Let XX be a Banach space and T¿:X¿XT¿:X¿X a family of invertible contractions, T¿=L¿+f¿T¿=L¿+f¿, where L¿L¿ is linear and f¿f¿ is nonlinear with f¿(0)=0f¿(0)=0. We give conditions for the existence of a family of global linearization maps H¿H¿, such that View the MathML sourceH¿°T¿°H¿-1=L¿, with a smooth dependence on ¿. The results depend strongly on the choice of some appropriate spaces of maps, adapted norms and the use of a specific fixed point theorem with smooth dependence on parameters
We establish sharp regularity estimates for solutions to Lu = f in Omega subset of R-n being the generator of any stable and symmetric Levy process. Such nonlocal operators L depend on a finite measure on Sn-1, called the spectral measure.; First, we study the interior regularity of solutions to Lu = f in B-1. We prove that if f is C-alpha then u belong to C alpha+2s whenever alpha + 2s is not an integer. In case f is an element of L-infinity we show that the solution u is C-2s when s not equal 1/2, and C2s - is an element of for all epsilon > 0 when s =1/2.; Then, we study the boundary regularity of solutions to Lu = f in Omega, u = 0 in R-n \ Omega, in C-1,C-1 domains Omega We show that solutions u satisfy u/d(s) is an element of Cs-is an element of (Omega) for all epsilon > 0, where d is the distance to partial derivative Omega. Finally, we show that our results are sharp by constructing two counterexamples.
Llibre, J.; Ramirez Inostroza, Rafael; Ramirez, V.; Sadovskaia, N. Journal of differential equations Vol. 260, num. 7, p. 5726-5760 DOI: 10.1016/j.jde.2015.12.019 Data de publicació: 2016-04-05 Article en revista
We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
Garcia Naranjo, L.; Marrero, J.; E. Pérez-Chavela; Rodriguez, M. Journal of differential equations Vol. 260, num. 7, p. 6375-6404 DOI: 10.1016/j.jde.2015.12.044 Data de publicació: 2016-01-13 Article en revista
We classify and analyze the stability of all relative equilibria for the two-body problem in the hyperbolic space of dimension 2 and we formulate our results in terms of the intrinsic Riemannian data of the problem.
We consider the problem of characterizing, for certain natural
number m, the local C^m-non-integrability near
elliptic fixed points of smooth planar measure preserving maps. Our
criterion relates this non-integrability with the existence of some
Lie Symmetries associated to the maps, together with the study of
the finiteness of its periodic points. One of the steps in the proof
uses the regularity of the period function on the whole period
annulus for non-degenerate centers, question that we believe that is
interesting by itself. The obtained criterion can be applied to
prove the local non-integrability of the Cohen map and of several
rational maps coming from second order difference equations.
We prove a radial symmetry result for bounded nonnegative solutions to the p-Laplacian semilinear equation -¿pu=f(u) posed in a ball of Rn and involving discontinuous nonlinearities f. When p=2 we obtain a new result which holds in every dimension n for certain positive discontinuous f. When p¿n we prove radial symmetry for every locally bounded nonnegative f. Our approach is an extension of a method of P.L. Lions for the case p=n=2. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.
We prove a radial symmetry result for bounded nonnegative solutions to the p-Laplacian semilinear equation −Δpu=f(u) posed in a ball of Rn and involving discontinuous nonlinearities f. When p=2 we obtain a new result which holds in every dimension n for certain positive discontinuous f. When p⩾n we prove radial symmetry for every locally bounded nonnegative f. Our approach is an extension of a method of P.L. Lions for the case p=n=2. It leads to radial symmetry combining the isoperimetric inequality and the Pohozaev identity.
We consider the monomial weight |x1|A1¿|xn|An in Rn, where Ai=0 is a real number for each i=1, n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by |x1|A1¿|xn|Andx, and they contain the best or critical exponent (which depends on A1, An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions.When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD=RA1+1×¿×RAn+1.
We consider the monomial weight |x1|A1⋯|xn|An in Rn, where Ai⩾0 is a real number for each i=1,…,n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by View the MathML source, and they contain the best or critical exponent (which depends on A1,…,An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions.
When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD=RA1+1×⋯×RAn+1.
Baldoma, I.; Fontich, E.; Guardia, M.; Martinez-seara, Tere Journal of differential equations Vol. 253, num. 12, p. 3304-3439 DOI: 10.1016/j.jde.2012.09.003 Data de publicació: 2012-12-15 Article en revista
This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by
the corresponding invariant manifolds.
We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space
at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates
quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local.
We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are
small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing
hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich
et al. (2011) .
In this paper we consider a representative a priori unstable Hamiltonian
system with 2 + 1/2 degrees of freedom and we apply
the geometric mechanism for diffusion introduced in [A. Delshams,
R. de la Llave, T.M. Seara, A geometric mechanism for diffusion in
Hamiltonian systems overcoming the large gap problem: heuristics
and rigorous verification on a model, Mem. Amer. Math. Soc.
179 (844) (2006), viii + 141 pp.], and generalized in [A. Delshams,
G. Huguet, Geography of resonances and Arnold diffusion in a priori
unstable Hamiltonian systems, Nonlinearity 22 (8) (2009) 1997–
2077]. We provide explicit, concrete and easily verifiable conditions
for the existence of diffusing orbits.
The simplification of the hypotheses allows us to perform the
straightforward computations along the proof and present the geometric
mechanism of diffusion in an easily understandable way.
In particular, we fully describe the construction of the scattering
map and the combination of two types of dynamics on a normally
hyperbolic invariant manifold.
In this article some qualitative and geometric aspects of non-smooth dynamical systems theory are discussed. The main aim of
this article is to develop a systematic method for studying local(and global) bifurcations in non-smooth dynamical systems. Our results deal with the classification and characterization of generic codimension-2 singularities of planar Filippov Systems as well as the presentation of the bifurcation diagrams and some dynamical consequences
We give an upper bound for the maximum number N of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves. Additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic. We also provide lower bounds for N.
For a polynomial planar vector field of degree n ≥ 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 + (n − 1)(n − 2)/2 when n is even, and (n − 1)(n − 2)/2 when n is odd. Furthermore, these upper bounds are reached.
We consider a non-autonomous system of ordinary differential equations. Assume that the time dependence is periodic with a very high frequency 1/¿, where ¿ is a small parameter and differentiability with respect to the parameter is lost when ¿ equals zero. We derive from Arenstorf's implicit function theorem a set of conditions to show the existence of periodic solutions. These conditions look formally like the standard analytic continuation method, namely, checking that a certain minor does not vanish. We apply this result to show the existence of a new class of periodic orbits of very large radii in the three-dimensional elliptic restricted three-body problem for arbitrary values of the masses of the primaries.