The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kähler manifold

For a compact Kähler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most ρ(X) of g-periodic prime divisors. When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic (a result in the spirit of the Dynamic Manin-Mumford conjecture as in Zhang (2006) [17]).     

A continuation theorem on periodic solutions of regular nonlinear systems and its application to the exact tracking problem for the inverted spherical pendulum

We present a continuation method to obtain a family of T-periodic solutions for a family of T-periodic systems.In particular, we present some sufficient analytical conditions, which have the advantage of being an easy application to some systems of interest in physics or engineering. We apply these conditions to the exact tracking problem for the inverted spherical pendulum.     

Conditional Fredholm determinant for the S-periodic orbits in Hamiltonian systems

For S being a symplectic orthogonal matrix on R2n, the S-periodic orbits in Hamiltonian systems are a solution which satisfies x(0)=Sx(T) for some period T. This paper is devoted to establishing the theory of conditional Fredholm determinant in studying the S-periodic orbits in Hamiltonian systems. First, we study the property of the conditional Fredholm determinant, such as the Fréchet differentiability, the splittingness for the cyclic type symmetric solutions. Also, we generalize the Hill formula originally gotten by Hill and Poincaré. More precisely, let M be the monodromy matrix of the S-periodic orbits, then we get the formula relating the characteristic polynomial of the matrix SM and the conditional Fredhom determinant. Moreover, we study the relation of the conditional Fredholm determinant and the relative Morse index. Applications to the problem of linear stability for the S-periodic orbits are given.     

Periodic products of groups

In this paper we provide an overview of the results relating to the n-periodic products of groups that have been obtained in recent years by the authors of the present paper, as well as some results obtained by other authors in this direction. The periodic products were introduced by S.I. Adian in 1976 to solve the Maltsev’s well-known problem. It was shown that the periodic products are exact, associative and hereditary for subgroups. They also possess some other important properties such as the Hopf property, the C*-simplicity, the uniform non-amenability, the SQ-universality, etc. It was proved that the n-periodic products of groups can uniquely be characterized by means of certain quite specific and simply formulated properties. These properties allow to extend to n-periodic products of various families of groups a number of results previously obtained for free periodic groups B(m, n). In particular,we describe the finite subgroups of n-periodic products, Also, we analyze and extend the simplicity criterion of n-periodic products obtained previously by S.I. Adian.     

Approximation by Fourier means and generalized moduli of smoothness

The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces L_p, 1 ≤ p ≤ +∞, of 2π-periodic pth integrable functions and in the space C of continuous 2π-periodic functions in terms of the generalized modulus of smoothness constructed froma 2π-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered.     

On periodic groups of odd period n ≥ 1003

In the paper, using the Adyan-Lysenok theorem claiming that, for any odd number n ≥ 1003, there is an infinite group each of whose proper subgroups is contained in a cyclic subgroup of order n, it is proved that the set of groups with this property has the cardinality of the continuum (for a given n). Further, it is proved that, for m ≥ k ≥ 2 and for any odd n ≥ 1003, the m-generated free n-periodic group is residually both a group of the above type and a k-generated free n-periodic group, and it does not satisfy the ascending and descending chain conditions for normal subgroups either.     

2��-periodic self-similar solutions for the anisotropic affine curve shortening problem

We study the existence of 2??-periodic positive solutions of the equation $$u_{\theta\theta}+u=\displaystyle{\frac{a(\theta)}{u^3}},$$ where a(??) is a positive smooth 2??-periodic function. A priori estimates and sufficient conditions for the existence of solutions of the equation are established.     

Stability properties of steady-states for a network of ferromagnetic nanowires

We investigate the problem of describing the possible stationary configurations of the magnetic moment in a network of ferromagnetic nanowires with length L connected by semiconductor devices, or equivalently, of its possible L-periodic stationary configurations in an infinite nanowire. The dynamical model that we use is based on the one-dimensional Landau–Lifshitz equation of micromagnetism. We compute all L-periodic steady-states of that system, define an associated energy functional, and these steady-states share a quantification property in the sense that their energy can only take some precise discrete values. Then, based on a precise spectral study of the linearized system, we investigate the stability properties of the steady-states.     

New periodic recurrences with applications

We develop two methods for constructing several new and explicit m-periodic difference equations. Then we apply our results to two different problems. Firstly we show that two simple natural conditions appearing in the literature are not necessary conditions for the global periodicity of the difference equations. Secondly we present the first explicit non-linear analytic potential differential system having a global isochronous center.     

Periodic solutions of nonlinear functional difference equations at nonresonance case

Sufficient conditions for the existence of at least one T-periodic solution of nonlinear functional difference equation

Δx(n)+a(n)x(n)=f(n,u(n)),     

Global branches of periodic solutions for forced delay differential equations on compact manifolds

We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x^′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs.     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

Global stability of periodic orbits of non-autonomous difference equations and population biology

Elaydi and Yakubu showed that a globally asymptotically stable(GAS) periodic orbit in an autonomous difference equation must in fact be a fixed point whenever the phase space is connected. In this paper we extend this result to periodic nonautonomous difference equations via the concept of skew-product dynamical systems. We show that for a k-periodic difference equation, if a periodic orbit of period r is GAS, then r must be a divisor of k. In particular sub-harmonic, or long periodic, oscillations cannot occur. Moreover, if r divides k we construct a non-autonomous dynamical system having minimum period k and which has a GAS periodic orbit with minimum period r. Our methods are then applied to prove a conjecture by J. Cushing and S. Henson concerning a non-autonomous Beverton-Holt equation which arises in the study of the response of a population to a periodically fluctuating environmental force such as seasonal fluctuations in carrying capacity or demographic parameters like birth or death rates.     

On S-asymptotically ω-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions for which exists ω>0 such that limt→∞(f(t+ω)−f(t))=0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.     

Monodromization and Difference Equations with Meromorphic Periodic Coefficients

We consider a system of two first-order difference equations in the complex plane. We assume that the matrix of the system is a 1-periodic meromorphic function having two simple poles per period and bounded as Im z → ±∞. We prove the existence and uniqueness of minimal meromorphic solutions, i.e., solutions having simultaneously a minimal set of poles and minimal possible growth as Im z → ±∞. We consider the monodromy matrix representing the shift-byperiod operator in the space of meromorphic solutions and corresponding to a basis built of two minimal solutions. We check that it has the same functional structure as the matrix of the initial system of equations and, in particular, is a meromorphic periodic function with two simple poles per period. This implies that the initial equation is invariant with respect to the monodromization procedure, that is, a natural renormalization procedure arising when trying to extend the Floquet–Bloch theory to difference equations defined on the real line or complex plane and having periodic coefficients. Our initial system itself arises after one renormalization of a self-adjoint difference Schrödinger equation with 1-periodic meromorphic potential bounded at ±i∞ and having two poles per period.     

New exact traveling and non-traveling wave solutions for (2 + 1)-dimensional Burgers equation

In this paper, some novel solitary wave solutions, including solitary-like wave solution, x-periodic soliton solution, y-periodic soliton solution, doubly periodic solution, rational solution, and new non-traveling wave solution, are obtained for (2 + 1)-dimensional Burgers equation by means of the generalized direct ansätz method and different test functions.     

On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity

It is well known that a scalar differential equation , where f(t,x) is continuous, T-periodic in t and weakly convex or concave in x has no, one or two T-periodic solutions or a connected band of T-periodic solutions. The last possibility can be excluded if f(t,x) is strictly convex or concave for some t in the period interval. In this paper we investigate how the actual number of T-periodic solutions for a given equation of this type in principle can be determined, if f(t,x) is also assumed to have a continuous derivative . It turns out that there are three cases. In each of these cases we indicate the monotonicity properties and the domain of values for the function P(ξ)=S(ξ)−ξ, where S(ξ) is the Poincaré successor function. From these informations the actual number of periodic solutions can be determined, since a zero of P(ξ) represents a periodic solution.     

On the existence of periodic solutions for a class of p-Laplacian system

By using generalized Borsuk theorem in coincidence degree theory, some criteria to guarantee the existence of ω-periodic solutions for a class of p-Laplacian system are derived.     

Optimal Recovery of Periodic Functions from Fourier Coefficients Given with an Error

We construct optimal methods of recovery of 2π-periodic functions analytic in a strip and its derivatives at a pointt∈ [0, 2π), using information about the Fourier coefficients given with an error in the uniform norm. The same problem is solved for the Sobolev spaceW^r₂.     

Selection problems of $${{\mathbb {Z}}}^2$$-periodic entropy solutions and viscosity solutions

\({{\mathbb {Z}}}^2\)-periodic entropy solutions of hyperbolic scalar conservation laws and \({{\mathbb {Z}}}^2\)-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate \({{\mathbb {Z}}}^2\)-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural \({{\mathbb {Z}}}^2\)-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.