Periodic Solutions Research Articles

(N,λ)-periodic solutions to abstract difference equations of convolution type

This work primarily focuses on (N,λ)-periodic sequences and their applications. To begin, we provide a brief overview of (N,λ)-periodic sequences and introduce several results. Secondly, in terms of applications and main objective, we establish sufficient criteria for both the existence and uniqueness of (N,λ)-periodic mild solutions for abstract difference equations of convolution type. Furthermore, we present illustrative examples to highlight our key findings.

Stability of Periodic solutions by Krasnoselskii fixed point theorem of neutral nonlinear system of dynamical equation with Variable Delays

The fixed point theorem is used in this study to provide stability results for the zero solution of a nonlinear neutral system of differential equations with functional delay.

The single-phase solution and Whitham modulation equations of the defocusing self-induced transparency system

Under investigation in this paper is the defocusing self-induced transparency system which can describe propagation characters of short pulses in Erbium doped fibers with a two-level medium. The single-phase periodic solution and Whitham modulation equations will be derived via finite-gap integration method and Whitham modulation theory. In addition, the oscillatory structure of dispersive shock waves will be constructed and analyzed.

Study of self-oscillations of a distributed rotor from a material with hereditarily elastic properties

Self-oscillatory solutions of dynamic equations of transverse vibrations of a distributed rotating ideally balanced rotor (shaft) on an isotropic support made of a material with nonlinear hereditary properties are studied. The derivation of dynamic equations (mathematical model), which are a system of two nonlinear partial differential equations with an infinite (integral) delay of the time argument, is given. A mathematical formulation of the initial–boundary value problem is formulated, its solvability and correctness are proven. The stability of the zero solution (stable rotation), the mechanisms of loss of stability, and the self-oscillatory solutions bifurcating in this case are studied. The possibility of bifurcation of periodic solutions (direct asynchronous precessions) and invariant two-dimensional tori (beat modes) is shown. The stability of self-oscillating solutions is determined by the parameters of the problem under consideration. The method of central manifolds of distributed systems and the theory of bifurcations were used as a research method. Asymptotic formulas are constructed for self-oscillating solutions.

Weyl almost periodic solutions in distribution to a mean-field stochastic differential equation driven by fractional Brownian motion

In this paper, we consider a class of mean-field stochastic differential equations driven by both Brownian motion and fractional Brownian motion. Using the Banach fixed point theorem and inequality technique, we obtain sufficient conditions for the existence and uniqueness of Weyl almost periodic solutions in distribution of the equations under consideration.

Bifurcation in an modified model of neutrophil cells with time delay

The hematological stem cells model is introduced with neutrophil dynamics of two department model setting forth. During the cells differentiation and proliferation process, the neutrophils are functioned with negative feedback with delay history, which contains delayed amplification coefficient. In more general view, the new introduction rate is given to replace the familiar Hill function which is helpful to understand the complex dynamics of neutrophils. The double Hopf bifurcation is calculated with the artificial handtools named DDE-Biftool, which is observed as the self-intersection of Hopf lines. The continuation of periodical solutions arising from Hopf points are done and the longer period solutions are manifested with multi-rhythm and bursting oscillation. The near dynamics of double Hopf points is simulated by DDE-Biftool with different route design, the multi-period attractors, quasi-periodical solutions and chaos are observed.

Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques

This study aims to explore the precise resolution of the nonlinear Benjamin Bona Mahony Burgers (BBMB) equation, which finds application in a variety of nonlinear scientific disciplines including fluid dynamics, shock generation, wave transmission, and soliton theory. Within this paper, we employ two versatile methodologies, specifically the extended exp(-Ψ(χ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\Psi (\\chi ))$$\\end{document} expansion technique and the novel Kudryashov method, to identify the exact soliton solutions of the nonlinear BBMB equation. The solutions we discovered involve trigonometric functions, hyperbolic functions, and rational functions. The uniqueness of this research lies in uncovering the bright soliton, kink wave solution, and periodic wave solution, and conducting stability analysis. Furthermore, the solutions’ graphical characteristics were explored through the utilization of the mathematical software Maple 2022 (https://maplesoft.com/downloads/selectplatform.aspx?hash=61ab59890f2313b2241fde3423fd975e). The system’s physical interpretation is defined through various types of graphs, including contour graphs, 3D-surface graphs, and line graphs, which use appropriate parameter values. These recommended techniques hold significant importance and are applicable in diverse nonlinear evolutionary equations found in the field of nonlinear sciences for illustrating nonlinear physical models.

Non-constant periodic solutions of the Ricker model with periodic parameters

In this paper, we study the Ricker model x n + 1 = x n exp ⁡ [ r n ( 1 − x n ) ] , n = 0 , 1 , 2 … , ( ∗ ) where x 0 ≥ 0 , and { r n } n = 0 ∞ is a sequence of positive ω-periodic numbers. We obtain sufficient conditions for equation (*) to have at least two non-constant periodic solutions by transforming the existence problem of non-constant ω-periodic solutions of (*) into the existence problem of positive fixed points except 1 of some function. For the special case of period-two parameters, we show that ( ∗ ) has at most two non-constant 2-periodic solutions, and give necessary and sufficient conditions for (*) to have no non-constant 2-periodic solutions, to have a unique non-constant 2-periodic solution which or the equilibrium 1 is semi-stable, and to have exactly two non-constant 2-periodic solutions, respectively. Finally, some specific examples are also provided to illustrate our theoretical results.

Soliton waves with optical solutions to the three-component coupled nonlinear Schrödinger equation

This study uses the modified Sardar sub-equation method to find novel soliton solutions to the nonlinear three-component coupled nonlinear Schrödinger equation (NLSE), which is used for pulse propagation in nonlinear optical fibers. Multi-component NLSE equations are widely used because they can represent a wide range of complex observable systems and more dynamic patterns of localized wave solutions. The optical solutions proposed in this study are novel and can be described using hyperbolic, trigonometric, and exponential functions. These solutions are categorized as bright, dark, singular, combo bright-singular, and periodic solutions. Some solutions’ dynamic behaviors are demonstrated by selecting appropriate physical parameter values. The results and computational analysis indicate that the techniques provided are simple, effective, and adaptable. They can be applied to a variety of nonlinear evolution equations, whether stable or unstable, and can be used in fields such as mathematics, mathematical physics, and applied sciences.

Exact analytical soliton solutions of the M-fractional Akbota equation

In this paper we explore the new analytical soliton solutions of the truncated M-fractional nonlinear (1+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional Akbota equation by applying the expa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp _a$$\\end{document} function technique, Sardar sub-equation and generalized kudryashov techniques. Akbota is an integrable equation which is Heisenberg ferromagnetic type equation and have much importance for the analysis of curve as well as surface geometry, in optics and in magnets. The obtained results are in the form of dark, bright, periodic and other soliton solutions. The gained results are verified as well as represented by two-dimensional, three-dimensional and contour graphs. The gained results are newer than the existing results in the literature due to the use of fractional derivative. The obtained results are very helpful in optical fibers, optics, telecommunications and other fields. Hence, the gained solutions are fruitful in the future study for these models. The used techniques provide the different variety of solutions. At the end, the applied techniques are simple, fruitful and reliable to solve the other models in mathematical physics.

On the number of limit cycles in piecewise smooth generalized Abel equations with many separation lines

This paper investigates generalized Abel equations of the form dx/dθ=A(θ)xp+B(θ)xq, where p, q∈Z≥2, p≠q, and A(θ) and B(θ) are piecewise trigonometrical polynomials of degree m with n−1∈N+ separation lines 0<θ1<θ2<⋯<θn−1<2π. The main objective is to obtain the maximum number of non-zero limit cycles (i.e., non-zero isolated periodic solutions) that the equation can have, denoted by Hθ1,θ2,…,θn−1(m), and to analyze how the number and location of separation lines {θi}i=1n−1 affect Hθ1,θ2,…,θn−1(m). By using the theories of Melnikov functions and ECT-systems, we obtain lower bounds for Hθ1,θ2,…,θn−1(m). Our result extend those of Huang et al. who studied the special case of n=2, and reveal that the lower bounds decrease in the presence of pairs of symmetrical separation lines.

Three Positive Periodic Solutions of Nonlinear Functional Difference Equations

Three Positive Periodic Solutions of Nonlinear Functional Difference Equations

Periodic second-order systems and coupled forced Van der Pol oscillators

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

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

Hopf Bifurcation and Turing Instability of a Delayed Diffusive Zooplankton–Phytoplankton Model with Hunting Cooperation

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results.

Bifurcation, Traveling Wave Solutions and Dynamical Analysis in the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-Dimensional Extended Vakhnenko–Parkes Equation

This paper is concerned with the bifurcation of the traveling wave solutions, as well as the dynamical behaviors and physical property of the soliton solutions of the (2+1)-dimensional extended Vakhnenko–Parkes (eVP) equation. Firstly, based on the traveling wave transformation, the planar dynamical system corresponding to the (2+1)-dimensional eVP equation is derived, and then the singularity type and trajectory map of this system are obtained and analyzed. Based on the bifurcation of this system, the analytical expression for the periodic wave solution is given and shown graphically. Secondly, the N-soliton solutions are obtained via the bilinear method, and some important physical quantities and asymptotic analysis of one-soliton and two-soliton solutions are discussed. The results obtained in this paper might be useful for understanding the propagation of high-frequency waves.

Dynamics of a stage-structured predator-prey system with fear-induced group defense in autonomous and nonautonomous settings.

In this investigation, we construct a predator-prey model that distinguishes between immature and mature prey, highlighting group defense strategies within the mature prey. First, we embark on exploring the positivity and boundedness of the solution, unraveling sustainable equilibrium points, and deducing their stability conditions. Upon further investigation, we observe that the system exhibits diverse bifurcations, including Hopf, saddle-node, transcritical, generalized Hopf, cusp, and Bogdanov-Takens bifurcations. The results reveal that heightened fear decreases mature prey density, potentially causing prey extinction beyond a certain threshold. Increased maturation rates lead to the coexistence of immature and mature prey populations and higher predator density. Stronger group defense boosts mature prey density, while weaker defense results in weak persistence. Lower values of the maturation rate of prey and the decline rate of predators sustain only the predator population, reliant on resources other than focal prey. Furthermore, our model demonstrates intriguing and diverse dynamical phenomena, including various forms of bistability across distinct bi-parameter planes. We also explore the dynamics of a related nonautonomous system, where certain parameters are considered to vary with time. In the seasonally forced model, we set out to define criteria regarding the existence and stability of positive periodic solutions. Numerical investigations into the seasonally forced model uncover a spectrum of dynamics, ranging from simple periodic solutions to higher periodicities, bursting patterns, and chaotic behavior.

Spatial patterns caused by the intracellular time delay in a super‐diffusive HBV model with logistic hepatocyte growth

A super‐diffusive HBV model with the intracellular time delay and logistic hepatocyte growth is studied. We discuss the local asymptotic stability of the unique positive equilibrium and prove that the intracellular time delay can cause spatial patterns. Furthermore, we find that system undergoes a Hopf bifurcation when the time delay increase to a critical value. By the normal form theory and the center manifold theorem, we investigate the stability and direction of the bifurcating periodic solutions. Finally, we present how the initial condition, time delay, superdiffusion coefficient and the exponent of superdiffusion affect spatial patterns by numerical simulations.

Periodic Solutions of the Euler–Bernoulli Quasilinear Vibration Equation for a Beam with an Elastically Fixed End

Periodic Solutions of the Euler–Bernoulli Quasilinear Vibration Equation for a Beam with an Elastically Fixed End

Dynamical behaviors of a network-based SIR epidemic model with saturated incidence and pulse vaccination

Pulse vaccination is an effective strategy to restrain the spread of infectious diseases. This paper proposes a network-based SIR epidemic model incorporating a saturated force of infection and vertical transmission along with pulse vaccination strategy and continuous treatment plan. Dynamical behaviors of the model are analyzed in virtue of the theory for impulsive differential equations. We give the boundedness of solutions and derive the expression of basic reproduction number R0. By the Floquet theorem and comparison principle, we show that the disease-free periodic solution is globally asymptotically stable when R0<1. Moreover, a sufficient condition for the uniform permanence of the system is also obtained. Finally, numerical analyses are given to substantiate the theoretical results and a case for Rubella is studied.