Topology of quasiperiodic functions on the plane

The article describes a topological theory of quasiperiodic functions on the plane. The development of this theory was started (in different terminology) by the Moscow topology group in early 1980s. It was motivated by the needs of solid state physics, as a partial (nongeneric) case of Hamiltonian foliations of Fermi surfaces with multivalued Hamiltonian function. The unexpected discoveries of their topological properties that were made in 1980s and 1990s have finally led to nontrivial physical conclusions along the lines of the so-called geometric strong magnetic field limit. A very fruitful new point of view comes from the reformulation of that problem in terms of quasiperiodic functions and an extension to higher dimensions made in 1999. One may say that, for single crystal normal metals put in a magnetic field, the semiclassical trajectories of electrons in the space of quasimomenta are exactly the level lines of the quasiperiodic function with three quasiperiods that is the dispersion relation restricted to a plane orthogonal to the magnetic field. General studies of the topological properties of levels of quasiperiodic functions on the plane with any number of quasiperiods were started in 1999 when certain ideas were formulated for the case of four quasiperiods. The last section of this work contains a complete proof of these results. Some new physical applications of the general problem were found recently.Comment: latex2e, 27 pages, 7 figure

Abelian Integral Method and its Application

Oscillation is a common natural phenomenon in real world problems. The most efficient mathematical models to describe these cyclic phenomena are based on dynamical systems. Exploring the periodic solutions is an important task in theoretical and practical studies of dynamical systems. Abelian integral is an integral of a polynomial differential 1-form over the real ovals of a polynomial Hamiltonian, which is a basic tool in complex algebraic geometry. In dynamical system theory, it is generalized to be a continuous function as a tool to study the periodic solutions in planar dynamical systems. The zeros of Abelian integral and their distributions provide the number of limit cycles and their locations. In this thesis, we apply the Abelian integral method to study the limit cycles bifurcating from the periodic annuli for some hyperelliptic Hamiltonian systems. For two kinds of quartic hyperelliptic Hamiltonian systems, the periodic annulus is bounded by either a homoclinic loop connecting a nilpotent saddle, or a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. For a quintic hyperelliptic Hamiltonian system, the periodic annulus is bounded by a more degenerate heteroclinic loop, which connects a nilpotent saddle to a hyperbolic saddle. We bound the number of zeros of the three associated Abelian integrals constructed on the periodic structure by employing the combination technique developed in this thesis and Chebyshev criteria. The exact bound for each system is obtained, which is three. Our results give answers to the open questions whether the sharp bound is three or four. We also study a quintic hyperelliptic Hamiltonian system with two periodic annuli bounded by a double homoclinic loop to a hyperbolic saddle, one of the periodic annuli surrounds a nilpotent center. On this type periodic annulus, the exact number of limit cycles via Poincar{\\u27e} bifurcation, which is one, is obtained by analyzing the monotonicity of the related Abelian integral ratios with the help of techniques in polynomial boundary theory. Our results give positive answers to the conjecture in a previous work. We also extend the methods of Abelian integrals to study the traveling waves in two weakly dissipative partial differential equations, which are a perturbed, generalized BBM equation and a cubic-quintic nonlinear, dissipative Schr\ {o}dinger equation. The dissipative PDEs are reduced to singularly perturbed ODE systems. On the associated critical manifold, the Abelian integrals are constructed globally on the periodic structure of the related Hamiltonians. The existence of solitary, kink and periodic waves and their coexistence are established by tracking the vanishment of the Abelian integrals along the homoclinic loop, heteroclinic loop and periodic orbits. Our method is novel and easily applied to solve real problems compared to the variational analysis

First-order perturbation for multi-parameter center families

Altres ajuts: Acord transformatiu CRUE-CSICIn the weak 16th Hilbert problem, the Poincaré-Pontryagin-Melnikov function, M(h), is used for obtaining isolated periodic orbits bifurcating from centers up to a first-order analysis. This problem becomes more difficult when a family of centers is considered. In this work we provide a compact expression for the first-order Taylor series of the function M(h,a) with respect to a, being a the multi-parameter in the unperturbed center family. More concretely, when the center family has an explicit first integral or inverse integrating factor depending on a. We use this new bifurcation mechanism to increase the number of limit cycles appearing up to a first-order analysis without the difficulties that higher-order studies present. We show its effectiveness by applying it to some classical examples

Integral characterization for Poincaré half-maps in planar linear systems

The intrinsic nature of a problem usually suggests a first suitable method to deal with it. Unfortunately, the apparent ease of application of these initial approaches may make their possible flaws seem to be inherent to the problem and often no alternative ways to solve it are searched for. For instance, since linear systems of differential equations are easy to integrate, Poincaré half-maps for piecewise linear systems are always studied by using the direct integration of the system in each zone of linearity. However, this approach is accompanied by two important defects: due to the different spectra of the involved matrices, many cases and strategies must be considered and, since the flight time appears as a new variable, nonlinear complicated equations arise. This manuscript is devoted to present a novel theory to characterize Poincaré half-maps in planar linear systems, that avoids the computation of their solutions and the problems it causes. This new perspective rests on the use of line integrals of a specific conservative vector field which is orthogonal to the flow of the linear system. Besides the obvious mathematical interest, this approach is attractive because it allows to simplify the study of piecewise-linear systems and deal with open problems in this field.Ministerio de Economía y Competitividad (Spain) / FEDER MTM2014-56272-C2-1-PMinisterio de Economía y Competitividad (Spain) / FEDER MTM2015-65608-PMinisterio de Economía y Competitividad (Spain) / FEDER MTM2017-87915-C2-1-PMinisterio de Economía y Competitividad (Spain) / FEDER PGC2018-096265-B-I00Consejería de Educación y Ciencia de la Junta de Andalucía (Spain) TIC-0130Consejería de Educación y Ciencia de la Junta de Andalucía (Spain) P12-FQM-1658Consejería de Educación y Ciencia de la Junta de Andalucía (Spain) P20_0116