Periodic solutions of Abel differential equations

For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have at most one limit cycle which appears through multiple Hopf bifurcation.     

Bifurcating time-periodic solutions of Navier-Stokes equations in infinite cylinders

Summary For the problem of hydrodynamical stability in an infinite cylindrical domain, we investigate all time-periodic solutions, not only spatially periodic ones, when a Hopf bifurcation occurs. When reflection symmetry is present, we show the existence of spatially quasiperiodic flows. We also show the existence of heteroclinic solutions connecting two symmetrically traveling waves that stay at each end of the cylinders (defect solutions). The technique we use rests on (i) a center manifold argument in a space of time-periodic vector fields, (ii) symmetry and normal form arguments for the reduced ordinary differential equation in two dimensions (without reflection symmetry) or in four dimensions (with reflection symmetry), and (iii) the integrability of the associated normal form. It then remains to prove a persistence result when we add the higher-order terms of the vector field.     

Time periodic solutions of a class of degenerate parabolic equations

1.IntroductionManypapershavebeendevotedtotheexistenceoftimeperiodicsolutionsforsemilinearparabolicequations,see[1--8].Atthesametime,thestudyofquasilinearperiodic-parabolicequationsalsoattractedmanyauthors,seealso[9--141.Inparticular,recentlyHess,PozioandTesei[13]usedthemonotonicitymethodstodealwiththeequationsonot=aam a(x,t)u,wherem>1andaisafunctionperiodicint,andMizoguchi[lllappliedtheLeray-Schauderdegreetheorytoinvestigatetheequationswithsuperlinearforcingtermwherem>1,hisapositiveperiodicf…     

Bifurcations of periodic solutions of delay differential equations

In this paper we develop Kaplan-Yorke's method and consider the existence of periodic solutions for some delay differential equations. We especially study Hopf and saddle-node bifurcations of periodic solutions with certain periods for these equations with parameters, and give conditions under which the bifurcations occur. We also give application examples and find that Hopf and saddle-node bifurcations often occur infinitely many times.     

Schaefer type theorem and periodic solutions of evolution equations

Two fixed point theorems for the sum of contractive and compact operators are obtained in this paper, which generalize and improve the corresponding results in [H. Schaefer, Über die methode der a priori-Schranken, Math. Ann. 129 (1955) 415-416; T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383-2390; V.I. Istrǎtescu, Fixed Point Theory, an Introduction, Reidel, Dordrecht, 1981; T.A. Burton, K. Colleen, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998) 23-31; D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press, Cambridge, 1980]. As the applications for the results, we obtain the existence of periodic solutions for some evolution equations with delay, which extend the corresponding results in [T.A. Burton, B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations 90 (1991) 357-396].     

Numbers of subnormal solutions for higher order periodic differential equations

In this paper, we estimate the number of subnormal solutions for higher order linear periodic differential equations, and estimate the growth of subnormal solutions and all other solutions. We also give a representation of subnormal solutions of a class of higher order linear periodic differential equations.     

Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system

We consider a delayed predator-prey system. We first consider the existence of local Hopf bifurcations, and then derive explicit formulas which enable us to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurcations, using the normal form theory and center manifold argument. Special attention is paid to the global existence of periodic solutions bifurcating from Hopf bifurcations. By using a global Hopf bifurcation result due to Wu [Trans. Amer. Math. Soc. 350 (1998) 4799], we show that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of delay. Finally, several numerical simulations supporting the theoretical analysis are also given.     

Multiplicity of positive periodic solutions to superlinear repulsive singular equations

In this paper, we study positive periodic solutions to the repulsive singular perturbations of the Hill equations. It is proved that such a perturbation problem has at least two positive periodic solutions when the anti-maximum principle holds for the Hill operator and the perturbation is superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.     

Global existence of periodic solutions in an infection model

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.     

Existence of positive periodic solutions of impulsive functional differential equations with two parameters

In this paper, we employ a well-known fixed-point index theorem to study the existence and non-existence of positive periodic solutions for the periodic impulsive functional differential equations with two parameters. Several existence and non-existence results are established.     

The number of periodic solutions of some analytic equations of Abel type

New results are proved to estimate the number of periodic solutions of a differential equation of Abel type by using a modification of a technique introduced by Ilyashenko. The main tool is an estimate on the number of zeros of a holomorphic function. A concrete example is analyzed but the results are presented to make the method flexible and applicable to other equations.     

Nontrivial periodic solutions to a type of delayed resonant differential equations

In this paper, we consider a type of delayed resonant differential equations. We focus on the existence of periodic solutions. Employing the Clark dual, we provide two sets of criteria on the existence of at least one periodic solution. In fact, the periodic solutions are critical points minimizing the dual functional of the coupled Hamiltonian system on certain subspaces of a Banach space.     

On analyticity rate estimates of the solutions to the Navier-Stokes equations in Bessel-potential spaces

On this paper spatial analyticity of solutions to the nonstationary incompressive Navier-Stokes flow in is established. The proof is based on the estimates for the higher order derivatives of solutions. These estimates imply not only the regularizing rates near t=0 but also decaying rates at t→∞, as long as the solution exists. Although basic strategy is similar to our previous work with Giga for Ln space, one can make the proof short using several tools from harmonic analysis.     

Existence results for periodic solutions of integro-dynamic equations on time scales

Using the topological degree method and Schaefer’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. Furthermore, we provide several applications to scalar equations, in which we develop a time scale analog of Lyapunov’s direct method and prove an analog of Sobolev’s inequality on time scales to arrive at a priori bound on all periodic solutions. Therefore, we improve and generalize the corresponding results in Burton et al. (Ann Mat Pura Appl 161:271–283, 1992)      

On periodic solutions of nonlinear integral equations modelling infectious disease on measure chain

The modelling of the spread of infectious disease is carried out for time t on a measure chain T. Our approach unifies the continuous case and the discrete case . The model is described by the integral equation

where x(t) represents the proportion of the population infected at time t, f(t,x(t)) denotes the proportion of the population newly infected per unit time, and τ is the length of time an individual remains infectious. Using the measure chain calculus, we shall develop criteria for the existence of a nontrivial and nonnegative periodic solution for the modelling equation. The criteria can be implemented numerically, for this we shall give an algorithm as well as illustrative examples.     

Minimal periods of periodic solutions of some Lipschitzian differential equations

A problem of finding lower bounds for periods of periodic solutions of a Lipschitzian differential equation, expressed in the supremum Lipschitz constant, is considered. Such known results are obtained for systems with inner product norms. However, utilizing the supremum norm requires development of a new technique, which is presented in this paper. Consequently, sharp bounds for equations of even order, both without delay and with arbitrary time-varying delay, are found. For both classes of system, the obtained bounds are attained in linear differential equations.