On the spectral flow in Lorentzian manifolds

In this paper we use functional analytical techniques to determine the differential equation satisfied by the eigenvalues of a smooth family of Fredholm operators, obtained from the index form along a Lorentzian geodesic. The formula is then applied to the study of the evolution of the index function, and, using a perturbation argument, we prove a version of the classical Morse index theorem for stationary Lorentzian manifolds. Received: January 31, 2000; in final form: March 13, 2002?Published online: February 20, 2003 The second author is partially sponsored by CNPq (Brazil), Grant 200615/01-7.     

A remark on the genericity of multiplicity results for forced oscillations on manifolds

We discuss the genericity of some multiplicity results for periodically perturbed autonomous first- and second-order ODEs on manifolds.?In particular, the genericity of the following property is investigated: if the differentiable manifold M is compact, then the equation _π=h(x,)+f(t,x,) on M has |χ(M)| geometrically distinct T-periodic solutions for any small enough T-periodic perturbing function f. Received: January 24, 2000; in final form: January 16, 2001?Published online: March 19, 2002     

Further results on global stability of third-order nonlinear differential equations

Sufficient conditions are established for the global stability of certain third-order nonlinear differential equations. Our result improves on Qian’s [C. Qian, On global stability of third-order nonlinear differential equations, Nonlinear Anal. 42 (2000) 651–661].     

On an area-preserving crystalline motion

The asymptotic behavior of solutions to an area-preserving crystalline motion is investigated in this paper. In this equation, the area enclosed by the solution polygon is preserved, while its circumference keeps on shrinking. By geometric consideration, establishing several isoperimetric inequalities and using the theory of dynamical systems, we show that the asymptotic shape of a solution polygon is the boundary of the Wulff shape. Received: 23 February 2000 / Accepted: 23 January 2001 / Published online: 23 April 2001     

Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems

This work deals with planar polynomial differential systems , . We give a set of necessary conditions for a system to have an invariant algebraic curve. These conditions are determined from the value of the cofactor at the singular points of the system, once considered in a compact space. We apply these results to show the non-Liouvillian integrability of several families of quadratic systems with an algebraic limit cycle.     

Polynomial differential systems having a given Darbouxian first integral

The Darbouxian theory of integrability allows to determine when a polynomial differential system in has a first integral of the kind f₁^λ₁?f_p^λ_pexp(g/h) where f_i, g and h are polynomials in , and for i=1,…,p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.     

Validity of the multiple scale method for very long intervals

For a class of nonlinear oscillation problems containing a small parameter, it is known that a two-scale method using timest and t gives results valid to any desired order for time (1/). We ask when results can be obtained which are valid for (1/²) or for allt > 0. We show that there is an obstruction to introducing a third time scale ² t, and give an example in which this obstruction does not vanish, so that a third scale cannot be introduced, even though the solution exists for all time. The obstruction does vanish if the first order averaged equation vanishes, in which case the three-scale solution actually involves onlyt and ² t and is valid for time (1/²). The obstruction also vanishes if a certain contracting or dissipative condition is met, but in this case the two-scale solution is already valid for all time and the third scale is not needed. These results correspond to known results for the method of averaging, but are here proved for the multiple scale method without use of averaging.     

On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Summary For the nonlinear system , which has a family { _h } of closed orbits, we consider perturbations of the type , whereP andQ are arbitrary polynomials. The abelian integralsA(h) corresponding to this family { _h } are investigated. By deriving differential equations forA(h) and proving monotonicity for quotients of abelian integrals, we obtain results on the number of zeros of abelian integrals and, hence, on the number of closed orbits _h which persist as limit cycles of the perturbed system (^*). In particular, a uniqueness theorem for limit cycles of (^*) with quadratic polynomialsP, Q is proved. Moreover, whenP, Q are of arbitrary degree, a lower bound for the possible number of limit cycles of (^*) is derived.     

Large volume growth and the topology of open manifolds

In this paper, we study complete noncompact Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We find some reasonable conditions to insure that this kind of manifolds are diffeomorphic to a Euclidean space or have finite topological type. Received: January 4, 2000; in final form: October 31, 2000 / Published online: 19 October 2001     

On Existence of Infinitely Many Homoclinic Solutions

 In this note we prove some sufficient condition for the existence of homoclinic solutions in nonautonomous ODE’s. As an application we show that there exist infinitely many (geometrically distinct) homoclinic solutions to the trivial 0 solution in the planar system

On a closed orbit of some constrained system

We study on what one calls a constrained system of ODEs on It consists of two ordinary differential equations and an algebraic equation with respect to three unknown functions. We seek closed orbits of such a system. A necessary and sufficient condition for the system to have non-trivial closed orbits is given. Elementary tools such as Lyapunov functions and Poincaré’s index theory are used in the proof of the result. As an application we consider a constrained system associated with a non-constraint system of ODEs called the modified Bonhöffer-van der Pol system.     

Almost periodic solutions of forced vectorial Liénard equations

We describe the set of bounded or almost periodic solutions of the following Liénard system: , where is almost periodic, is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R^N. Then, we state a result of existence and uniqueness of almost periodic solution.     

Constructing cubature formulae by the method of reproducing kernel

Summary. We examine the method of reproducing kernel for constructing cubature formulae on the unit ball and on the triangle in light of the compact formulae of the reproducing kernels that are discovered recently. Several new cubature formulae are derived. Received April 15, 1998 / Revised version received November 24, 1998 / Published online January 27, 2000     

Oscillation constant of second-order non-linear self-adjoint differential equations

Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini and Villari. Received: January 5, 2001?Published online: June 11, 2002     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.     

On interfaces between incompatible wells and elastic deformations of infinite cylinders

We consider the minimization problem for the functional where is an infinitely long cylinder. The density is polyconvex and assumed to be 0 on a set of wells and positive elsewhere. We show that the gradients of solutions with finite energy have to approach one component for and one component for , if the number of components is finite (among other conditions). Moreover, for certain pairs of distinct components we construct nontrivial minimizers within the class of solutions approaching the given components. We follow ideas developed in the variational study of heteroclinic connections for Lagrangian systems and we put special emphasis on multiplicity of such interface solutions. We discuss an application in the theory of nonlinear elasticity, where such solutions are called semi-necks. When a two-dimensional infinite hyperelastic strip is stretched along its infinite direction it may occur that for a given tensile load many homogeneous deformations are possible. In such a case we show by infimizing the energy functional the existence of configurations that tend asymptotically to two different homogeneous deformations. Received: 1 March 2000 / Accepted: 4 December 2000 / Published online: 4 May 2001     

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

Non-isochronicity of the center in polynomial Hamiltonian systems

In 2002 Jarque and Villadelprat proved that planar polynomial Hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial Hamiltonian systems of even degree. Recently, it was proved that a planar polynomial Hamiltonian system is non-isochronous if a quantity, denoted by M_2m−2, can be computed such that M_2m−2≤0. As a corollary of this criterion, the open question was answered for those systems with only even degree nonlinearities. In this paper we consider the case of M_2m−2>0 and give a new criterion for non-isochronicity. Applying the new criterion, we also answer the open question for some cases in which some terms of odd degree are included.