Existence Of Multiple Periodic Solutions Research Articles

Multiple Hopf Bifurcations of Four Coupled van der Pol Oscillators with Delay

In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results.

A Maupertuis-type principle in relativistic mechanics and applications

We provide a Maupertuis-type principle for the following system of ODE, of interest in special relativity: ddtmx˙1-|x˙|2/c2=∇V(x),x∈Ω⊂Rn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\ extrm{d}}{{\ extrm{d}}t}\\left( \\frac{m\\dot{x}}{\\sqrt{1-|\\dot{x}|^2/c^2}}\\right) =\ abla V(x),\\qquad x\\in \\Omega \\subset \\mathbb {R}^n, \\end{aligned}$$\\end{document}where m, c > 0 and V: Omega rightarrow mathbb {R} is a function of class C^1. As an application, we prove the existence of multiple periodic solutions with prescribed energy for a relativistic N-centre type problem in the plane.

Multiple periodic solutions of second order parameter-dependent equations via rotation numbers

<abstract><p>We investigate the existence of multiple periodic solutions for a class of second order parameter-dependent equations of the form $ x''+f(t, x) = sp(t) $. We compare the behavior of its solutions with suitable linear and piecewise linear equations near positive infinity and infinity. Furthermore, in this context, the nonlinearity $ f $ does not satisfy the usual sign condition, and the global existence of solutions for the Cauchy problem associated to the equation is not guaranteed. Our approach is based on the Poincaré-Birkhoff twist theorem, a rotation number approach and the phase-plane analysis. Our result generalizes the result in Fonda and Ghirardelli <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> for second order parameter-dependent equations.</p></abstract>

Multiplicity of periodic solutions for weakly coupled parametrized systems with singularities

<abstract><p>We prove the existence of multiple periodic solutions for weakly coupled parametrized systems with a singularity of repulsive type at the origin and linear growth at infinity. The proof is based on a higher dimensional Poincaré-Birkhoff theorem and the phase-plane analysis of the solutions.</p></abstract>

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.

Multiple Periodicity in a Predator–Prey Model with Prey Refuge

We consider a delayed prey–predator model incorporating a refuge with a non-monotone functional response. It is supposed that prey can live in the predatory region and prey refuge, respectively. Based on Mawhin’s coincidence degree and nontrivial estimation techniques for a priori bounds of unknown solutions to the operator equation Lv=λNv, we prove the existence of multiple periodic solutions. Finally, an example demonstrates the feasibility of our main results.

Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect

<p style='text-indent:20px;'>The bistable dynamics of a modified Nicholson's blowflies delay differential equation with Allee effect is analyzed. The stability and basins of attraction of multiple equilibria are studied by using Lyapunov-LaSalle invariance principle. The existence of multiple periodic solutions are shown using local and global Hopf bifurcations near positive equilibria, and these solutions generate long transient oscillatory patterns and asymptotic stable oscillatory patterns.</p>

Border Collision Bifurcations and Chaos in a Ring of Three Unidirectionally Coupled Nonmonotonic Piecewise Constant Neurons

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

Periodic solutions for second-order difference equations with quadratic\u2013supquadratic condition

In this paper, we consider the existence of multiple periodic solutions for a class of second-order difference equations with quadratic–supquadratic growth condition at infinity. Moreover, we give three examples to illustrate our main result.

Multiple Periodic Solutions of a Class of Fractional Laplacian Equations

Abstract In this paper, we study the existence of multiple periodic solutions for the following fractional equation: ( - Δ ) s ⁢ u + F ′ ⁢ ( u ) = 0 , u ⁢ ( x ) = u ⁢ ( x + T ) x ∈ ℝ . (-\Delta)^{s}u+F^{\prime}(u)=0,\qquad u(x)=u(x+T)\quad x\in\mathbb{R}. For an even double-well potential, we establish more and more periodic solutions for a large period T. Without the evenness of F we give the existence of two periodic solutions of the problem. We make use of variational arguments, in particular Clark’s theorem and Morse theory.

Existence of Multiple Periodic Solutions for Cubic Nonautonomous Differential Equation

In this article, approaches to estimate the number of periodic solutions of ordinary differential equation are considered. Conditions that allow determination of periodic solutions are discussed. We investigated focal values for first-order differential nonautonomous equation by using the method of bifurcation analysis of periodic solutions from a fine focus Z=0. Keeping in focus the second part of Hilbert’s sixteenth problem particularly, we are interested in detecting the maximum number of periodic solution into which a given solution can bifurcate under perturbation of the coefficients. For some classes like C7,7,C8,5,C8,6,C8,7, eight periodic multiplicities have been observed. The new formulas ξ10 and ϰ10 are constructed. We used our new formulas to find the maximum multiplicity for class C9,2. We have succeeded to determine the maximum multiplicity ten for class C9,2 which is the highest known multiplicity among the available literature to date. Another challenge is to check the applicability of the methods discussed which is achieved by presenting some examples. Overall, the results discussed are new, authentic, and novel in its domain of research.

Existence of multiple periodic solutions to a semilinear wave equation with x-dependent coefficients

AbstractThis paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

On variational and topological methods in nonlinear difference equations

In this paper, first we survey the recent progress in usage of the critical point theory to study the existence of multiple periodic and subharmonic solutions in second order difference equations and discrete Hamiltonian systems with variational structure. Next, we propose a new topological method, based on the application of the equivariant version of the Brouwer degree to study difference equations without an extra assumption on variational structure. New result on the existence of multiple periodic solutions in difference systems (without assuming that they are their variational) satisfying a Nagumo-type condition is obtained. Finally, we also put forward a new direction for further investigations.

Multiple periodic solutions for two classes of nonlinear difference systems involving classical (ϕ1,ϕ2)-Laplacian

In this paper, we investigate the existence of multiple periodic solutions for two classes of nonlinear difference systems involving $(\phi_1,\phi_2)$-Laplacian. First, by using an important critical point theorem due to B. Ricceri, we establish an existence theorem of three periodic solutions for the first nonlinear difference system with $(\phi_1,\phi_2)$-Laplacian and two parameters. Moreover, for the second nonlinear difference system with $(\phi_1,\phi_2)$-Laplacian, by using the Clark's Theorem, we obtain a multiplicity result of periodic solutions under a symmetric condition. Finally, two examples are given to verify our theorems.

Multiple periodic solutions for Γ-symmetric Newtonian systems

The existence of periodic solutions in Γ-symmetric Newtonian systems x¨=−∇f(x) can be effectively studied by means of the Γ×O(2)-equivariant gradient degree with values in the Euler ring U(Γ×O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U(Γ×O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ×O(2)), and the reduced Γ×O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.

Existence of multiple periodic solutions to asymptotically linear wave equations in a ball

This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation $$\begin{aligned} u_{tt}-\Delta u = g(t,x,u) \end{aligned}$$ in a n-dimensional ball with radius R, where \(n>1\) and g(t, x, u) is radially symmetric in x and T-periodic in time. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of R and T. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to construct at least three radially symmetric solutions with time period T, when T is a rational multiple of R and g(t, x, u) satisfies some monotonicity and asymptotically linear conditions.

Multiple periodic solutions for a class of second-order neutral functional differential equations

In this paper we consider a class of second-order neutral functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of Z_{2} group index theory and variational methods. The main result is also illustrated with an example.

Multiple Periodic Solutions for a Class of Second-Order Neutral Impulsive Functional Differential Equations

In this paper, we study a class of second-order neutral impulsive functional differential equations. Under certain conditions, we establish the existence of multiple periodic solutions by means of critical point theory and variational methods. We propose an example to illustrate the applicability of our result.

Global existence of periodic solutions in an infection model

Abstract In this paper, a HTLV-I infection model with CTL immune response is considered. Taking the immune delay as a bifurcation parameter we investigate the global existence of periodic solutions of this model which shows existence of multiple periodic solutions theoretically.

MULTIPLE PERIODIC SOLUTIONS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS ACROSS RESONANCE

In this paper we study the existence of multiple periodic solutions of second-order ordinary differential equations. New results of multiplicity of periodic solutions are obtained when the nonlinearity may cross multiple consecutive eigenvalues. The arguments are proceeded by a combination of variational and degree theoretic methods.