Existence Of Periodic Solutions Research Articles

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).

On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales

Abstract This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.

Mathematical modeling and bifurcation analysis of a delayed eco-epidemiological model with disease in predator and linear harvesting

ABSTRACT This study proposes a three-dimensional prey-predator model with disease in predators, including the time delay due to the gestation of the predator population. The positive invariance of the solutions and the existence of equilibria in the proposed system have been explained. We also analyse the local stability of all feasible equilibrium points for the delayed and non-delayed systems. Additionally, the system’s Hopf bifurcation with regard to several parameters has been investigated. Further, the existence of periodic solutions through Hopf bifurcation is shown with respect to time delay, and it is also shown that time delay is crucial in order to control the dynamics of the system. Explicit formulas are presented to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solution using the normal form theory and the centre manifold theorem. The outcome of the theoretical investigation is ultimately validated through numerical simulation.

Numerical-analytic successive approximation method for the investigation of periodic solutions of nonlinear integro-differential systems with piecewise constant argument of generalized type

In this paper, we focus on investigating the existence and approximation of periodic solutions for a nonlinear integro-differential system with a piecewise alternately advanced and retarded argument of generalized type, referred to as DEPCAG. The argument is a general step function, and we obtain criteria for the existence of periodic solutions for such equations. &#x0D; Our approach involves converting the given DEPCAG into an equivalent integral equation and using a new approach for periodic solutions. We construct appropriate mappings and employ a numerical-analytic method to investigate periodic solutions of the ordinary differential equation given by A. M. Samoilenko. &#x0D; Additionally, we use the contraction mapping principle to demonstrate the existence of a unique periodic solution.

Existence results for positive periodic solutions to first order neutral differential equations with distributed deviating arguments

We take into account the first order nonlinear neutral differential equation with distributed deviating arguments.&#x0D; Using Krasnoselskii's fixed point theorem, we give some new criteria for the existence of positive periodic solutions to this equation. The theorems we have established are illustrated by an example.

Periodic solutions for nonlinear systems of Ode's with generalized variable exponents operators

Ordinary and partial differential equations with operators containing the p-Laplace operator, the so called ϕ-Laplace operator, operators with variable exponents and the double phase operators have been studied thoroughly for at least the last 20 years and the amount of papers dealing with these subjects is huge. In this paper we deal with the problem of the existence of periodic solutions to some system of ODE's involving a fairly general operator generalizing in this form corresponding results for the areas of research just mentioned. In our results neither the function generating the differential operator nor the right hand side function in the boundary value problem considered are necessarily equal to the gradient of a function. Our main result is a here is a continuation theorem based on the Leray-Schauder degree theory which allows us to obtain the existence of solutions for the boundary value problem considered.

Existence and bifurcation of periodic solutions to the Lp-Minkowski problem with indefinite weight

This paper is devoted to the existence and bifurcation of positive periodic solutions for Lp-Minkowski problem with indefinite weight. We provide new sufficient conditions for the existence of at least one positive periodic solution. The main tools are Leray-Schauder alternative principle and a global continuation theorem by Manásevich-Mawhin. Using the numerical bifurcation theory, we study the dynamic behaviors of periodic solutions in the cases of indefinite and positive weight, where the weight term is a simple sinusoidal function. In the positive weight case, the multiplicity of positive periodic solutions is detected for the first time in the Lp-Minkowski problem, which is generated by a saddle-node bifurcation.

Исследование периодических решений системы обыкновенных дифференциальных уравнений с квазиоднородной нелинейностью

The article considers a system of ordinary differential equations in which the main nonlinear part, which is a quasi-homogeneous mapping, is distinguished. The question of the existence of periodic solutions is investigated. Consideration of a quasi-homogeneous mapping allows us to generalize previously known results on the existence of periodic solutions for a system of ordinary differential equations with the main positively homogeneous non-linearity. An a priori estimate for periodic solutions is proved under the condition that the corresponding unperturbed system of equations with a quasi-homogeneous right-hand side does not have nonzero bounded solutions. Under the conditions of an a priori estimate, the following results were obtained: 1) the invariance of the existence of periodic solutions under continuous change (homotopy) of the main quasi-homogeneous non-linear part was proved; 2) the problem of homotopy classification of two-dimensional quasi-homogeneous mappings satisfying the a priori estimation condition has been solved; 3) a criterion for the existence of periodic solutions for a two-dimensional system of ordinary differential equations with the main quasi-homogeneous non-linearity is proved.

Positivity and Periodicity in Nonlinear Neutral Mixed Type Levin-Nohel Integro-Differential Equations

In this work, we give sufficient conditions for the existence of periodic and positive periodic solutions for a nonlinear neutral mixed type Levin-Nohel integro-differential equation with variable delays by using Krasnoselskii’s fixed point theorem. Also, we obtain the existence of a unique periodic solution of the posed equation by means of the contraction mapping principle. As an application, we give an example to illustrate our results. Previous results are extended and generalized.

Positive periodic solution for enterprise cluster model with feedback controls and time-varying delays on time scales

&lt;abstract&gt;&lt;p&gt;This paper aims to study a class of enterprise cluster models with feedback controls and time-varying delays on time scales. Based on periodic time scales theory and the fixed point theorem of strict-set-contraction, some new sufficient conditions for the existence of positive periodic solutions are obtained. Finally, two examples are presented to verify the validity and applicability of the main results in this paper.&lt;/p&gt;&lt;/abstract&gt;

Periodic solutions in reversible systems in second order systems with distributed delays

&lt;abstract&gt;&lt;p&gt;In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.&lt;/p&gt;&lt;/abstract&gt;

Resonance with Landesman-Lazer conditions for parameter-dependent equations: a multiplicity result via the Poincaré-Birkhoff theorem

&lt;p&gt;We investigate the resonance problem and prove the existence of multiple periodic solutions to a second order parameter-dependent equation $ x''+f(t, x) = sp(t) $. We weaken the usual requirement on the sublinearity of perturbations when $ |x| $ becomes large; and develop a more general method to investigate the rotational characterizations of the Landesman-Lazer conditions. Furthermore, $ f $ does not satisfy the common sign condition, and even the global existence of the solution is not guaranteed.&lt;/p&gt;

A new approach to mathematical modeling of chemical synapses

The purpose of this work is to study a new mathematical model of a ring neural network with unidirectional chemical connections, which is a singularly perturbed system of differential-difference equations with delay. Methods. A combination of analytical and numerical methods is used to study the existence and stability of special periodic solutions in this system, the so-called traveling waves. Results. The proposed methods make it possible to show that the ring system under study allows the number of stable traveling waves to increase with the number of oscillators in the network. Conclusion. In this article, we rethink and refine the previously proposed method of mathematical modeling of chemical synapses. On the one hand, it was possible to fully take into account the requirement of the Volterra structure of the corresponding equations and, on the other hand, the hypothesis of saturating conductivity. This makes it possible to observe the principle of uniformity: the new mathematical model is based on the same principles as the previously proposed model of electrical synapses.

Gompertz models with periodical treatment and applications to prostate cancer.

In this paper, Gompertz type models are proposed to understand the temporal tumor volume behavior of prostate cancer when a periodical treatment is provided. Existence, uniqueness, and stability of periodic solutions are established. The models are used to fit the data and to forecast the tumor growth behavior based on prostate cancer treatments using capsaicin and docetaxel anticancer drugs. Numerical simulations show that the combination of capsaicin and docetaxel is the most efficient treatment of prostate cancer.

A study of periodic solutions of several types of nonlinear models in biomathematics

Abstract Biomathematics is a cross-discipline formed by the interpenetration of mathematics with life sciences, biology, and other disciplines, and biomathematical models provide an effective tool for solving problems in the above application areas. Our aim in this paper is to combine mathematical analytical tools and numerical simulation methods to investigate the existence and steady state of periodic solutions in different nonlinear models. Time lags with both discrete and distributed characteristics are introduced into the Lotka-Volterra predator-feeder system, and based on the discussion of the central manifold theorem and canonical type theory, it is proved that the branching periodic solution exists when the discrete time lag parameter τ &gt; τ 0. In the SEIRS infectious disease model with nonlinear incidence term and vertical transmission, the global stability of the disease-free equilibrium point and the local asymptotic stability of the endemic equilibrium point are analyzed through the computation and discussion of the fundamental regeneration number R 0 (p, q). A class of convergence-growth models with nonlinear sensitivity functions is studied, and the global boundedness of classical solutions and their conditions are demonstrated based on global dynamics. A mathematical generalization of the muscular vascular model is made by introducing a centralized parameter, the relationship between periodic solutions and chaotic phenomena is explored utilizing a systematic equivalence transformation, and the equation of the homoscedastic orbitals is deduced to be z 2 = x 2 ( A - 1 2 x 2 ) {z^2} = {x^2}\left( {A - {1 \over 2}{x^2}} \right) .

A mathematical model of flavescence dorée in grapevines by considering seasonality

&lt;p&gt;This paper presents a mathematical model to describe the spread of flavescence dorée, a disease caused by the bacterium &lt;italic&gt;Candidatus Phytoplasma vitis&lt;/italic&gt;, which is transmitted by the insect vector &lt;italic&gt;Scaphoideus titanus&lt;/italic&gt; in grapevine crops. The key contribution of this work is the derivation of conditions under which positive periodic solutions exist. These conditions are based on the assumption that key factors such as recruitment rates, disease transmission, and vector infectivity vary periodically, thus reflecting seasonal changes. The existence of these periodic solutions is proven using the degree theory, and numerical examples are provided to support the theoretical findings. This model aims to enhance the understanding of the epidemiological dynamics of flavescence dorée and contribute to developing better control strategies to manage the disease in grapevines.&lt;/p&gt;