Number Of Periodic Solutions Research Articles

Mirrored dynamics of a wild mosquito population suppression model with Ricker-type survival probability and time delay.

Here, we formulated a delayed mosquito population suppression model including two switching sub-equations, in which we assumed that the growth of the wild mosquito population obeys the Ricker-type density-dependent survival function and the release period of sterile males equals the maturation period of wild mosquitoes. For the time-switched delay model, to tackle with the difficulties brought by the non-monotonicity of its growth term to its dynamical analysis, we employed an essential transformation, derived an auxiliary function and obtained some expected analytical results. Finally, we proved that under certain conditions, the number of periodic solutions and their global attractivities for the delay model mirror that of the corresponding delay-free model. The findings can boost a better understanding of the impact of the time delay on the creation/suppression of oscillations harbored by the mosquito population dynamics and enhance the success of real-world mosquito control programs.

The Existence of Multiple Periodic Solutions to a Class of Fourth-Order Difference Equations

In this paper, we study the multiplicity of periodic solutions for a class of fourth-order difference equations. We estimate a lower bound on the number of periodic solutions when the nonlinearity is asymptotic both at the origin and at infinity. Our results generalize the reference’ results.

Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules

Chirality is an indispensable geometric property in the world that has become invariably interlocked with life. The main goal of this paper is to study the nonlinear dynamic behavior and periodic vibration characteristic of a two-coupled-oscillator model in the optics of chiral molecules. We systematically discuss the stability and local dynamic behavior of the system with two pairs of identical conjugate complex eigenvalues. In particular, the existence and number of periodic solutions are investigated by establishing the curvilinear coordinate and constructing a Poincaré map to improve the Melnikov function. Then, we verify the accuracy of the theoretical analysis by numerical simulations, and take a comprehensive look at the nonlinear response of multiple periodic motion under certain conditions. The results might be of important significance for the vibration control, safety stability and design optimization for chiral molecules.

Complex Methods for Bounds on the Number of Periodic Solutions with an Application to a Neural Model

The qualitative behavior of the differential equation , where x is real and f is periodic in t, is determined by the attracting periodic solutions. This article gives an exposition of a method of Ilyashenko that uses complex analysis to determine an upper bound on the number of periodic solutions. The method is then applied to a well-known model of a single neuron or pool of neurons.

Rogue Waves generation by Using higher order rational solutions of Discrete Nonlinear Schrödinger Equation

The higher-order discrete nonlinear Schrödinger equation has been demonstrated to be related to cubic-quintic nonlinearity. This executes the specified number of periodic solutions. The localized solution of the homogeneous cubic-quintic discrete nonlinear Schrödinger equation has been researched. By employing scaling transformations, it was possible to convert cubic-quintic discrete nonlinear Schrödinger equations to constant-coefficient higher-order rational solutions with discrete nonlinear Schrödinger equations (DNLSE). We investigated our solution in the periodic form of the gain/loss term and discovered some intriguing results. Even the solution is interconnected, and its features are extensively exploited in a variety of physical phenomena, including optical waveguides, B-E condensates in optical media, non-linearity in photonic crystals, Kerr-type nonlinearity and photo refracting medium.

Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules


 
 
 We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.
 
 

Effects of oblateness of the primaries on natural periodic orbit-attitude behaviour of satellites in three body problem

• Equations of Motion for the Perturbed Circular Restricted Three Body Problem (P-CR3BP) with considering A 2 - A 6 primaries oblate coefficients have derived for the first time. • It was shown that considering primaries zonal harmonic coefficients j 2 - j 6 has a significant effect on satellite orbit-attitude behavior. It is observed that considering A 2 - A 6 primaries oblate coefficients amplify the effect of oblate perturbations and the zonal harmonic coefficients after j 6 have negligible effect. • Proper initial guess of periodic solution of perturbed model identified by Poincaré maps. • As an innovation, finding the appropriate value of inertia ratio K to find periodic response of orbit-attitude was performed using Poincaré mapping for the first time. • It was shown for the first time, by more accurately simulating the study environment by considering oblate perturbations, the density of points on the islands created on Poincaré map appeared to increase, and in fact, it became easier to identify the initial guess of attitude states. • Increasing the inertia ratio and increasing the orbital period reduces the chances of finding periodic responses. This study was done to investigate the effect of perturbations of both oblate massive primaries on the periodic orbit-attitude behavior of satellites at the three-body problem. Governing equations of the perturbed coupled orbit-attitude were derived using principles of the Lagrangian mechanics. As initial conditions of the coupled model play a key role in finding periodic responses of both orbit and attitude, so the perturbed circular restricted three-body problem (P-CR3BP) coupled orbit-attitude correction algorithm was proposed to correct the initial guess of the coupled model. The correction algorithm requires an appropriate initial guess vector as the number of periodic solutions is limited. This initial guess vector consists of two parts: orbital and attitude dynamic state parameters. A suitable initial guess was suggested for orbital parameters considering a small error at the initial conditions of the unperturbed periodic orbits. These initial conditions were extracted by another orbital correction algorithm to correct orbital states of periodic orbits in the unperturbed circular restricted three-body problem (U-CR3BP). Furthermore, the initial guess of attitude dynamic parameters was identified by the Poincaré mapping method. Considering oblate perturbations would change the initial conditions of periodic orbits at P-CR3BP relative to U-CR3BP. Also, because the proposed model of the coupled orbit-attitude governing equations is such that only orbital state parameters influence attitude dynamic parameters, so the behavior of attitude dynamics will also change by altering orbital motion. Comparison of patterns in Poincaré maps, satellites angular velocity, and initial conditions of periodic coupled orbit-attitude response in U-CR3BP and P-CR3BP indicated the effect of both oblate primaries perturbations. Consideration of these perturbations resulted in a more accurate simulation of the study environment that helps to understand the natural motion of the satellite.

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

Restricted three body problem considering the perturbations of both oblate massive primaries

This paper was conducted to investigate the periodic solution of the Perturbed Circular Restricted Three Body Problem (P-CRTBP) orbit with respect to the perturbations effect of the both oblate massive primaries. The governing equations of the satellite motion in this problem has been extracted employing Lagrangian mechanics. Since, the problem may have no closed-formed solution and the numerical methods must be utilized to address this type of issue, so the problem could have different periodic or non-periodic solutions depending on the initial conditions. For this purpose, an algorithm named “orbital correction algorithm” has been utilized to achieve the appropriate initial conditions of the P-CRTBP periodic orbit state variables. Since, the number of periodic solutions is restricted; the suitable initial guess vector as the inputs of the orbit correction algorithm increases the probability of achieving more accurate initial conditions. Suitable initial guess could be selected from the third-order approximation of the Unperturbed Circular Restricted Three Body Problem’s (U-CRTBP) equations of motion. Considering the effect of these synthetic perturbations, could be considered as a stepping stone to describe the spacecraft's natural motion in such problems. Since, the number of periodic solutions is restricted; the suitable initial guess vector as the inputs of the orbit correction algorithm increases the chances of achieving more accurate initial conditions.

Periodic Solutions of Discontinuous Duffing Equations

In this paper, we investigate the multiplicity problem of periodic solutions for a class of periodically forced Duffing equations allowing for discontinuities. By using a generalized form of the Poincare–Birkhoff theorem due to Ding (Proc Am Math Soc 88:341–346, 1983), we demonstrate that the discontinuous equation has an infinite number of periodic solutions with large amplitude.

Solitary Waves and Periodic Waves in a Perturbed KdV Equation

In this paper, we consider a perturbed Korteweg–de Vries (KdV) equation with weak dissipation and Marangoni effects. Main attention is focused on the existence conditions of periodic and solitary wave solutions of the perturbed KdV equation. Based on bifurcation theory of dynamic system and geometric singular perturbation method, the parameter conditions and wave speed conditions for the existence of one periodic solution, one solitary solution and the coexistence of a solitary solution and infinite number of periodic solutions are given. By using Chebyshev criterion to analyze the ratio of Abelian integrals, the monotonicity of wave speed is proved, and the upper and lower bounds of wave speed are obtained.

Bifurcation Features of Periodic Solutions of the Mackey–Glass Equation

Bifurcations of periodic solutions of the well-known Mackey–Glass equation from its unique equilibrium state under varying equation parameters are considered. The equation is used as a mathematical model of variations in the density of white blood cells (neutrophils). Written in dimensionless variables, the equation contains a small parameter multiplying the derivative, which makes this equation singular. It is shown that the behavior of solutions to the equation with initial data from a fixed neighborhood of the equilibrium state in the equation phase space is described by a countable system of nonlinear ordinary differential equations. This system has a minimal structure and is called the normal form of the equation in the neighborhood of the equilibrium state. One fast variable and a countable number of slow variables can be extracted from this system of equations. As a result, the averaging method can be applied to the obtained system. It is shown that the equilibrium states of the averaged system of equations in slow variables are associated with periodic solutions of the same stability type in the original equation. The possibility of simultaneous bifurcation of a large number of periodic solutions (multistability bifurcation) is shown. It is also shown that, with a further increase in the bifurcation parameter, each of the periodic solutions exhibits the transition to a chaotic attractor through a series of period-doubling bifurcations. Thus, the behavior of the solutions of the Mackey–Glass equation is characterized by chaotic multistability.

A New Method to Study the Periodic Solutions of the Ordinary Differential Equations Using Functional Analysis

In this paper, a new theorems of the derived numbers method to estimate the number of periodic solutions of first-order ordinary differential equations are formulated and proved. Approaches to estimate the number of periodic solutions of ordinary differential equations are considered. Conditions that allow us to determine both upper and lower bounds for these solutions are found. The existence and stability of periodic problems are considered.

Almost Periodic Solutions of First-Order Ordinary Differential Equations

Approaches to estimate the number of almost periodic solutions of ordinary differential equations are considered. Conditions that allow determination for both upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are studied. The novelty of this paper lies in the fact that the use of apparatus derivatives allows for the reduction of restrictions on the degree of smoothness of the right parts. In our work, regarding the number of periodic solutions of equations first order, we don’t require a high degree of smoothness and no restriction on the smoothness of the second derivative of the Schwartz equation. We have all of these restrictions lifted. Our new form presented also emphasizes this novelty.

Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

Given two polynomials P,q we consider the following question: “how large can the index of the first non-zero moment m˜k=∫abPkq be, assuming the sequence is not identically zero?” The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: K⩽2+deg⁡q+3(deg⁡P−1)2. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation y′=py2+εqy3 for p,q polynomials and ε≪1. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed 5+deg⁡q+3deg2⁡p. This is the first such bound depending solely on the degrees of the Abel equation.

Magnetorheological damping and semi-active control of an autoparametric vibration absorber

A numerical study of an application of magnetorheological (MR) damper for semi-active control is presented in this paper. The damper is mounted in the suspension of a Duffing oscillator with an attached pendulum. The MR damper with properties modelled by a hysteretic loop, is applied in order to control of the system response. Two methods for the dynamics control in the closed-loop algorithm based on the amplitude and velocity of the pendulum and the impulse on–off activation of MR damper are proposed. These concepts allow the system maintaining on a desirable attractor or, if necessary, to change a position from one attractor to another. Additionally, the detailed bifurcation analysis of the influence of MR damping on the number of periodic solutions and their stability is shown by continuation method. The influence of MR damping on the chaotic behavior is studied, as well.

Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameterσassociated with the intensity of harvesting is explored. Asσgrows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.

Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

Bifurcation of the periodic motion in nonlinear delayed oscillators

We use the residue harmonic balance scheme to study the periodic motions of a class of second-order delay-differential equations with cubic nonlinearities near and after Hopf bifurcation. The multiple solutions are found by homotopy continuation. Then, the approximation to any desired accuracy for a specific solution is captured by solving linear equations iteratively. The second-order solutions give good predictions for the frequency and amplitude, which are verified by the Runge–Kutta numerical solutions. Two typical examples, the temporal dynamics of the delay Liénard oscillator and the delay feedback Duffing system, are studied and compared. The results show how to trace analytically the relevant effect of the stiffness coefficient and the time delay on the dynamics and on the number of periodic solutions, even for large values of the bifurcation parameters.

Bifurcation analysis of delay-induced periodic oscillations

In this paper we consider a generic differential equation with a cubic nonlinearity and delay. This system, in the absence of delay, is known to undergo an oscillatory instability. The addition of the delay is shown to result in the creation of a number of periodic solutions with constant amplitude and a constant frequency; the number of solutions increases with the size of the delay. Indeed, for many physical applications in which oscillatory instabilities are induced by a delayed response or feedback mechanism, the system under consideration forms the underlying backbone for a mathematical model. Our study showcases the effectiveness of performing a numerical bifurcation analysis, alongside the use of analytical and geometrical arguments, in investigating systems with delay. We identify curves of codimension-one bifurcations of periodic solutions. We show how these curves interact via codimension-two bifurcation points: double singularities which organise the bifurcations and dynamics in their local vicinity.