Existence Of Periodic Solutions Research Articles

Research on Periodic Solutions of Non-autonomous Second-order Hamiltonian Systems

In the field of nonlinear science, Hamiltonian systems are crucial models extensively applied in engineering, physics, and biology. The existence of periodic solutions in non-autonomous second-order Hamiltonian systems, as a relatively specialized form, has been a focal and challenging research area. This paper employs variational methods to analyze the existence of periodic solutions in such systems, aiming to provide insights for related research endeavors.

Mathematical model of mosquito population dynamics with constants and periodic releases of Wolbachia-infected males

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter N , called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when N ≤ 1 , while if N > 1 , the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes M i crit to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, T ∗ the threshold on the release period and Λ ∗ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results.

Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x″(t)+f(x(t))x′(t)+φ(t)xμ(t)−1xγ(t)=e(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x''(t)+f(x(t))x'(t)+\\varphi (t)x^{\\mu}(t)-\\frac{1}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+∞)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:(0,+\\infty )\\rightarrow R$\\end{document} is continuous, which may have a singularity at the origin, the sign of φ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varphi (t)$\\end{document}, e(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e(t)$\\end{document} is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when μ∈[0,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in [0,+\\infty )$\\end{document}.

Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay

Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

Study on the Solutions of Impulsive Integrodifferential Equations of Mixed Type Based on Infectious Disease Dynamical Systems

Since ancient times, infectious diseases have been a major source of harm to human health. Therefore, scientists have established many mathematical models in the history of fighting infectious diseases to study the law of infection and then analyzed the practicability and effectiveness of various prevention and control measures, providing a scientific basis for human prevention and research of infectious diseases. However, due to the great differences in the transmission mechanisms and modes of many diseases, there are many kinds of infectious disease dynamic models, which make the research more and more difficult. With the continuous progress of infectious disease research technology, people have adopted more ways to prevent and interfere with the derivation and spread of infectious disease, which will make the state of infectious disease system change in an instant. The mutation of this state can be described more scientifically and reasonably by the mathematical impulse dynamic system, which makes the research more practical. Based on this, a time-delay differential system model of infectious disease under impulse effect was established by means of impulse differential equation theory. A class of periodic boundary value problems for impulsive integrodifferential equations of mixed type with integral boundary conditions was studied. The existence of periodic solutions of these equations was obtained by using the comparison theorem, upper and lower solution methods, and the monotone iteration technique. Finally, combined with the practical application, the established time-delay differential system model was applied to the prediction of the stability and persistence of the infectious disease dynamic system, and the correctness of the conclusion was further verified. This study provides some reference for the prevention and treatment of infectious diseases.

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 .

Twist periodic solutions of the electron beam focusing system with unshielded cathode

In this article, we consider the electron beam focusing system guided by an axially symmetric periodic magnetic field with unshielded cathode. Based on the third‐order approximation method and some quantitative methods, we establish two different sufficient conditions for the existence of twist periodic solutions of the system. Such twist periodic solutions are stable in the sense of Lyapunov. Some results in the literature are generalized and improved.

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 $1$.