Foundations of Computational Mathematics, Nov 17, 2016
In this paper we present and illustrate a general methodology to apply KAM theory in particular p... more In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to ε = 0.9716), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
Programa de doctorat de Matem atica Aplicada i An alisi. Bienni 92-94. Mem oria presentada pera a... more Programa de doctorat de Matem atica Aplicada i An alisi. Bienni 92-94. Mem oria presentada pera aspirar al grau de Doctor en Ci encies Matem atiques perla Universitat de Barcelona. Certi co que la present mem oria ha estat realitzada per n' Alex Haro i Provinciale, i dirigida per mi. Barcelona, 3 d e juliol de 1998. Carles Sim o i Torres A la Cristina ' 1:839286755: We shall refer to it as the golden vector. Its rst convergents are
The solution of the Grad–Shafranov (GS) equation, which is at the basis of Tokamak plasma MHD equ... more The solution of the Grad–Shafranov (GS) equation, which is at the basis of Tokamak plasma MHD equilibrium computation, in the fixed boundary formulation first requires the description of the plasma boundary in terms of a few physical quantities. Limiter plasmas are characterized by a boundary that is a closed magnetic surface tangent to a material surface for which there is a simple parametrization in terms of those quantities. Divertor plasmas are, by contrast, characterized by a boundary that is not a closed flux surface nor tangent to any material surface, but contains saddle point(s) at which the boundary has cusp(s) of 90◦. For such a boundary the limiter plasma parametrization is no longer valid. We investigate whether there is an analogous simple parametrization of divertor plasma boundaries.
Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) play... more Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.
In this paper we present an a-posteriori KAM theorem for the existence of an (n − d)-parameters f... more In this paper we present an a-posteriori KAM theorem for the existence of an (n − d)-parameters family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω ∈ R d , of type (γ, τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n − d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω, ωp), with ωp ∈ R n−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size ρ, and the corresponding error in the functional equation is ε. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ −2 ρ −2τ −1 ε is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ −4 ρ −4τ ε is small enough. The approach is suitable to perform computer assisted proofs. Contents 1. Introduction 2. The setting and the KAM theorem 2.1. Basic notation 2.2. Symplectic setting 2.3. Hamiltonian systems and first integrals in involution 2.4. Invariant tori 2.5. Linearized dynamics and reducibility 2.6. A (modified) quasi-Newton method 2.7. Analytic setting 2.8. The KAM theorem 3. Proof of the KAM theorem 3.1. Some lemmas to control approximate geometric properties 3.2. One step of the iterative procedure 3.3. Convergence of the iterative process 4. Acknowledgements References Appendix A. An auxiliary lemma to control the inverse of a matrix Appendix B. Compendium of constants involved in the KAM theorem
Chapter 8. Bifurcation theory for KAM tori 8.1. Classification of KAM invariant tori 8.2. Local e... more Chapter 8. Bifurcation theory for KAM tori 8.1. Classification of KAM invariant tori 8.2. Local equivalence of Bifurcations diagrams Chapter 9. The close-to-integrable case 9.1. The integrable case 9.2. Persistence of invariant tori 9.3. Unfolding non-twist tori 9.4. The Birkhoff potential and the potential of an invariant torus Appendix A. Hamiltonian vector fields A.1. One-bite small divisor equations A.2. Automatic reducibility of invariant tori A.3. Families of Hamiltonians and moment map A.4. Potential and moment of an invariant FLD A.5. Transformed Tori Theorem A.6. A parametric KAM Theorem
In this paper we present and illustrate a general methodology to apply KAM theory in particular p... more In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to ε = 0.9716), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satis... more A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a-posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a-posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations.
This paper presents a methodology to study non-twist invariant circles and their bifurcations for... more This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps. We recall that nontwist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g. against external noise) and hence, they appear as design goals in some applications. Our methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous aposteriori theorems which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer assisted proofs. Our algorithms are also guaranteed to converge up to the breakdown of the invariant circles, then they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, they do not require symmetries and can run without continuous manual adjustments. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include estimates for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. These numerical explorations lead to some new mathematical conjectures.
In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invari... more In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n − d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.
A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satis... more A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a-posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a-posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations.
Communications in Nonlinear Science and Numerical Simulation
Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) play... more Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.
Discrete and Continuous Dynamical Systems - Series B, 2006
In this paper we develop several numerical algorithms for the computation of invariant manifolds ... more In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori. The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34]. The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties. The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.
Communications in Nonlinear Science and Numerical Simulation
The goal of this paper is to present a methodology for the computation of invariant tori in Hamil... more The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincaré maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant tori and their invariant bundles around librational equilibrium points of the Restricted Three Body Problem. The invariant bundles are the first order approximations of the corresponding invariant manifolds, commonly known as the whiskers, which are very important in the dynamical organization and have important applications in space mission design.
ABSTRACT The area that has probably received most applications of qualitative dynamical systems i... more ABSTRACT The area that has probably received most applications of qualitative dynamical systems is the study of business cycles and economic fluctuations. The majority of applications have employed the standard theory of bifurcations which deals with dynamical systems defined on open sets. In particular, this implies that all relevant trajectories, including steady states, must be contained in the interior of the state space. In the majority of cases the resulting planar economies explore the asymptotic behavior of an iterative process and provide a comprehensive description of the geometric structures arising from the evolution of the system. In general these applications just consider the evolution of the associated dynamical system near the steady state(s) characterizing the resulting invariant manifolds just locally. Typically the resulting dynamical systems are not able to generate eigenvalues in the unit circle for all the relevant parameter values. When center manifolds are present, some eigenvalues have moduli 1, the study of global dynamics is a bit more challenging and new numerical considerations are required. In particular, the center manifold theorem allows the dynamics to be projected onto the center manifold without loosing any significant aspect of the dynamics. Thus, dynamics near a stationary co-dimension one bifurcation can be described by an effective dynamics in a one-dimensional subspace. The dynamics projected onto the center manifold can be represented by the so called normal forms through a nonlinear transformation of the phase space variables. The basic goal of this paper is to apply computational techniques from the dynamical system literature in order to study the asymptotic behavior of an iterative process that has a center manifold, while providing a comprehensive description of the geometric structures arising from the evolution of an economy where private and public circulating liabilities coexist. In particular, we consider the model by Azar
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015
We present a computer-assisted numerical study of different bifurcations of saddle invariant tori... more We present a computer-assisted numerical study of different bifurcations of saddle invariant tori in quasiperiodic area preserving twist maps. We detect three bifurcation scenarios. In the first scenario, the smooth bifurcation, the regularity of the invariant torus is preserved, and the stable and unstable invariant bundles collide smoothly. In the other scenarios, the spiky and folding breakdowns, the invariant torus looses smoothness, and the invariant bundles collide non-smoothly. In the former, the C(1) seminorm of the torus does not blow up while in the latter it does. Numerics suggest that after the breakdowns non-uniformly hyperbolic invariant objects persist. These are qualitatively different depending on the type of breakdown. Finally, using anti-integrable limit theory, we provide a proof of existence of non-uniformly hyperbolic invariant objects for systems very far from the integrable regime.
Foundations of Computational Mathematics, Nov 17, 2016
In this paper we present and illustrate a general methodology to apply KAM theory in particular p... more In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to ε = 0.9716), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
Programa de doctorat de Matem atica Aplicada i An alisi. Bienni 92-94. Mem oria presentada pera a... more Programa de doctorat de Matem atica Aplicada i An alisi. Bienni 92-94. Mem oria presentada pera aspirar al grau de Doctor en Ci encies Matem atiques perla Universitat de Barcelona. Certi co que la present mem oria ha estat realitzada per n' Alex Haro i Provinciale, i dirigida per mi. Barcelona, 3 d e juliol de 1998. Carles Sim o i Torres A la Cristina ' 1:839286755: We shall refer to it as the golden vector. Its rst convergents are
The solution of the Grad–Shafranov (GS) equation, which is at the basis of Tokamak plasma MHD equ... more The solution of the Grad–Shafranov (GS) equation, which is at the basis of Tokamak plasma MHD equilibrium computation, in the fixed boundary formulation first requires the description of the plasma boundary in terms of a few physical quantities. Limiter plasmas are characterized by a boundary that is a closed magnetic surface tangent to a material surface for which there is a simple parametrization in terms of those quantities. Divertor plasmas are, by contrast, characterized by a boundary that is not a closed flux surface nor tangent to any material surface, but contains saddle point(s) at which the boundary has cusp(s) of 90◦. For such a boundary the limiter plasma parametrization is no longer valid. We investigate whether there is an analogous simple parametrization of divertor plasma boundaries.
Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) play... more Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.
In this paper we present an a-posteriori KAM theorem for the existence of an (n − d)-parameters f... more In this paper we present an a-posteriori KAM theorem for the existence of an (n − d)-parameters family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω ∈ R d , of type (γ, τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n − d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω, ωp), with ωp ∈ R n−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of size ρ, and the corresponding error in the functional equation is ε. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ −2 ρ −2τ −1 ε is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ −4 ρ −4τ ε is small enough. The approach is suitable to perform computer assisted proofs. Contents 1. Introduction 2. The setting and the KAM theorem 2.1. Basic notation 2.2. Symplectic setting 2.3. Hamiltonian systems and first integrals in involution 2.4. Invariant tori 2.5. Linearized dynamics and reducibility 2.6. A (modified) quasi-Newton method 2.7. Analytic setting 2.8. The KAM theorem 3. Proof of the KAM theorem 3.1. Some lemmas to control approximate geometric properties 3.2. One step of the iterative procedure 3.3. Convergence of the iterative process 4. Acknowledgements References Appendix A. An auxiliary lemma to control the inverse of a matrix Appendix B. Compendium of constants involved in the KAM theorem
Chapter 8. Bifurcation theory for KAM tori 8.1. Classification of KAM invariant tori 8.2. Local e... more Chapter 8. Bifurcation theory for KAM tori 8.1. Classification of KAM invariant tori 8.2. Local equivalence of Bifurcations diagrams Chapter 9. The close-to-integrable case 9.1. The integrable case 9.2. Persistence of invariant tori 9.3. Unfolding non-twist tori 9.4. The Birkhoff potential and the potential of an invariant torus Appendix A. Hamiltonian vector fields A.1. One-bite small divisor equations A.2. Automatic reducibility of invariant tori A.3. Families of Hamiltonians and moment map A.4. Potential and moment of an invariant FLD A.5. Transformed Tori Theorem A.6. A parametric KAM Theorem
In this paper we present and illustrate a general methodology to apply KAM theory in particular p... more In this paper we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real-analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to ε = 0.9716), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.
A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satis... more A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a-posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a-posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations.
This paper presents a methodology to study non-twist invariant circles and their bifurcations for... more This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps. We recall that nontwist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g. against external noise) and hence, they appear as design goals in some applications. Our methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous aposteriori theorems which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer assisted proofs. Our algorithms are also guaranteed to converge up to the breakdown of the invariant circles, then they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, they do not require symmetries and can run without continuous manual adjustments. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include estimates for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. These numerical explorations lead to some new mathematical conjectures.
In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invari... more In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n − d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.
A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satis... more A fundamental question in Dynamical Systems is to identify regions of phase/parameter space satisfying a given property (stability, linearization, etc). Given a family of analytic circle diffeomorphisms depending on a parameter, we obtain effective (almost optimal) lower bounds of the Lebesgue measure of the set of parameters that are conjugated to a rigid rotation. We estimate this measure using an a-posteriori KAM scheme that relies on quantitative conditions that are checkable using computer-assistance. We carefully describe how the hypotheses in our theorems are reduced to a finite number of computations, and apply our methodology to the case of the Arnold family. Hence we show that obtaining non-asymptotic lower bounds for the applicability of KAM theorems is a feasible task provided one has an a-posteriori theorem to characterize the problem. Finally, as a direct corollary, we produce explicit asymptotic estimates in the so called local reduction setting (à la Arnold) which are valid for a global set of rotations.
Communications in Nonlinear Science and Numerical Simulation
Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) play... more Estimating the norm of the solution of the linear difference equation u(θ) − u(θ + ω) = v(θ) plays a fundamental role in KAM theory. Optimal (in certain sense) estimates for the solution of this equation were provided by Rüssmann in the mid 70's. The aim of this paper is to compare the sharpness of these classic estimates with more specific estimates obtained with the help of the computer. We perform several experiments to quantify the improvement obtained when using computer assisted estimates. By comparing these estimates with the actual norm of the solution, we can analyze the different sources of overestimation, thus encouraging future improvements.
Discrete and Continuous Dynamical Systems - Series B, 2006
In this paper we develop several numerical algorithms for the computation of invariant manifolds ... more In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori. The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34]. The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties. The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.
Communications in Nonlinear Science and Numerical Simulation
The goal of this paper is to present a methodology for the computation of invariant tori in Hamil... more The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincaré maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant tori and their invariant bundles around librational equilibrium points of the Restricted Three Body Problem. The invariant bundles are the first order approximations of the corresponding invariant manifolds, commonly known as the whiskers, which are very important in the dynamical organization and have important applications in space mission design.
ABSTRACT The area that has probably received most applications of qualitative dynamical systems i... more ABSTRACT The area that has probably received most applications of qualitative dynamical systems is the study of business cycles and economic fluctuations. The majority of applications have employed the standard theory of bifurcations which deals with dynamical systems defined on open sets. In particular, this implies that all relevant trajectories, including steady states, must be contained in the interior of the state space. In the majority of cases the resulting planar economies explore the asymptotic behavior of an iterative process and provide a comprehensive description of the geometric structures arising from the evolution of the system. In general these applications just consider the evolution of the associated dynamical system near the steady state(s) characterizing the resulting invariant manifolds just locally. Typically the resulting dynamical systems are not able to generate eigenvalues in the unit circle for all the relevant parameter values. When center manifolds are present, some eigenvalues have moduli 1, the study of global dynamics is a bit more challenging and new numerical considerations are required. In particular, the center manifold theorem allows the dynamics to be projected onto the center manifold without loosing any significant aspect of the dynamics. Thus, dynamics near a stationary co-dimension one bifurcation can be described by an effective dynamics in a one-dimensional subspace. The dynamics projected onto the center manifold can be represented by the so called normal forms through a nonlinear transformation of the phase space variables. The basic goal of this paper is to apply computational techniques from the dynamical system literature in order to study the asymptotic behavior of an iterative process that has a center manifold, while providing a comprehensive description of the geometric structures arising from the evolution of an economy where private and public circulating liabilities coexist. In particular, we consider the model by Azar
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015
We present a computer-assisted numerical study of different bifurcations of saddle invariant tori... more We present a computer-assisted numerical study of different bifurcations of saddle invariant tori in quasiperiodic area preserving twist maps. We detect three bifurcation scenarios. In the first scenario, the smooth bifurcation, the regularity of the invariant torus is preserved, and the stable and unstable invariant bundles collide smoothly. In the other scenarios, the spiky and folding breakdowns, the invariant torus looses smoothness, and the invariant bundles collide non-smoothly. In the former, the C(1) seminorm of the torus does not blow up while in the latter it does. Numerics suggest that after the breakdowns non-uniformly hyperbolic invariant objects persist. These are qualitatively different depending on the type of breakdown. Finally, using anti-integrable limit theory, we provide a proof of existence of non-uniformly hyperbolic invariant objects for systems very far from the integrable regime.
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