidimensional version of this projection. This could include extending the concepts of projection and dimension reduction to multiple random variables, as well as analyzing how these projections can simplify multidimensional stochastic differential equations. As an application, we will prove the existence of almost periodic solutions to n-dimensional stochastic differential equations using the Smith projection. In fact, we will generalize the result from the article [M.-A. Boudref and A. Berboucha, Existence of almost periodic solutions of stochastic differential equations with periodic coefficients, Random Oper. Stoch. Equ. 25 2017, 1, 57–70] to dimensions n > 1 {n>1} .

Periodic solutions to integro-differential equations: variational formulation, symmetry, and regularity

. We consider nonconstant periodic constrained minimizers of semilinear elliptic equations for integro-differential operators in ℝ. We prove that, after an appropriate translation, each of them is necessarily an even function which is decreasing in half its period. In particular, it has only two critical points in half its period, the absolute maximum and minimum. If these statements hold for all nonconstant periodic solutions, and not only for constrained minimizers, remains as an open problem. Our results apply to operators with kernels in two different classes: kernels K which are convex and kernels for which K(τ 1∕2) is a completely monotonic function of τ. This last new class arose in our previous work on nonlocal Delaunay surfaces in ℝ n . Due to their symmetry of revolution, it gave rise to a 1d problem involving an operator with a nonconvex kernel. Our proofs are based on a not so well-known Riesz rearrangement inequality on the circle 𝕊 1 established in 1976. We also put in evidence a new regularity fact which is a truly nonlocal-semilinear effect and also occurs in the nonperiodic setting. Namely, for nonlinearities in C β and when 2 s + β < 1 (2s being the order of the operator), the solution is not always C 2 s + β − ϵ for all ϵ > 0.

Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems

The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type ddtmx˙1-|x˙|2/c2=-αx|x|3+ε∇xU(t,x),x∈Rd\\{0},with d=2 or d=3, bifurcating, for ε small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that U is T-periodic in time) and the case of the fixed-energy problem (assuming that U is independent of time) are considered.

Numerical approximation and dynamics of periodic solution in distribution of stochastic differential equations

Numerical approximation and dynamics of periodic solution in distribution of stochastic differential equations

Nonlinear dynamic wave properties of travelling wave solutions in in (3+1)-dimensional mKdV-ZK model.

The (3+1)-dimensional mKdV-ZK model is an important framework for studying the dynamic behavior of waves in mathematical physics. The goal of this study is to look into more generic travelling wave solutions (TWSs) for the generalized ion-acoustic scenario in three dimensions. These solutions exhibit a combination of rational, trigonometric, hyperbolic, and exponential solutions that are concurrently generated by the new auxiliary equation and the unified techniques. We created numerous soliton solutions, including kink-shaped soliton solutions, anti-kink-shaped solutions, bell-shaped soliton solutions, periodic solutions, and solitary soliton solutions, for various values of the free parameters in the produced solutions. The attained solutions are displayed geometrically in the surface plot (3-D), contour, and combined two-dimensional (2-D) figures. The combined 2-D figure would make it easier to understand the impact of the speed of the wave. Based on time, the influence of the nonlinear parameter β on wave type is comprehensively investigated using various figures, demonstrating the significant impact of nonlinearity. These graphical representations are based on specific parameter settings, which help to grasp the model's intricate general behavior. However, the results of this research are compared with the outcomes obtained in published literature executed by other scholars. The results indicate the approach's effectiveness and reliability, making it suitable for widespread use in a range of sophisticated nonlinear models. These techniques successfully generate inventive soliton solutions for various nonlinear models, which are crucial in mathematical physics.

Dynamics and soliton solutions of the perturbed Schrödinger-Hirota equation with cubic-quintic-septic nonlinearity in dispersive media

&lt;p&gt;This study systematically investigates the dynamics of the perturbed Schrödinger-Hirota equation with cubic-quintic-septic nonlinearity under spatiotemporal dispersion, providing insights into soliton propagation in dispersive media. We begin by examining the system's phase portrait and chaotic behavior, followed by the derivation of exact traveling wave solutions, including optical solitons and periodic solutions, using an enhanced algebraic method. The findings are vividly illustrated through three-dimensional and two-dimensional graphical simulations, which analyze the impact of key parameters on the solutions. This study not only presents a variety of optical soliton solutions, but also clarifies the underlying dynamics, offering theoretical guidance for fiber optic communication systems and holding significant applied value for achieving more efficient and reliable optical communications.&lt;/p&gt;

Existence of periodic traveling wave solutions of a family of generalized Burgers-Fisher equations

Existence of periodic traveling wave solutions of a family of generalized Burgers-Fisher equations

Existence and Nonexistence of Periodic Solutions for a Class of Fourth‐Order Partial Difference Equations

Existence and Nonexistence of Periodic Solutions for a Class of Fourth‐Order Partial Difference Equations

The applicability limits of the lowest-order substitute model for a cantilever beam system hard-impacting a moving base.

This paper examines the circumstances under which a one-degree-of-freedom approximate system can be employed to predict the dynamics of a cantilever beam comprising an elastic element with a significant mass and a concentrated mass embedded at its end, impacting a moving rigid base. A reference model of the system was constructed using the finite element method, and an approximate lowest-order model was proposed that could be useful in engineering practice for rapidly ascertaining the dynamics of the system, particularly for predicting both periodic and chaotic motions. The number of finite elements in the reference model was determined based on the calculated values of natural frequencies, which were found to correspond to the values of natural frequencies derived from the application of analytical formulas. The precision of the parameter identification and the outcomes yielded by the substitute model were validated through the calculation of the regions of stable periodic solutions using the analytical Peterka method. Subsequently, the qualitative and quantitative limits of the substitute model's applicability were determined. The quantitative limits were delineated through the utilization of Lyapunov exponents and characteristics associated with the energy dissipation due to impacts and the average number of impacts per excitation period. These characteristics provide a foundation for the introduction of global distance measures of the dynamic behavior of diverse systems within a specified range of the control parameter.

Existence and multiplicity results for non-autonomous second-order Hamiltonian systems

Abstract In this article, using the least action principle and minimax methods in critical point theory, some existence and multiplicity results for periodic solutions of second-order Hamiltonian systems are obtained.

Mathematical model for managing vector-borne pathogen outbreaks in chickens using impulsive vaccination and drug treatment

AbstractIn this paper, we propose an epidemic mathematical model with an impulsive vaccination strategy to predict outbreaks in chickens caused by vectors. The analysis of the model is divided into two parts: one considering impulsive vaccination and the other without it. We determine the basic reproduction number of disease transmission and analyze the stability conditions of the proposed model for both disease-free and endemic equilibria, addressing both local and global stability. The results reveal that the disease will die out when the basic reproduction number is less than one. Numerical simulations demonstrate that impulsive vaccination significantly reduces the number of exposed and infected chickens, leading to disease eradication in approximately 270 days, compared to over 360 days without impulsive vaccination. The existence and non-negativity of the model solutions are also investigated. The susceptible population is considered to be vaccinated. We find that the periodic solution of the disease-free equilibrium is locally asymptotically stable under specific conditions outlined in the proposed theorem. This highlights the effectiveness of impulsive vaccination strategies in controlling disease transmission.

Dynamic modeling of the Insulin-Glucose-Glucocorticoid impulsive control system

This paper introduces a class of insulin-glucose-glucocorticoid impulsive systems in the treatment of patients with diabetes to consider the effect of glucocorticoids. The existence and uniqueness of the positive periodic solution of the impulsive model at double fixed time is confirmed for type 1 diabetes mellitus (T1DM) using the function. Further, the global asymptotic stability of the positive periodic solution is achieved following Floquet multiplier theory and comparison principle. Additionally, the permanence of the system is confirmed in type 2 diabetes mellitus (T2DM) via the comparison theorem. Numerical analysis verifies the results of theoretical calculations and indicates that combining therapeutic strategies under hormonal interactions with the dose of exogenous insulin and glucocorticoid medicines within organisms provides more reasonable clinical strategies.

Bifurcation and Exact Solutions of a Concatenation Physical Model

To find the exact explicit solution of the concatenation model, the corresponding differential system of the amplitude component is used, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007], its corresponding traveling wave system is solved and analyzed, obtaining the corresponding phase portraits and showing the dynamical behavior of the amplitude component. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons, are found explicitly.

Approximate analytic periodic solution of circular Sitnikov restricted four-body problem

Approximate analytic periodic solution of circular Sitnikov restricted four-body problem

Fractional Solitons in Optical Twin-Core Couplers with Kerr Law Nonlinearity and Local M-Derivative Using Modified Extended Mapping Method

This study focuses on optical twin-core couplers, which facilitate light transmission between two closely aligned optical fibers. These couplers operate based on the principle of coupling, allowing signals in one core to interact with those in the other. The Kerr effect, which describes how a material’s refractive index changes in response to the intensity of light, induces the nonlinear behavior essential for generating solitons—self-sustaining wave packets that preserve their shape and speed. In our research, we employ fractional derivatives to investigate how fractional-order variations influence wave propagation and soliton dynamics. By utilizing the modified extended mapping method (MEMM), we derive solitary wave solutions for the equations governing the behavior of optical twin-core couplers under Kerr nonlinearity. This methodology produces novel fractional traveling wave solutions, including dark, bright, singular, and combined bright–dark solitons, as well as hyperbolic, Jacobi elliptic function (JEF), periodic, and singular periodic solutions. To enhance understanding, we present physical interpretations through contour plots and include both 2D and 3D graphical representations of the results.

The profile of soliton molecules for integrable complex coupled Kuralay equations

Abstract This study focuses on mathematically exploring the Kuralay equation, which is applicable in diverse fields such as nonlinear optics, optical fibers, and ferromagnetic materials. This study aims to investigate various soliton solutions and analyze the integrable motion of the induced space curves. This study employs traveling wave transformation, converting the partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are derived utilizing both the generalized Jacobi elliptic function expansion (JEFE) method and novel extended direct algebraic (EDA) methods. The outcomes encompass a diverse range of soliton solutions, including double periodic waves, shock wave solutions, kink-shaped soliton solutions, solitary waves, bell-shaped solitons, and periodic wave solutions, obtained using Mathematica. In contrast, the EDA method produces dark, bright, singular, combined dark-bright solitons, dark-singular combined solitons, solitary wave solutions, etc. Visual representation of these soliton solutions is accomplished through 3D, 2D, and contour graphics, with a meticulous selection of parametric values. The graphical presentation underscores the influence of the parameters on the soliton propagation.