About a conjecture for uniformly isochronous polynomial centers
We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.
A survey of isochronous centers
In this survey we give an overview of the results obtained in the study of isochronous centers of vector fields in the plane. This paper consists of two parts. In the first one (sections 2--8), we review some general techniques that proved to be useful in the study of isochronicity. In the second one (sections 9--16), we try to give a picture of the state of the art at the moment this review was written. In section 2, we give some basic definitions about centers, isochronous centers, first integrals, integrating factors, particular algebraic solutions, and other related concepts. In this sections we also give some general theorems about centers and isochronous centers, and we give a brief account of the evolution of the researches in this field. In the successive sections we focus on various methods that have been used in attacking the isochonicity problem. We start with linearizations in section 3, stating Poincar\'e's classical theorem and some of its consequences. Section 4 is devoted to describe the procedure that leads to define and compute isochronous constants. In section 5, commutators are introduced, and basic facts about couples of commuting systems are described. Classical theorems about systems obtained from complex ordinary differential equations are collected in section 6. Hamiltonian systems are considered in section 7, where their connection to the study of the Jacobian Conjecture is showed, too. Section 8 is concerned with systems having constant angular speed with respect to some coordinate system. The second part starts with section 9, that is devoted to recent results about second order differential equations not immediately reducible to hamiltonian systems. This section also contains the characterization of isochronous centers of reversible Li\'enard systems. In section 10 we list all fundamental results about isochronous centers of quadratic systems. Next section contains results about cubic systems with homogeneous nonlinearities. Sections 12 is devoted to cubic reversible systems. In section 13 we collect results about quartic and quintic systems with homogeneous nonlinearities. A class of particular cubic systems, with degenerate infinity is considered in section 14. Finally, section 15 is devoted to Kukles system. All the sections of the second part, and some of the first part, contain tables, where the main features of the considered systems are collected. When possible, for every class of systems we have written the system in rectangular and polar coordinates, and we have reported a first integral, a commutator, a linearization and a reciprocal integrating factor. The bibliography contains references both to papers devoted to the study of isochronicity and to papers concerned with integrability of plane systems and the study of the period function of centers. We have tried to make the bibliography so complete as possible for what is concerned with isochronicity. We have made no effort to make it complete for papers about integrability and the study of the period function. We apologize for possible mistakes and encourage the readers to communicate us any corrections.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.