Periodic Solutions Research Articles

Periodic fractional Ambrosetti–Prodi for one-dimensional problem with drift

We prove Ambrosetti–Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order s∈(0,1). We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are C1,α under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical. We finish the work obtaining Ambrosetti-Prodi-type results for a problem with singular nonlinearities.

Asteroid spin and shape properties from Gaia DR3 photometry

The third data release of in June 2022, included the first large sample of sparse photometric data for more than 150\,000 Solar System objects (SSOs), mainly asteroids. The SSO photometric data can be processed to derive information on the physical properties for a large number of objects, including spin properties, surface photometric behaviour in a variety of illumination conditions, and overall shape. After selecting a set of 22\,815 objects for which an adequate number of accurate photometric measurements had been obtained by we applied the 'genetic' algorithm of photometric inversion developed by the Data Processing and Analysis Consortium to process SSO photometric data. Given the need to minimise the required data processing time, the algorithm was set to adopt a simple triaxial ellipsoid shape model. Our results show that in spite of the limited variety of observing circumstances and the limited numbers of measurements per object at present (in the majority of cases no greater than $40$ and still far from the number expected at the end of the mission of about $60$ - $70$), the proportion of correct determinations for the spin period among the observed targets is about $85$<!PCT!>. This percentage is based on a comparison with reliable literature data following a moderate filtering procedure developed to remove dubious solutions. The analysis performed in this paper is important in the context of developing further improvements to the adopted data reduction procedure. This includes the possible development of better solution filtering procedures that take into account, for each object, the possible presence of multiple, equivalent spin period solutions that have not been systematically investigated in this preliminary application.

ON THE PERIODIC SOLITON SOLUTIONS FOR FRACTIONAL SCHRÖDINGER EQUATIONS

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed gEDAM technique entails creating nonlinear ordinary differential equations via a fractional complex transformation, which are then solved to acquire soliton solutions. Several 3D and contour graphs of the soliton solutions reveal periodicity in the wave profiles that offer crucial perspectives into the behavior of the system. The work sheds light on the dynamics of FSEs by displaying numerous families of periodic soliton solutions and their intricate relationships. These results hold significance not only for comprehending the dynamics of FSEs but also for nonlinear fractional partial differential equation applications.

Global insights into seed dispersal mutualism with cooperative efforts and environmental variability

Seed dispersal mutualisms assume a crucial role in sustaining overall ecosystem functionality. This study extends the foundational framework of seed dispersal mutualism by incorporating animal cooperation and environmental variability. Through the integration of stability theory and phase analysis, we investigate the global dynamics of seed dispersal mutualism in deterministic environments. Theoretical findings reveal that the deterministic system exclusively exhibits equilibrium dynamics, ruling out periodic solutions and chaotic phenomena. Notably, in cases of multiple coexisting equilibria, adjacent equilibria display distinct stability properties. Furthermore, employing stochastic analysis and Lyapunov exponents, we ascertain the uniqueness of global positive solutions in the stochastic model, along with stochastic ultimate boundedness, and precise thresholds for the stochastic persistence and extinction. These results illuminate the theoretical mechanisms governing transitions in seed dispersal mutualism across various stable population levels and underscore the insights into how essential biological factors, including animal cooperation efforts and environmental variability, impact the long-term dynamics of seed dispersal mutualism.

Mathematical Analysis for the Influence of Seasonality on Chikungunya Virus Dynamics

In this article, we discuss a mathematical system modelling Chikungunya virus dynamics in a seasonal environment with general incidence rates. We establish the existence, uniqueness, positivity and boundedness of a periodic orbit. We show that the global dynamics is determined using the basic reproduction number denoted by R0 and calculated using the spectral radius of a linear integral operator. We show the global stability of the disease free periodic solution if R0<1 and we show also the persistence of the disease if R0>1 where the trajectories converge to a periodic orbit. Finally, we display some numerical examples confirming the theoretical findings.

Abundant New Optical Soliton Solutions to the Biswas–Milovic Equation with Sensitivity Analysis for Optimization

This study extensively explores the Biswas–Milovic equation (BME) with Kerr and power law nonlinearity to extract the unique characteristics of optical soliton solutions. These optical soliton solutions have different applications in the field of precision in optical switching, applications in waveguide design, exploration of nonlinear optical effects, imaging precision, reduced intensity fluctuations, suitability for optical signal processing in optical physics, etc. Through the powerful (G′/G, 1/G)-expansion analytical method, a variety of soliton solutions are expressed in three distinct forms: trigonometric, hyperbolic, and rational expressions. Rigorous validation using Mathematica software ensures precision, while dynamic visual representations vividly portray various soliton patterns such as kink, anti-kink, singular soliton, hyperbolic, dark soliton, and periodic bright soliton solutions. Indeed, a sensitivity analysis was conducted to assess how changes in parameters affect the exact solutions, aiding in the understanding of system behavior and informing decision-making, especially in accurately designing or analyzing real-world optical phenomena. This investigation reveals the significant influence of parameters λ, τ, c, B, and Κ on the precise solutions in Kerr and power law nonlinearities within the BME. Notably, parameter λ exhibits consistently high sensitivity across all scenarios, while parameters τ and c demonstrate pronounced sensitivity in scenario III. The outcomes derived from this method are distinctive and carry significant implications for the dynamics of optical fibers and wave phenomena across various optical systems.

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

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.

A general and efficient harmonic balance method for nonlinear dynamic simulation

This paper presents a general and efficient harmonic balance (GE-HB) method to expeditiously evaluate the periodic responses of nonlinear systems. By employing the fast Fourier transform (FFT) technique, it rapidly computes harmonic expansion coefficients of the nonlinear terms and their corresponding constant tensor matrices. Notably, the nonlinear part of the Jacobian matrix is just the product of the coefficients and the constant tensor matrices, loop computations are no longer requisite for calculating the Jacobian matrix. This, facilitates accelerated determination of the nonlinear portion of the Jacobian matrix, thus obtaining the periodic responses of the system in an efficient manner. Furthermore, the constant tensor matrices are exclusively contingent upon the harmonic bases of the system's periodic response, devoid of reliance on the equation's specific formulation, facilitates obtaining the Jacobian matrix in a general manner. Simulation results on a 20-degree-of-freedom nonlinear oscillator chain system demonstrate that the efficiency of the GE-HB method is about tenfold that of the harmonic balance-alternating frequency/time domain (HB-AFT) method and twenty-fivefold that of the fourth-order Runge-Kutta (RK4) method. Further, the simulation of a 284-degree-of-freedom nonlinear rotor system exhibits the GE-HB method's computational efficiency surpassing HB-AFT method by over 500 times and Newmark method by over 3000 times. Moreover, in these two examples, the GE-HB method captures all solution branches, including unstable periodic solutions, providing comprehensive insights into periodic responses of the systems. The GE-HB method proposed here has the potential to explore dynamic characteristics for high-dimensional complex nonlinear systems in engineering.

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.

Interaction between soliton and periodic solutions and the stability analysis to the Gilson–Pickering equation by bilinear method and $$\exp (-\theta (\alpha ))$$-function approach arising plasma physics

Interaction between soliton and periodic solutions and the stability analysis to the Gilson–Pickering equation by bilinear method and $$\exp (-\theta (\alpha ))$$-function approach arising plasma physics

Persistence of a competition model of plankton allelopathy in time–space periodic environment

This work is devoted to the study of a competition model of plankton allelopathy imposed in time-space periodic environment. We prove that the system admits positive periodic solutions under certain conditions. We further obtain some sufficient conditions for the uniqueness and global stability of the positive periodic solution, which shows that the model is persistent. The main tools for our arguments are comparison theorems based on the maximum principle, sub- and supersolutions method, and an iteration method, which also permit the treatment of some more general reaction–diffusion models in periodic environment.

Periodic Solutions in a Simple Delay Differential Equation

A simple-form scalar differential equation with delay and nonlinear negative periodic feedback is considered. The existence of several types of slowly oscillating periodic solutions is shown with the same and double periods of the feedback coefficient. The periodic solutions are built explicitly in the case with piecewise constant nonlinearities involved. The periodic dynamics are shown to persist under small perturbations of the equation, which make it smooth. The theoretical results are verified through extensive numerical simulations.

Diffusion and distributed delay effects in a predator–prey system: A mathematical analysis

This study explores a nonautonomous model of prey-predator dynamics, featuring continuous distributed delay and diffusion, with time-dependent parameters. Our model depicts two ecological patches hosting two species: predators remain localized in one patch, while prey can disperse between both. Our analysis concentrates on ensuring the model’s consistent presence and achieving global stability. Additionally, we derive necessary conditions for periodic solutions by developing a pertinent Lyapunov function.

Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.

Bright, dark, and periodic soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation with fourth-order nonlinearity and dispersion

Bright, dark, and periodic soliton solutions for the (2+1)-dimensional nonlinear Schrödinger equation with fourth-order nonlinearity and dispersion

Bifurcation Analysis of a Class of Two-Delay Lotka–Volterra Predation Models with Coefficient-Dependent Delay

In this paper, a class of two-delay differential equations with coefficient-dependent delay is studied. The distribution of the roots of the eigenequation is discussed, and conditions for the stability of the internal equilibrium and the existence of Hopf bifurcation are obtained. Additionally, using the normal form method and the central manifold theory, the bifurcation direction and the stability for the periodic solution of Hopf bifurcation are calculated. Finally, the correctness of the theory is verified by numerical simulation.

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

In this paper, we study a class of nonlinear differential equations with multiple distributed delays. We reduced the problem of search for periodic solutions of the distributed delay differential equations to that of finding zeros of an operator on some Banach space. Utilizing the equivariant degree theory, we obtain some sufficient conditions to guarantee the existence of multiplicity of periodic solutions. Also, we give some information about the symmetry of the periodic solutions. At last, we give an example to illustrate our result.

Resonant soliton molecules, asymmetric solitons and the other diverse wave solutions to the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation for liquid with gas bubbles

Under the present study, we focus on developing some exact solutions of the (3 + 1)-dimensional generalized Kudryashov-Sinelshchikov equation (KSE) for the liquid with gas bubbles. First, the resonant soliton molecules on the different planes are extracted via imposing the velocity resonance condition on the N-soliton solutions that constructed via the Hirota bilinear method. Besides, the asymmetric solitons are also derived by choosing the appropriate parameters. In the end, the diverse kinds of travelling wave solutions including the bright wave, dark wave, singular wave and periodic wave solutions that expressed by the trigonometric, hyperbolic and rational functions are constructed by employing three effective approaches, namely the variational method, Sub-equation method and Wang’s direct mapping method-II. The profiles of the attained solutions are depicted to present the dynamical performances. As far as we all know, the results of this research are all new and can enable us figure out the physical properties of the (3 + 1)-dimensional generalized KSE better.