Existence Of Periodic Solutions Research Articles

Three-Body 3D-Kepler Electromagnetic Problem—Existence of Periodic Solutions

The main purpose of the present paper is to prove the existence of periodic solutions of the three-body problem in the 3D Kepler formulation. We have solved the same problem in the case when the three particles are considered in an external inertial system. We start with the three-body equations of motion, which are a subset of the equations of motion (previously derived by us) for any number of bodies. In the Minkowski space, there are 12 equations of motion. It is proved that three of them are consequences of the other nine, so their number becomes nine, as much as the unknown trajectories are. The Kepler formulation assumes that one particle (the nucleus) is placed at the coordinate origin. The motion of the other two particles is described by a neutral system with respect to the unknown velocities. The state-dependent delays arise as a consequence of the finite vacuum speed of light. We obtain the equations of motion in spherical coordinates and split them into two groups. In the first group all arguments of the unknown functions are delays. We take their solutions as initial functions. Then, the equations of motion for the remaining two particles must be solved to the right of the initial point. To prove the existence–uniqueness of a periodic solution, we choose a space consisting of periodic infinitely smooth functions satisfying some supplementary conditions. Then, we use a suitable operator which acts on these spaces and whose fixed points are periodic solutions. We apply the fixed point theorem for the operators acting on the spaces of periodic functions. In this manner, we show the stability of the He atom in the frame of classical electrodynamics. In a previous paper of ours, we proved the existence of spin functions for plane motion. Thus, we confirm the Bohr and Sommerfeld’s hypothesis for the He atom.

The soliton and periodic solutions for the perturbed sine–cosine–Gordon equation

In the paper, the existence of soliton and periodic solutions is studied for the perturbed sine–cosine–Gordon equation. The corresponding traveling wave system is converted to a regularly Hamilton system via using bifurcation theory of differential equations and the geometric singular perturbation theory. Then by using Melnikov's method and symbolic computation, periodic wave solutions, solitary solutions, and kink and antikink solutions are obtained. The wave speed selection principles are given, respectively. Numerical simulations are performed to illustrate our theoretical analysis.

On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems

Abstract The main goal of this article is to study the existence and uniqueness of periodic solutions for the implicit problem with nonlinear fractional differential equation involving the Caputo tempered fractional derivative. The proofs are based upon the coincidence degree theory of Mawhin. To show the efficiency of the stated result, two illustrative examples will be demonstrated.

Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks

In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.

Periodic solutions in a 2D-symmetric Hamiltonian system through reduction and averaging method

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of π / 2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow.

Magnetoacoustic waves in spin-1/2 dense quantum degenerate plasma: nonlinear dynamics and dissipative effects

Abstract Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

Existence of Periodic Solutions for Second-Order Ordinary p-Laplacian Systems

In this paper, we study the variational principle and the existence of periodic solutions for a new class of second-order ordinary p-Laplacian systems. The variational principle is given by making use of two methods. We obtain three existence theorems of periodic solutions to this problem on various sufficient conditions on the potential function F(t,x) or nonlinearity ∇F(t,x). Four examples are presented to illustrate the feasibility and effectiveness of our results.

Periodic solution problems of neutral-type stochastic neural networks with time-varying delays

This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems.

Existence of Periodic Solutions for a Class of p-Laplacian Systems with Delay

Existence of Periodic Solutions for a Class of p-Laplacian Systems with Delay

Periodic Solutions in Kolmogorov-Type Predator–Prey Systems

AbstractWe consider a class of prey-predator models, i.e., a Kolmogorov-type system. We are interested in their dynamics when a certain parameter (that can be viewed as the death rate of the predator) changes from zero value to positive. By utilizing alternative but simple techniques, including a sub- and super-solutions method, we establish the existence of periodic solutions when some conditions are satisfied. We also prove that the solutions are bounded by a non-periodic trajectory when the parameter vanishes. We give an example to illustrate our results.

Periodic solutions in distribution for stochastic lattice differential equations

In this article, we consider stochastic lattice differential equations (SLDEs) in weighted space $l^2_\rho$ of infinite sequences. We establish the well-posedness of solutions and prove the existence of periodic solutions in distribution. An example is given to illustrate the validity of our results.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/25/abstr.html

On the Hill Discriminant of Lamé's Differential Equation

Lamé's differential equation is a linear differential equation of the second order with a periodic coefficient involving the Jacobian elliptic function ${\rm sn}$ depending on the modulus $k$, and two additional parameters $h$ and $\nu$. This differential equation appears in several applications, for example, the motion of coupled particles in a periodic potential. Stability and existence of periodic solutions of Lamé's equations is determined by the value of its Hill discriminant $D(h,\nu,k)$. The Hill discriminant is compared to an explicitly known quantity including explicit error bounds. This result is derived from the observation that Lamé's equation with $k=1$ can be solved by hypergeometric functions because then the elliptic function ${\rm sn}$ reduces to the hyperbolic tangent function. A connection relation between hypergeometric functions then allows the approximation of the Hill discriminant by a simple expression. In particular, one obtains an asymptotic approximation of $D(h,\nu,k)$ when the modulus $k$ tends to $1$.

Periodic Solutions to Nonlinear Second-Order Difference Equations with Two-Dimensional Kernel

In this work, we provide conditions for the existence of periodic solutions to nonlinear, second-order difference equations of the form y(t+2)+by(t+1)+cy(t)=g(y(t)), where b and c are real parameters, c≠0, and g:R→R is continuous.

Existence of periodic solutions and stability for a nonlinear system of neutral differential equations

In this article, we study the existence and uniqueness of periodic solutions, and stability of the zero solution to the nonlinear neutral system $$ \frac{d}{dt}x(t)=A(t)h\big(x(t-\tau_1(t))\big)+\frac{d}{dt}Q\big(t,x(t-\tau_2(t))\big) +G\big(t,x(t),x(t-\tau_2(t))\big). $$ We use integrating factors to transform the neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution. Our results generalize the corresponding results in the existing literature. An example is given to illustrate our results.For more information see https://ejde.math.txstate.edu/Volumes/2024/21/abstr.html

Dynamical properties, chirped solutions, and chaotic behaviors of the extended nonlinear Schrödinger equation

In this paper, we focus on an extended nonlinear Schrödinger equation describing the pulse propagation in a nonlinear Schrödinger equation with self‐steepening and magneto‐optic effects. The existence of periodic and solitary solutions are proved based on the bifurcation method, and also, the Hamiltonian properties and the classification of its equilibrium points are obtained. The chirped solutions of the extended nonlinear Schrödinger equations are obtained by using the complete discrimination system for polynomial method, and under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model. In particular, we consider the external perturbation terms to analyze the chaotic behavior of this equation.

Periodicity, stability, and synchronization of solutions of hybrid coupled dynamic equations with multiple delays

This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries.

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension n⩾3. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space Rn. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear systems corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear systems on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear systems to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

On the qualitative and quantitative analysis for two fourth–order difference equations

Our aim in the present paper is to derive the closed-form solutions for the two fourth-order difference equations xn+1=xn-2xn-3axn+bxn-3,n≥0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{n+1}=\\frac{x_{n-2}x_{n-3}}{ax_{n}+bx_{n-3}}, \\ n\\ge 0, \\end{aligned}$$\\end{document}and xn+1=xn-2xn-3-axn+bxn-3,n≥0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{n+1}=\\frac{x_{n-2}x_{n-3}}{-ax_{n}+bx_{n-3}}, \\ n\\ge 0, \\end{aligned}$$\\end{document}with positive arbitrary real parameters a, b and arbitrary real initial conditions, as well as study the qualitative behaviors for each. For the first equation, we show that every admissible solution converges to a period-3 solution when a+b=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a+b=1$$\\end{document}. For the second equation, we show that every admissible solution converges to zero if b>2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b>2$$\\end{document} when b2≥4a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^2\\ge 4a$$\\end{document}. When b2<4a\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b^2<4a$$\\end{document}, we show the existence of periodic solutions under certain conditions. We introduce the forbidden sets as well as provide some illustrative examples for the above-mentioned equations.

A qualitative investigation of some rational difference equations

In this paper, we study some qualitative properties of the solutions for the following difference equation% \[ y_{n+1}=\frac{\alpha +\alpha _{0} y_{n}^{r}+\alpha _{1} y_{n-1}^{r}+\cdots+\alpha _{k} y_{n-k}^{r}}{\beta +\beta _{0} y_{n}^{r}+\beta _{1} y_{n-1}^{r}+\cdots+\beta _{k} y_{n-k}^{r}}% ,~~~~n\geq 0,\tag{I}\label{i} \] where \(r, \alpha , \alpha _{0}, \alpha _{1},\ldots, \alpha _{k}, \beta , \beta _{0}, \beta _{1},\ldots, \beta _{k}\in (0,\infty )\) and \(k \) is a non-negative integer number. We find the equilibrium points for the considered equation. Then classify these points in terms of local stability or not. We investigate the boundedness and the global stability of the solutions for the considered equation. Also, we study the existence of periodic solutions of Eq. (I).