EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions
http://hdl.handle.net/2117/3227
Fri, 15 Jan 2021 21:23:41 GMT2021-01-15T21:23:41ZExponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
http://hdl.handle.net/2117/331406
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.
Wed, 04 Nov 2020 16:48:56 GMThttp://hdl.handle.net/2117/3314062020-11-04T16:48:56ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereWe study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.The dynamics of digits: calculating Pi with Galperin's billiards
http://hdl.handle.net/2117/330414
The dynamics of digits: calculating Pi with Galperin's billiards
Aretxabaleta, Xabier M.; Gonchenko, Marina; Harshman, Nathan L.; Jackson, Steven Glenn; Olshanii, Maxim; Astrakharchik, Grigori
In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number p . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of p in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be p itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of p with Galperin billiards, including curious cases with irrational number bases.
Mon, 19 Oct 2020 11:42:39 GMThttp://hdl.handle.net/2117/3304142020-10-19T11:42:39ZAretxabaleta, Xabier M.Gonchenko, MarinaHarshman, Nathan L.Jackson, Steven GlennOlshanii, MaximAstrakharchik, GrigoriIn Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number p . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of p in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be p itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of p with Galperin billiards, including curious cases with irrational number bases.Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
http://hdl.handle.net/2117/135375
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequencies
Wed, 26 Jun 2019 08:42:03 GMThttp://hdl.handle.net/2117/1353752019-06-26T08:42:03ZDelshams Valdés, AmadeuGonchenko, MarinaGutiérrez Serrés, PereWe study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors<k, ¿>,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequenciesOn local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps
http://hdl.handle.net/2117/129971
On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps
Gonchenko, Marina; Gonchenko, Sergey; Ovsyannikov, Ivan; Vieiro, Arturo
We study the 1:4 resonance for the conservative cubic Hénon maps C^± with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues¿±i and for 4-periodic orbits. While for C^–, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map C^+ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by \pi/4. For both maps, several bifurcations are detected and illustrated.
Fri, 01 Mar 2019 08:23:12 GMThttp://hdl.handle.net/2117/1299712019-03-01T08:23:12ZGonchenko, MarinaGonchenko, SergeyOvsyannikov, IvanVieiro, ArturoWe study the 1:4 resonance for the conservative cubic Hénon maps C^± with positive and negative cubic terms. These maps show up different bifurcation structures both for fixed points with eigenvalues¿±i and for 4-periodic orbits. While for C^–, the 1:4 resonance unfolding has the so-called Arnold degeneracy [the first Birkhoff twist coefficient equals (in absolute value) to the first resonant term coefficient], the map C^+ has a different type of degeneracy because the resonant term can vanish. In the last case, non-symmetric points are created and destroyed at pitchfork bifurcations and, as a result of global bifurcations, the 1:4 resonant chain of islands rotates by \pi/4. For both maps, several bifurcations are detected and illustrated.A numerical method for computing initial conditions of Lagrangian invariant tori using the frequency map
http://hdl.handle.net/2117/89912
A numerical method for computing initial conditions of Lagrangian invariant tori using the frequency map
Villanueva Castelltort, Jordi; Luque Jiménez, Alejandro
We present a numerical method for computing initial conditions of Lagrangian quasi-periodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi-periodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L-4 in a RTBP. (C) 2016 Elsevier B.V. All rights reserved.
Wed, 14 Sep 2016 11:28:13 GMThttp://hdl.handle.net/2117/899122016-09-14T11:28:13ZVillanueva Castelltort, JordiLuque Jiménez, AlejandroWe present a numerical method for computing initial conditions of Lagrangian quasi-periodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi-periodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L-4 in a RTBP. (C) 2016 Elsevier B.V. All rights reserved.Traveling wave solutions in a half-space for boundary reactions
http://hdl.handle.net/2117/77011
Traveling wave solutions in a half-space for boundary reactions
Cabré Vilagut, Xavier; Consul Porras, M. Nieves; Mande Nieto, José Vicente
We prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of unbalanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveling fronts. Our existence results are obtained via a variational method, while the uniqueness of the speed and of the front rely on a comparison principle and the sliding method.
Tue, 22 Sep 2015 11:21:36 GMThttp://hdl.handle.net/2117/770112015-09-22T11:21:36ZCabré Vilagut, XavierConsul Porras, M. NievesMande Nieto, José VicenteWe prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of unbalanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveling fronts. Our existence results are obtained via a variational method, while the uniqueness of the speed and of the front rely on a comparison principle and the sliding method.A note on symplectic and Poisson linearization of semisimple Lie algebra actions
http://hdl.handle.net/2117/26960
A note on symplectic and Poisson linearization of semisimple Lie algebra actions
Miranda Galcerán, Eva
In this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson setting
Mon, 23 Mar 2015 16:46:48 GMThttp://hdl.handle.net/2117/269602015-03-23T16:46:48ZMiranda Galcerán, EvaIn this note we prove that an analytic symplectic action of a semisimple Lie algebra can be locally linearized in Darboux coordinates. This result yields simultaneous analytic linearization for Hamiltonian vector fields in a neighbourhood of a common zero. We also provide an example of smooth non-linearizable Hamiltonian action with semisimple linear part. The smooth analogue only holds if the semisimple Lie algebra is of compact type. An analytic equivariant b-Darboux theorem for b-Poisson manifolds and an analytic equivariant Weinstein splitting theorem for general Poisson manifolds are also obtained in the Poisson settingGeneric bifurcations of low codimension of planar Filippov Systems
http://hdl.handle.net/2117/26671
Generic bifurcations of low codimension of planar Filippov Systems
Martínez-Seara Alonso, M. Teresa; Guàrdia Munarriz, Marcel; Teixeira, Marco Antonio
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
Thu, 12 Mar 2015 10:45:38 GMThttp://hdl.handle.net/2117/266712015-03-12T10:45:38ZMartínez-Seara Alonso, M. TeresaGuàrdia Munarriz, MarcelTeixeira, Marco AntonioIn 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 consequencesA distributed attitude control law for formation flying based on the Cucker-smale model
http://hdl.handle.net/2117/26670
A distributed attitude control law for formation flying based on the Cucker-smale model
Paita, Fabrizio; Gómez Muntané, Gerard; Masdemont Soler, Josep
In this paper we consider the attitude synchronization problem for a swarm of spacecrafts flying in formation.
Starting from previous works on consensus dynamics, we construct a distributed attitude control law and derive analytically
sufficient conditions for the formation to converge asymptotically towards a synchronized, non–accelerating state (possibly defined a priori). Moreover, motivated by the results obtained on a particular consensus model, first introduced by F. Cucker and S. Smale to modellize the translational dynamics of flocks, we numerically explore
the dependence of the convergence process on the dimension of the formation and the relative initial conditions of the spacecrafts. Finally, we generalize the class of weights defined by the previous authors in order to dampen the aforementioned effects, thus making our control law suitable for very large formations.
Thu, 12 Mar 2015 09:06:25 GMThttp://hdl.handle.net/2117/266702015-03-12T09:06:25ZPaita, FabrizioGómez Muntané, GerardMasdemont Soler, JosepIn this paper we consider the attitude synchronization problem for a swarm of spacecrafts flying in formation.
Starting from previous works on consensus dynamics, we construct a distributed attitude control law and derive analytically
sufficient conditions for the formation to converge asymptotically towards a synchronized, non–accelerating state (possibly defined a priori). Moreover, motivated by the results obtained on a particular consensus model, first introduced by F. Cucker and S. Smale to modellize the translational dynamics of flocks, we numerically explore
the dependence of the convergence process on the dimension of the formation and the relative initial conditions of the spacecrafts. Finally, we generalize the class of weights defined by the previous authors in order to dampen the aforementioned effects, thus making our control law suitable for very large formations.Matrices positivas y aplicaciones
http://hdl.handle.net/2117/26622
Matrices positivas y aplicaciones
García Planas, María Isabel
Mon, 09 Mar 2015 12:14:12 GMThttp://hdl.handle.net/2117/266222015-03-09T12:14:12ZGarcía Planas, María Isabel