Uniqueness Of Periodic Solutions Research Articles

Analyzing the dynamical sensitivity and soliton solutions of time-fractional Schrödinger model with Beta derivative

In physical domains, Beta derivatives are necessary to comprehend wave propagation across various nonlinear models. In this research work, the modified Sardar sub-equation approach is employed to find the soliton solutions of (1+1)-dimensional time-fractional coupled nonlinear Schrödinger model with Beta fractional derivative. These models are fundamental in real-world applications such as control systems, processing of signals, and fiber optic networks. By using this strategy, we are able to obtain various unique optical solutions, including combo, dark, bright, periodic, singular, and rational wave solutions. In addition, We address the sensitivity analysis of the proposed model to investigate the truth that it is extremely sensitive. These studies are novel and have not been performed before in relation to the nonlinear dynamic features of these solutions. We show these behaviors in 2-D, contour 3-D structures across the associated physical characteristics. Our results demonstrate that the proposed approach offers useful results for producing solutions of nonlinear fractional models in application of mathematics and wave propagation in fiber optics.

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

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. 
 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. 
 Additionally, we use the contraction mapping principle to demonstrate the existence of a unique periodic solution.

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.

Periodic solution analysis of a population dynamics system model for pulsating organisms

Abstract Population dynamics has a wide range of applications in ecological theory, especially in the fields of plant and animal conservation and the management and development of ecological environments. Periodic solution analysis of a population dynamics model for pulsating organisms. The influence of impulsive dynamics on the periodic solution of the system is investigated in this paper, which considers several types of population dynamics systems with impulsive effects. First, the impulsive differential modeling of the model of a constantator in a polluted environment considering time-lagged growth response and impulsive inputs proves that only ̄t needs to be sufficiently large to have x(t) > m x, such that, the constantator seeks a unique periodic solution for microbial extinction and persistent survivability. Next, a model of integrated pest control is modeled to find, a periodic solution for pest extinction and the existence of (0, I* (t)) is globally stable. Then, a Lur’e system with impulsive biodynamics is explored, modeled with uncertain parameters, and simulated with Chua’s circuit system to determine that the state trajectory lines all eventually converge to 0 and have stable periodic solutions. Finally, the Beddington-DeAngelis predator-prey model with impulsive effects is used to argue, using correlation priming, for the existence of z i * ( t ) = exp { x i * ( t ) } z_i^*\left( t \right) = \exp \left\{ {x_i^*\left( t \right)} \right\} , i = 1, 2, such that z * ( t ) = ( z 1 * ( t ) , z 2 * ( t ) ) T {z^*}\left( t \right) = {\left( {z_1^*\left( t \right),\,z_2^*\left( t \right)} \right)^T} there is a positive ω − periodic solution for this system.

New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

Reaction–Diffusion Problems on Time-Periodic Domains

AbstractReaction–diffusion equations are studied on bounded, time-periodic domains with zero Dirichlet boundary conditions. The long-time behaviour is shown to depend on the principal periodic eigenvalue of a transformed periodic-parabolic problem. We prove upper and lower bounds on this eigenvalue under a range of different assumptions on the domain, and apply them to examples. The principal eigenvalue is considered as a function of the frequency, and results are given regarding its behaviour in the small and large frequency limits. A monotonicity property with respect to frequency is also proven. A reaction–diffusion problem with a class of monostable nonlinearity is then studied on a periodic domain, and we prove convergence to either zero or a unique positive periodic solution.

Subsonic time-periodic solution to the compressible Euler equations triggered by boundary conditions

In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term β(t,x)ρ|u|αu in a bounded domain, which can be used to describe gas transmission in a nozzle. The model is imposed a subsonic time-periodic boundary condition. The main results reveal that the time-periodic boundary can trigger an unique subsonic time-periodic smooth solution and this unique periodic solution is stable under small perturbations on initial and boundary data. To get the existence of subsonic time-periodic solution, we use the linear iterative skill and transfer the boundary value problem into two initial value ones by using the hyperbolic property of the system, then the corresponding linearized system can be decoupled. The uniqueness is a direct by-product of the stability. There is no small assumptions on coefficient β(t,x).

A nonlocal diffusion competition model with seasonal succession and free boundaries

In this paper, we investigate a nonlocal diffusion competition model with seasonal succession and free boundaries, which can be used to describe the dynamics of two species in an environment alternating between bad and good seasons. We first show the existence and uniqueness of global solution, and then focus on the long time behaviors of two species, namely, if h∞−g∞<∞ (the two species cannot spread over R), then they must die out in the long run; If h∞=−g∞=∞ (the two species spread to infinity as t→∞ at both fronts and survives in the new environment), then they must tend to the unique positive time periodic solution. Moreover, some sufficient conditions for spreading and vanishing are also given.

A mosquito population suppression model with a saturated Wolbachia release strategy in seasonal succession

Releasing Wolbachia-infected male mosquitoes to suppress wild female mosquitoes through cytoplasmic incompatibility has shown great promise in controlling and preventing mosquito-borne diseases. To make the release logistically and economically feasible, we propose a saturated release strategy, which is only implemented during the epidemic season of mosquito-borne diseases. Under this assumption, the model becomes a seasonally switching ordinary differential equation model. The seasonal switch brings rich dynamics, including the existence of a unique periodic solution or exactly two periodic solutions, which are proved by using the qualitative property of the Poincaré map. Sufficient conditions are also obtained for determining the stability of the periodic solutions.

Analysis of tumor-immune functional responses in a mathematical model of neoantigen cancer vaccines

Cancer neoantigen vaccines have emerged as a promising approach to stimulating the immune system to fight cancer. We propose a simple model including key elements of cancer-immune interactions and conduct a phase plane analysis to understand the immunological mechanisms of cancer neoantigen vaccines. Analytical results are obtained for two widely used functional forms that represent the killing rate of tumor cells by immune cells: the law of mass action (LMA) and the dePillis-Radunskaya Law (LPR). Using the LMA, our results reveal that a slowly growing tumor can escape the immune surveillance and that there is a unique periodic solution. The LPR offers richer dynamics, in which tumor elimination and uncontrolled tumor growth are both present. We show that tumor elimination requires sufficient number of initial activated T cells in relationship to the malignant cells, which lends support to using the neoantigen cancer vaccine as an adjuvant therapy after the primary tumor is surgically removed or treated using radiotherapy. We also derive a sufficient condition for uncontrolled tumor growth under the assumption of the LPR. The juxtaposition of analyses with these two different choices for the killing rate function highlights their importance on model behavior and biological implications, by which we hope to spur further theoretical and experimental work to understand mechanisms underlying different functional forms for the killing rate.

Optimal Stabilization of Periodic Orbits

In this contribution the optimal stabilization problem of periodic orbits is studied in a symplectic geometry setting. For this, the stable manifold theory for the point stabilization case is generalized to the case of periodic orbit stabilization. Sufficient conditions for the existence of a normally hyperbolic invariant manifold (NHIM) of the Hamiltonian system are derived. It is shown that the optimal control problem has a solution if the related periodic Riccati equation has a unique periodic solution. For the analysis of the stable and unstable manifold a coordinate transformation is used which is moving along the orbit. As an example, an optimal control problem is considered for a spring-mass oscillator system, which should be stabilized at a certain energy level.

Existence of traveling waves in a delayed convecting shallow water fluid model

&lt;abstract&gt;&lt;p&gt;This paper investigates a delayed shallow water fluid model that has not been studied in previous literature. Applying geometric singular perturbation theory, we prove the existence of traveling wave solutions for the model with a nonlocal weak delay kernel and local strong delay convolution kernel, respectively. When the convection term contains a nonlocal weak generic delay kernel, the desired heteroclinic orbit is obtained by using Fredholm theory and linear chain trick to prove the existence of two kink wave solutions under certain parametric conditions. When the model contains local strong delay convolution kernel and weak backward diffusion, under the same parametric conditions to the previous case, the corresponding traveling wave system can be reduced to a near-Hamiltonian system. The existence of a unique periodic wave solution is established by proving the uniqueness of zero of the Melnikov function. Uniqueness is proved by utilizing the monotonicity of the ratio of two Abelian integrals.&lt;/p&gt;&lt;/abstract&gt;

Global Stability Analysis of a Certain Second-Order Rational Difference Equation with Nonlinear Terms

In this paper we observed the global dynamics and the occurrence of a certain bifurcation for the corresponding values of a certain rational difference equation of the second order with analyzed quadratic terms. The analysis of the local stability of the unique equilibrium point, as well as the unique periodic solution of period two, was performed in detail. The constraint of the equations on both sides for the corresponding values of the parameters is proved and on this basis the global stability is analyzed. The existence of Neimark-Sacker bifurcation with respect to the arrangement of equilibrium points has been proven. Thus, the basins of attraction have been determined in full for all the positive values of the parameters and all the positive initial conditions.

Dynamical behavior and optimal impulse control analysis of a stochastic rumor spreading model

The Internet era has brought great convenience to our life and communication. Meanwhile, it also makes a bunch of rumors propagate faster and causes even more harm to human life. Therefore, it is necessary to perform effective control mechanisms to minimize the negative social impact from rumors. Thereout, firstly, we formulate a rumor spreading model considering psychological factors and thinking time, then, we add white noise (i.e., stochastic interference) and two pulse control strategies which denote education mechanism and refutation mechanism into the model. Secondly, we obtain the global positive solutions and demonstrate the global exponential stability of the unique positive periodic rumor-free solution. Thirdly, we discuss the extinction and persistence of rumor. Moreover, we use Pontriagin’s minimum principle to explore the optimal impulse control. Finally, several numerical simulations are carried out to verify the effectiveness and availability of the theoretical analysis. We conclude that the pulse control strategies have a great influence on controlling rumor spreading, and different control strategies should be adopted under different transmission scenarios.

A Generalized Approach of the Gilpin–Ayala Model with Fractional Derivatives under Numerical Simulation

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we showed that the classical nonlinear fractional model is bounded. The Banach contraction mapping principle was used to prove that the model has a unique positive asymptotical periodic solution. We provide an example and numerical simulation to inspect the correctness and availability of our essential outcomes.

The existence and uniqueness of periodic solutions for nonlinear neutral first order differential equation with functional delay

We employ the fixed point theorem of Krasnoselskii, to show the existence and uniqueness of periodic solutions of the nonlinear neutral differential equation:&#x0D; By modifying the given neutral differential equation into an equivalent integral equation using lemma (2.1). This is done by constructing appropriate operators, one is a contraction and the other is compact, which allow us to prove the existence of periodic solutions. In addition, we used the Banach fixed-point theorem to guarantee a unique periodic solution.

A periodic trachoma epidemic model on heterogeneous networks

In this paper, considering the impact of environmental change on trachoma transmission, we establish a periodic trachoma epidemic model on heterogeneous networks, which is a non-autonomous model. We prove theoretically the existence of basic reproduction number shown by R0. And then, we demonstrate that the disease-free periodic solution is globally asymptotically stable if R0<1, and there exists a unique positive periodic solution which is also is globally asymptotically stable if R0>1. The method obtained in this paper can be seen as an extension for proving the global stability of the high-dimensional periodic epidemic model on heterogeneous networks. And our results provide theoretical basis for decision-makers to make control strategies.

DYNAMICS OF AN SIRS EPIDEMIC MODEL WITH PERIODIC INFECTION RATE ON A SCALE-FREE NETWORK

Influenced by seasonal changes, the infection rate of many infectious diseases fluctuates in cycles. In this paper, we propose and investigate an SIRS model on a scale-free network. To model seasonality, we assume that the infection rate is periodic. The existence and positivity of solutions of the proposed model are proved and the basic reproduction number [Formula: see text] is defined. The global stability of steady states is determined by rigorous mathematical analysis. When [Formula: see text], the disease-free equilibrium [Formula: see text] is globally asymptotically stable. When [Formula: see text], the system has a unique positive periodic solution [Formula: see text], and [Formula: see text] is globally asymptotically stable. Numerical simulations are performed to support our theoretic results, and the effects of various parameters on the amplitude and mean of infected individuals are studied. The sensitivity of parameters of the basic reproduction number [Formula: see text] is solved by the Sobol global sensitivity analysis method, and the results show that the effects of the parameters [Formula: see text] and [Formula: see text] on [Formula: see text] are remarkable.

Modeling transport of extended interacting objects with drop-off phenomenon.

We study a deterministic framework for important cellular transport phenomena involving a large number of interacting molecules called the excluded flow of extended interacting objects with drop-off effect (EFEIOD). This model incorporates many realistic features of biological transport process including the length of biological “particles” and the fact that they can detach along the biological ‘tracks’. The flow between the consecutive sites is unidirectional and is described by a “soft” simple exclusion principle and by repelling or attracting forces between neighboring particles. We show that the model admits a unique steady-state. Furthermore, if the parameters are periodic with common period T, then the steady-state profile converge to a unique periodic solution of period T. Simulations of the EFEIOD demonstrate several non-trivial effects of the interactions on the system steady-state profile. For example, detachment rates may help in increasing the steady-state flow by alleviating traffic jams that can exist due to several reasons like bottleneck rate or interactive forces between the particles. We also analyze the special case of our model, when there are no forces exerted by neighboring particles, and called it as the ribosome flow model of extended objects with drop-off effect (RFMEOD), and study the sensitivity of its steady-state to variations in the parameters.