Rafael de la Llave - Academia.edu (original) (raw)

Papers by Rafael de la Llave

Research paper thumbnail of Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

We study the problem of instability in the following a priori unstable Hamiltonian system with a ... more We study the problem of instability in the following a priori unstable Hamiltonian system with a time-periodic perturbation Hε(p, q, I ,φ, t ) = h(I )+ n ∑ i=1 ± ( 1 2 p2 i +Vi (qi ) ) +εH1(p, q, I ,φ, t ), where (p, q) ∈ Rn ×Tn , (I ,φ) ∈ Rd ×Td with n,d ≥ 1, Vi are Morse potentials, and ε is a small nonzero parameter. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H1. Indeed, the set of admissible H1 is C ω dense and C 3 open. The proof also works for arbitrarily small Vi . Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3,∞)∪ {∞,ω}.

Research paper thumbnail of Invariant manifolds for analytic difference equations

We use a modification of the parameterization method to study invariant manifolds for difference ... more We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider include not only the classical strong stable and unstable manifolds but also manifolds associated to non-resonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational we present sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations can not be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We also present extensions to finite differentiable difference equations.

Research paper thumbnail of Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

Journal of Nonlinear Science

We consider a mechanical system consisting of n penduli and a d-dimensional generalized rotator s... more We consider a mechanical system consisting of n penduli and a d-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic, we allow for rather general time-dependence. The strength of the perturbation is given by a parameter ε P R. For all |ε| sufficiently small, the augmented flow-obtained by making the time into a new variable-has a p2d`1q-dimensional normally hyperbolic locally invariant manifoldΛε. We define a Melnikov-type vector, which gives the first order expansion of the displacement of the stable and unstable manifolds ofΛ0 under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds ofΛε, W s pΛεq and W u pΛεq, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of W s pΛεq and W u pΛεq can be explicitly computed, up to the first order, in terms of the Melnikov-type vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order Opεq in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general-type orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the nondegeneracy conditions on the Melnikov potential are generic.

Research paper thumbnail of Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps

Discrete and Continuous Dynamical Systems, 2012

We prove the existence of certain analytic invariant manifolds associated with fixed points of an... more We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.

Research paper thumbnail of Perturbation and numerical methods for computing the minimal average energy

We investigate the differentiability of minimal average energy associated to the functionals S ε ... more We investigate the differentiability of minimal average energy associated to the functionals S ε (u) = R d 1 2 |∇u| 2 + εV(x, u) dx, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter ε, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.

Research paper thumbnail of Percival Lagrangian approach to the Aubry–Mather theory

We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence... more We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions We also compare the proofs and results with those obtained in other formalisms.

Research paper thumbnail of On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

Discrete & Continuous Dynamical Systems - A

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [Fed75, S... more We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [Fed75, Section 5.11] to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in S 1 × S 2 , with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in T 3 for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of "integrable" and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

Research paper thumbnail of A rigorous partial justification of Greene's criterion

Journal of Statistical Physics, 1992

We prove several theorems that lend support to Greene's criterion for the existence or not of inv... more We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications.

Research paper thumbnail of Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors

Journal of Statistical Physics, 1992

We study numerically the complex domains of validity for KAM theory in generalized standard mappi... more We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits.

Research paper thumbnail of Cohomology Equations and Commutators of Germs of Contact Diffeomorphisms

Transactions of The American Mathematical Society, 1989

Research paper thumbnail of On Irwin's Proof Of The Pseudostable Manifold Theorem

. We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This prepri... more . We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This preprint is available from the math-physics electronic preprints archive. Send e-mail to mp arc@math.utexas.edufor instructions2Supported in part by National Science Foundation Grants3e-mail address: llave@math.utexas.edu4e-mail address: wayne@math.psu.edu- 2 -1. IntroductionIn [Ir1], Irwin introduced a very clever method to prove the stable manifold theorem near hyperbolicpoints. The proof ...

Research paper thumbnail of On Irwin’s proof of the pseudostable manifold theorem

Mathematische Zeitschrift, 1995

. We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This prepri... more . We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This preprint is available from the math-physics electronic preprints archive. Send e-mail to mp arc@math.utexas.edufor instructions2Supported in part by National Science Foundation Grants3e-mail address: llave@math.utexas.edu4e-mail address: wayne@math.psu.edu- 2 -1. IntroductionIn [Ir1], Irwin introduced a very clever method to prove the stable manifold theorem near hyperbolicpoints. The proof ...

Research paper thumbnail of Planelike minimizers in periodic media

Communications on Pure and Applied Mathematics, 2001

We show that given an elliptic integrand J in R d which is periodic under integer translations, g... more We show that given an elliptic integrand J in R d which is periodic under integer translations, given any plane in R d , there is at least one minimizer of J which remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to R d /Z d the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations.

Research paper thumbnail of Interfaces of Ground States in Ising Models with Periodic Coefficients

Journal of Statistical Physics, 2005

We study the interfaces of ground states of ferromagnetic Ising models with external fields. We s... more We study the interfaces of ground states of ferromagnetic Ising models with external fields. We show that, if the coefficients of the interaction and the magnetic field are periodic, the magnetic field has zero flux over a period and is small enough, then for every plane, we can find a ground state whose interface lies at a bounded distance of the plane. This bound on the width of the interface can be chosen independent of the plane. We also study the average energy of the plane-like interfaces as a function of the direction. We show that there is a well defined thermodynamic limit for the energy of the interface and that it enjoys several convexity properties.

Research paper thumbnail of A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

Journal of Dynamics and Differential Equations, 2009

We prove the existence of a smooth center manifold for several partial differential equations, in... more We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.

Research paper thumbnail of Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

We study the problem of instability in the following a priori unstable Hamiltonian system with a ... more We study the problem of instability in the following a priori unstable Hamiltonian system with a time-periodic perturbation Hε(p, q, I ,φ, t ) = h(I )+ n ∑ i=1 ± ( 1 2 p2 i +Vi (qi ) ) +εH1(p, q, I ,φ, t ), where (p, q) ∈ Rn ×Tn , (I ,φ) ∈ Rd ×Td with n,d ≥ 1, Vi are Morse potentials, and ε is a small nonzero parameter. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H1. Indeed, the set of admissible H1 is C ω dense and C 3 open. The proof also works for arbitrarily small Vi . Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3,∞)∪ {∞,ω}.

Research paper thumbnail of Invariant manifolds for analytic difference equations

We use a modification of the parameterization method to study invariant manifolds for difference ... more We use a modification of the parameterization method to study invariant manifolds for difference equations. We establish existence, regularity, smooth dependence on parameters and study several singular limits, even if the difference equations do not define a dynamical system. This method also leads to efficient algorithms that we present with their implementations. The manifolds we consider include not only the classical strong stable and unstable manifolds but also manifolds associated to non-resonant spaces. When the difference equations are the Euler-Lagrange equations of a discrete variational we present sharper results. Note that, if the Legendre condition fails, the Euler-Lagrange equations can not be treated as a dynamical system. If the Legendre condition becomes singular, the dynamical system may be singular while the difference equation remains regular. We present numerical applications to several examples in the physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We also present extensions to finite differentiable difference equations.

Research paper thumbnail of Global Melnikov Theory in Hamiltonian Systems with General Time-Dependent Perturbations

Journal of Nonlinear Science

We consider a mechanical system consisting of n penduli and a d-dimensional generalized rotator s... more We consider a mechanical system consisting of n penduli and a d-dimensional generalized rotator subject to a time-dependent perturbation. The perturbation is not assumed to be either Hamiltonian, or periodic or quasi-periodic, we allow for rather general time-dependence. The strength of the perturbation is given by a parameter ε P R. For all |ε| sufficiently small, the augmented flow-obtained by making the time into a new variable-has a p2d`1q-dimensional normally hyperbolic locally invariant manifoldΛε. We define a Melnikov-type vector, which gives the first order expansion of the displacement of the stable and unstable manifolds ofΛ0 under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds ofΛε, W s pΛεq and W u pΛεq, respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of W s pΛεq and W u pΛεq can be explicitly computed, up to the first order, in terms of the Melnikov-type vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order Opεq in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions and give, at the same time, the effects on periodic, quasi-periodic, or general-type orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the nondegeneracy conditions on the Melnikov potential are generic.

Research paper thumbnail of Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps

Discrete and Continuous Dynamical Systems, 2012

We prove the existence of certain analytic invariant manifolds associated with fixed points of an... more We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.

Research paper thumbnail of Perturbation and numerical methods for computing the minimal average energy

We investigate the differentiability of minimal average energy associated to the functionals S ε ... more We investigate the differentiability of minimal average energy associated to the functionals S ε (u) = R d 1 2 |∇u| 2 + εV(x, u) dx, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter ε, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.

Research paper thumbnail of Percival Lagrangian approach to the Aubry–Mather theory

We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence... more We present some streamlined proofs of some of the basic results in Aubry-Mather theory (existence of quasi-periodic minimizers, multiplicity results when there are gaps among minimizers) based on the study of hull functions. We present results in arbitrary number of dimensions We also compare the proofs and results with those obtained in other formalisms.

Research paper thumbnail of On a remarkable example of F. Almgren and H. Federer in the global theory of minimizing geodesics

Discrete & Continuous Dynamical Systems - A

We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [Fed75, S... more We present an exposition of a remarkable example attributed to Frederick Almgren Jr. in [Fed75, Section 5.11] to illustrate the need of certain definitions in the calculus of variations. The Almgren-Federer example, besides its intended goal of illustrating subtle aspects of geometric measure theory, is also a problem in the theory of geodesics. Hence, we wrote an exposition of the beautiful ideas of Almgren and Federer from the point of view of geodesics. In the language of geodesics, Almgren-Federer example constructs metrics in S 1 × S 2 , with the property that none of the Tonelli geodesics (geodesics which minimize the length in a homotopy class) are Class-A minimizers in the sense of Morse (any finite length segment in the universal cover minimizes the length between the end points; this is also sometimes given other names). In other words, even if a curve is a minimizer of length among all the curves homotopic to it, by repeating it enough times, we get a closed curve which does not minimize in its homotopy class. In that respect, the example is more dramatic than a better known example due to Hedlund of a metric in T 3 for which only 3 Tonelli minimizers (and their multiples) are Class-A minimizers. For dynamics, the example also illustrates different definitions of "integrable" and clarifies the relation between minimization and hyperbolicity and its interaction with topology.

Research paper thumbnail of A rigorous partial justification of Greene's criterion

Journal of Statistical Physics, 1992

We prove several theorems that lend support to Greene's criterion for the existence or not of inv... more We prove several theorems that lend support to Greene's criterion for the existence or not of invariant circles in twist maps. In particular, we show that some of the implications of the criterion are correct when the Aubry-Mather sets are smooth invariant circles or uniformly hyperbolic. We also suggest a simple modification that can work in the case that the Aubry-Mather sets have nonzero Lyapunov exponents. The latter is based on a closing lemma for sets with nonzero Lyapunov exponents, which may have several other applications.

Research paper thumbnail of Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors

Journal of Statistical Physics, 1992

We study numerically the complex domains of validity for KAM theory in generalized standard mappi... more We study numerically the complex domains of validity for KAM theory in generalized standard mappings. We compare methods based on Padé approximants and methods based on the study of periodic orbits.

Research paper thumbnail of Cohomology Equations and Commutators of Germs of Contact Diffeomorphisms

Transactions of The American Mathematical Society, 1989

Research paper thumbnail of On Irwin's Proof Of The Pseudostable Manifold Theorem

. We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This prepri... more . We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This preprint is available from the math-physics electronic preprints archive. Send e-mail to mp arc@math.utexas.edufor instructions2Supported in part by National Science Foundation Grants3e-mail address: llave@math.utexas.edu4e-mail address: wayne@math.psu.edu- 2 -1. IntroductionIn [Ir1], Irwin introduced a very clever method to prove the stable manifold theorem near hyperbolicpoints. The proof ...

Research paper thumbnail of On Irwin’s proof of the pseudostable manifold theorem

Mathematische Zeitschrift, 1995

. We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This prepri... more . We simplify and extend Irwin's proof of the pseudostable manifold theorem.1This preprint is available from the math-physics electronic preprints archive. Send e-mail to mp arc@math.utexas.edufor instructions2Supported in part by National Science Foundation Grants3e-mail address: llave@math.utexas.edu4e-mail address: wayne@math.psu.edu- 2 -1. IntroductionIn [Ir1], Irwin introduced a very clever method to prove the stable manifold theorem near hyperbolicpoints. The proof ...

Research paper thumbnail of Planelike minimizers in periodic media

Communications on Pure and Applied Mathematics, 2001

We show that given an elliptic integrand J in R d which is periodic under integer translations, g... more We show that given an elliptic integrand J in R d which is periodic under integer translations, given any plane in R d , there is at least one minimizer of J which remains at a bounded distance from this plane. This distance can be bounded uniformly on the planes. We also show that, when folded back to R d /Z d the minimizers we construct give rise to a lamination. One particular case of these results is minimal surfaces for metrics invariant under integer translations.

Research paper thumbnail of Interfaces of Ground States in Ising Models with Periodic Coefficients

Journal of Statistical Physics, 2005

We study the interfaces of ground states of ferromagnetic Ising models with external fields. We s... more We study the interfaces of ground states of ferromagnetic Ising models with external fields. We show that, if the coefficients of the interaction and the magnetic field are periodic, the magnetic field has zero flux over a period and is small enough, then for every plane, we can find a ground state whose interface lies at a bounded distance of the plane. This bound on the width of the interface can be chosen independent of the plane. We also study the average energy of the plane-like interfaces as a function of the direction. We show that there is a well defined thermodynamic limit for the energy of the interface and that it enjoys several convexity properties.

Research paper thumbnail of A Smooth Center Manifold Theorem which Applies to Some Ill-Posed Partial Differential Equations with Unbounded Nonlinearities

Journal of Dynamics and Differential Equations, 2009

We prove the existence of a smooth center manifold for several partial differential equations, in... more We prove the existence of a smooth center manifold for several partial differential equations, including ill posed equations with unbounded nonlinearities. We also prove smooth dependence on parameters with respect to some perturbations, including unbounded ones. More concretely, we prove an abstract theorem and present applications to several concrete equations: ill posed Boussinesq, equation and system and nonlinear Laplace equations in cylindrical domains. We also consider the effect of some geometric structures.