Existence Of Periodic Solutions Research Articles

Approximations of random periodic solutions for path‐dependent stochastic differential equations with finite/infinite delay

The aim of this work is to show the numerical approximation of random periodic solutions for path‐dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous‐time Euler–Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path‐dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean‐square sense of a numerical approximation to path‐dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path‐dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure.

On the periodic behavior of the generalized Chazy differential equation.

We consider the periodic behavior of the generalized Chazy differential equation x⃛+|x|qx¨+k|x|qxx˙2=0, where q is a positive integer and k is a real number. We give a pure analysis on the existence of non-trivial periodic solutions for k=q+1 and the non-existence of them for k≠q+1. Our method is based on considering the projections of the orbits onto the phase plane (x,x˙). We find that a non-trivial periodic solution of the equation is equivalent to a closed curve formed by two equilibrium points and two orbits with some specific constraints in the corresponding planar system and that the existence of such closed curves can be obtained by the existence of real zeros of some returning map. Our conclusion covers all q, which completes a recent result.

New Results Regarding Positive Periodic Solutions of Generalized Leslie–Gower-Type Population Models

In this paper, we focus on the existence of positive periodic solutions of generalized Leslie–Gower-type population models. Using the topological degree, we provide sufficient conditions to demonstrate the existence of positive periodic solutions to the considered models. It is interesting that the positive periodic solutions in this paper are general positive functions, not e exponential functions, which generalizes and improves the existing results. We note that due to the symmetrical property of periodic solutions, the results of this paper provide a deeper understanding of the periodic behavior of biological populations. Two numerical examples show the effectiveness of our main results.

On Small-Amplitude Asymmetric Water Waves

AbstractWe generalize the method used by Mæhlen and Seth (Asymmetric traveling wave solutions of the capillary–gravity Whitham equation, 2023, arxiv:2312.15343) used to prove the existence of small-amplitude asymmetric solutions to the capillary–gravity Whitham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetric solutions for this class of equations. The result is then applied to an infinite depth capillary–gravity Whitham equation and an infinite depth capillary–gravity Babenko equation to show a nonexistence result of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary–gravity Babenko equation.

Periodic and Quasi-Periodic Orbits in the Collinear Four-Body Problem: A Variational Analysis

This paper investigated the periodic and quasi-periodic orbits in the symmetric collinear four-body problem through a variational approach. We analyze the conditions under which homographic solutions minimize the action functional. We compute the minimal value of the action functional for these solutions and show that, for four equal masses organized in a linear configuration, these solutions are the minimizers of the action functional. Additionally, we employ numerical experiments using Poincaré sections to explore the existence and stability of periodic and quasi-periodic solutions within this dynamical system. Our results provide a deeper understanding of the variational principles in celestial mechanics and reveal complex dynamical behaviors, crucial for further studies in multi-body problems.

Dynamics of a pituitary–adrenal model with distributed time delays

This paper investigates the effects of distributed time delays on the dynamics of the pituitary–adrenal axis. Our two-dimensional model incorporates distributed time delays with general delay kernels, specifically illustrating interactions between adrenocorticotropic hormone (ACTH) and cortisol (CORT). We derive general theoretical results for general delay kernels, exemplified by Dirac and weak Gamma kernels, revealing stability transitions characterized by Hopf bifurcations. We establish conditions for the local asymptotic stability of the unique equilibrium and discuss the existence of periodic solutions. Numerical simulations demonstrate that periodic oscillations appear for appropriate values of the average time delays. Including an external input results in both ultradian and circadian rhythms, highlighting the system’s dynamic responsiveness to external stimuli.

Oscillations in a tumor-immune system interaction model with immune response delay.

In this paper we consider a tumor-immune system interaction model with immune response delay, in which a nonmonotonic function is used to describe immune response to the tumor burden and a time delay is used to represent the time for the immune system to respond and take effect. It is shown that the model may have one, two or three tumor equilibria, respectively, under different conditions. Time delay can only affect the stability of the low tumor equilibrium and local Hopf bifurcation occurs when the time delay passes through a critical value. The direction and stability of the bifurcating periodic solutions are also determined. Moreover, the global existence of periodic solutions is established by using a global Hopf bifurcation theorem. We also observe the existence of relaxation oscillations and complex oscillating patterns driven by the time delay. Numerical simulations are presented to illustrate the theoretical results.

Dynamics of autonomous and nonautonomous Leslie–Gower model for the impacts of fear and its carry‐over effects including predator harvesting

Recently, certain biological field experiments and research findings have revealed that predators not only reduce prey populations through direct predation (lethal effect) but also indirectly affect the growth rate of prey by inducing fear; these nonlethal effects can be carried over seasons or generations. In this paper, we proposed an autonomous and a nonautonomous Leslie–Gower model incorporating some biological factors such as fear and its carry‐over effects, prey refuge, and nonlinear predator harvesting. In the autonomous model, first, we examined the well‐posedness, positivity solutions, and their boundedness. Also, it is shown that new equilibrium points emerge and disappear with the change in intrinsic growth rate of predator. Further, we compute and analyzed the local and global stability at the interior equilibrium points. Furthermore, Hopf bifurcation and direction and stability of limit cycle at interior equilibrium points are established. Moreover, the sensitivity analysis of the biological parameters is carried out numerically with two statistical methods via Latin hypercube sampling and partial rank correlation coefficients. It was found that at the small values of fear factor, carry‐over effect, and refuge behavior of prey parameters, the autonomous system exhibits oscillatory behavior, whereas for small values of prey birth rate and harvesting effort, the system remains stable. However, when intrinsic growth rate of predator increases from a low value to a higher value, the system shows transition from stability to instability and back to stability. We also explore the effects of seasonal variations in biological parameters in the nonautonomous model. Additionally, we investigate the theoretical existence of positive periodic solutions using the continuation theorem. The influences of seasonal variations in some parameters such as prey's birth rate, fear, and its carry‐over effects; refuge behavior of prey; intrinsic growth rate of predator; and harvesting effort are then numerically investigated. The results reveal the emergence of positive periodic solutions and bursting patterns in the seasonally forced model.

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.

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

On a priori estimate of periodic solutions of a system of nonlinear ordinary differential equations of the second order

The question of a priori estimation of periodic solutions for a system of non-linear ordinary differential equations of the second order with a distinguished main positively homogeneous part is investigated. In this question, the well-known methods for deriving an a priori estimation of periodic solutions for similar systems of first-order ordinary differential equations are not directly applicable.Combining these methods with the idea of qualitative research of singularly perturbed ordinary differential equations, conditions are found that provide an a priori estimation of periodic solutions for the system of second-order equations.The conditions for the a priori estimation are formulated in terms of the properties of the main positively homogeneous part of the system of equations. The existence of periodic solutions is proved to be invariant under a continuous change of the main positively homogeneous part and the conditions of a priori estimation are preserved. Based on the results obtained, in the future, it is possible to investigate the existence of periodic solutions.

The analysis of traveling wave structures and chaos of the cubic–quartic perturbed Biswas–Milovic equation with Kudryashov's nonlinear form and two generalized nonlocal laws

The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.

Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time

Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time

Periodic and subharmonic solutions in the motion of a bead on a rotating circular hoop

We establish the necessary conditions for the existence and multiplicity of periodic and subharmonic solutions to a second-order nonlinear ordinary differential equation (ODE). This ODE describes the motion of a bead on a rotating circular hoop subjected to a constant angular velocity ω and a T-periodic forcing. Our approach involves estimating bounds for the angular velocity and period using upper and lower solution methods.

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}.

Dual effects of additional food supply and threshold control on the dynamics of a Leslie–Gower model with pest herd behavior

Bemisia tabaci is a major agricultural pest and recognized by the United Nations Food and Agriculture Organization (FAO) as the second largest pest in the world. This paper aims to study the management of Bemisia tabaci from theoretical aspects. Firstly, considering the omnivorous nature of natural enemy species and pest herd behavior, a Leslie–Gower type pest-natural enemy model with additional food supply and non-differentiable rational uptake function is put forward, and the effects of additional food supply and aggregation efficiency on the dynamical behavior of the system are analyzed. Secondly, for the slowness of pest control by natural enemy species, a threshold control strategy is adopted to prevent the spread of Bemisia tabaci by spraying a certain intensity of insecticides and releasing a certain number of natural enemies. The complex dynamics of the system induced by the threshold control are discussed such as the existence of predator-extinction periodic solution, the existence and stability of coexisting order-1 periodic solution. Thirdly, from the perspective of cost control, an optimization problem is constructed based on the existence of the order-1 periodic solution, and the optimal control threshold, control strength and control frequency are determined. At last, to illustrate the theoretical results more intuitively, the numerical simulations in MATLAB program are presented to illustrate the effectiveness and practical significance of the control measures. The results of this study provide a reference for the scientific and reasonable management of Bemisia tabaci and is helpful to realize the sustainable development of agricultural production.

Exponential stability of periodic solutions for a hematopoiesis model with two time delays

In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.

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