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http://hdl.handle.net/2117/127
Fri, 01 Aug 2014 08:30:36 GMT2014-08-01T08:30:36Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoNonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
http://hdl.handle.net/2117/22391
Title: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Authors: Cabré Vilagut, Xavier; Sire, Yannick
Abstract: This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.http://hdl.handle.net/2117/22391Cabré Vilagut, Xavier; Sire, YannicknoThis is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.Estructuras A-infinito en la opérada de cactus
http://hdl.handle.net/2117/22097
Title: Estructuras A-infinito en la opérada de cactus
Authors: Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, Andrew
Abstract: Diversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterhttp://hdl.handle.net/2117/22097Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, AndrewnoDiversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterJ2 effect and elliptic inclined periodic orbits in the collision three-body problem
http://hdl.handle.net/2117/21117
Title: J2 effect and elliptic inclined periodic orbits in the collision three-body problem
Authors: Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaume
Abstract: The existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.http://hdl.handle.net/2117/21117Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaumenocollision restricted three-body problem, periodic orbits, symmetric orbits, critical inclination, continuation methodThe existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.Central cohomology operations and K-theory
http://hdl.handle.net/2117/23645
Title: Central cohomology operations and K-theory
Authors: Gálvez Carrillo, Maria Immaculada; Whitehouse, Sarah
Abstract: For stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP<n> for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.Tue, 29 Jul 2014 08:35:40 GMThttp://hdl.handle.net/2117/236452014-07-29T08:35:40ZGálvez Carrillo, Maria Immaculada; Whitehouse, SarahnoK-theory, Operations, CobordismFor stable degree 0 operations, and also for additive unstable operations of bidegree (0, 0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP<n> for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory.Observation of the internal states of a PEMFC anode gas channel
http://hdl.handle.net/2117/23623
Title: Observation of the internal states of a PEMFC anode gas channel
Authors: Luna Pacho, Julio Alberto; Batlle Arnau, Carles; Kunusch, Cristian; Riera Colomer, Jordi; Sarmiento Carnevali, María Laura; Serra, Maria
Abstract: In this work a non-linear, model-based state observer is developed. Dynamics of the system described by a distributed parameter, partial differential equations model. Forward and backward discretization of the model to take advantage of the boundary conditions. Simulation results are presented to show the performance of the observer. Comparison between the linear and nonlinear approaches of the observers.Fri, 25 Jul 2014 11:46:12 GMThttp://hdl.handle.net/2117/236232014-07-25T11:46:12ZLuna Pacho, Julio Alberto; Batlle Arnau, Carles; Kunusch, Cristian; Riera Colomer, Jordi; Sarmiento Carnevali, María Laura; Serra, Marianocontrol nonlinearities, observability
PEMFC, distrubuted model, nonlinear observers, super-twisting, sliding modeIn this work a non-linear, model-based state observer is developed. Dynamics of the system described by a distributed parameter, partial differential equations model. Forward and backward discretization of the model to take advantage of the boundary conditions. Simulation results are presented to show the performance of the observer. Comparison between the linear and nonlinear approaches of the observers.Simulation of cable dynamics for moored ocean platforms: modeling aids design of large, underwater power cable
http://hdl.handle.net/2117/23616
Title: Simulation of cable dynamics for moored ocean platforms: modeling aids design of large, underwater power cable
Authors: Prat Farran, Joana d'Arc; Zaragoza Monroig, M. Luisa; Río Fernandez, Joaquín delFri, 25 Jul 2014 10:56:15 GMThttp://hdl.handle.net/2117/236162014-07-25T10:56:15ZPrat Farran, Joana d'Arc; Zaragoza Monroig, M. Luisa; Río Fernandez, Joaquín delnoDiscrete-time static output-feedback H-infinity controller design for vehicle suspensions
http://hdl.handle.net/2117/23579
Title: Discrete-time static output-feedback H-infinity controller design for vehicle suspensions
Authors: Palacios Quiñonero, Francisco; Rubió Massegú, Josep; Rossell Garriga, Josep Maria; Karimi, Hamid Reza
Abstract: This paper provides a direct and practical presentation of a novel methodology for static output feedback controller design. The proposed design strategy has been successfully applied in the fields of control systems for seismic protection of large
buildings and multi-building structures, control of offshore wind
turbines, and active control of vehicle suspensions. The positive
results obtained in these initial applications clearly indicate that
this approach could be an effective tool in a large variety of
control problems, for which an LMI formulation of the statefeedback
version of the problem is available. The main objective
of the paper is to facilitate a brief and friendly presentation of the
main ideas involved in the new design methodology. To this end,
a discrete-time static output-feedback H1 controller is designed
for a simplified quarter-car suspension system. Numerical simulations
indicate that the proposed controller exhibits a remarkably
good behavior when compared with the corresponding statefeedback
H1 controller.Mon, 21 Jul 2014 11:46:38 GMThttp://hdl.handle.net/2117/235792014-07-21T11:46:38ZPalacios Quiñonero, Francisco; Rubió Massegú, Josep; Rossell Garriga, Josep Maria; Karimi, Hamid RezanoThis paper provides a direct and practical presentation of a novel methodology for static output feedback controller design. The proposed design strategy has been successfully applied in the fields of control systems for seismic protection of large
buildings and multi-building structures, control of offshore wind
turbines, and active control of vehicle suspensions. The positive
results obtained in these initial applications clearly indicate that
this approach could be an effective tool in a large variety of
control problems, for which an LMI formulation of the statefeedback
version of the problem is available. The main objective
of the paper is to facilitate a brief and friendly presentation of the
main ideas involved in the new design methodology. To this end,
a discrete-time static output-feedback H1 controller is designed
for a simplified quarter-car suspension system. Numerical simulations
indicate that the proposed controller exhibits a remarkably
good behavior when compared with the corresponding statefeedback
H1 controller.Generalizing the Steiner-Lehmus theorem using the Gröbner cover
http://hdl.handle.net/2117/23574
Title: Generalizing the Steiner-Lehmus theorem using the Gröbner cover
Authors: Montes Lozano, Antonio; Recio Muñiz, Tomás
Abstract: In this note we present an application of a new tool (the Gröbner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Gröbner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, Steiner-Lehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Gröbner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind.Mon, 21 Jul 2014 08:52:22 GMThttp://hdl.handle.net/2117/235742014-07-21T08:52:22ZMontes Lozano, Antonio; Recio Muñiz, TomásnoAutomatic deduction, Automatic discovery, Comprehensive Gröbner system, Elementarygeometry, Gröbner coverIn this note we present an application of a new tool (the Gröbner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Gröbner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, Steiner-Lehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Gröbner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind.Three-dimensional simulation of crack propagation in ferroelectric polycrystals: Effect of combined toughening mechanisms
http://hdl.handle.net/2117/23536
Title: Three-dimensional simulation of crack propagation in ferroelectric polycrystals: Effect of combined toughening mechanisms
Authors: Abdollahi Hosnijeh, Amir; Arias Vicente, Irene
Abstract: We simulate the fracture processes of ferroelectric polycrystals in three dimensions using a phase-field model. In this model, the grain boundaries, cracks and ferroelectric domain walls are represented in a diffuse way by three phase-fields. We thereby avoid the difficulty of tracking the interfaces in three dimensions. The resulting model can capture complex interactions between the crack and the polycrystalline and ferroelectric domain microstructures. The simulation results show the effect of the microstructures on the fracture response of the material. Crack deflection, crack bridging, crack branching and ferroelastic domain switching are observed to act as the main fracture toughening mechanisms in ferroelectric polycrystals. Our fully 3-D simulations illustrate how the combination of these mechanisms enhances the fracture toughness of the material, and pave the way for further systematic studies, including fracture homogenization.Thu, 17 Jul 2014 08:57:13 GMThttp://hdl.handle.net/2117/235362014-07-17T08:57:13ZAbdollahi Hosnijeh, Amir; Arias Vicente, IrenenoPolycrystals, Ferroelectricity, Fracture, Phase-field models, Finite element analysis, PHASE-FIELD SIMULATION, ENERGY-RELEASE RATE, R-CURVE BEHAVIOR, ELECTRIC-FIELDS, ELECTROMECHANICAL FRACTURE, BRITTLE-FRACTURE, COMPUTER-SIMULATION, COMPUTATIONAL MODEL, BOUNDARY-CONDITIONS, DOMAIN EVOLUTIONWe simulate the fracture processes of ferroelectric polycrystals in three dimensions using a phase-field model. In this model, the grain boundaries, cracks and ferroelectric domain walls are represented in a diffuse way by three phase-fields. We thereby avoid the difficulty of tracking the interfaces in three dimensions. The resulting model can capture complex interactions between the crack and the polycrystalline and ferroelectric domain microstructures. The simulation results show the effect of the microstructures on the fracture response of the material. Crack deflection, crack bridging, crack branching and ferroelastic domain switching are observed to act as the main fracture toughening mechanisms in ferroelectric polycrystals. Our fully 3-D simulations illustrate how the combination of these mechanisms enhances the fracture toughness of the material, and pave the way for further systematic studies, including fracture homogenization.El e-portafolio del estudiante en Mahara-Moodle y Google Sites
http://hdl.handle.net/2117/23534
Title: El e-portafolio del estudiante en Mahara-Moodle y Google Sites
Authors: García Planas, María Isabel; Taberna Torres, Judit
Abstract: Es bien conocido el potencial que tiene el uso del e-portfolio del estudiante para hacer visible tanto para el mismo como para los demás, de cómo y hasta que nivel ha logrado sus objetivos.
Para realizar un e-portafolio se pueden utilizar distintas aplicaciones, entre ellas se encuentran Google Sites y Mahara.
Google Sites permite de forma sencilla editando una plantilla previamente preparada por el profesor, crear una página web como muestra de sus trabajos desarrollados a lo largo de sus estudios.
Mahara es un sistema de e-portafolio que puede conectarse a Moodle y el cual puede ser controlado por el estudiante y puede ser visible por el grupo. Debido a la interoperabilidad de Mahara especialmente con Moodle, este permite que desde la plataforma de Atenea se puedan interconectar el profesorado con el e-portafolio del estudiante.
En este trabajo presentamos los resultados obtenidos desde el año 2011 hasta la actualidad en las investigaciones realizadas sobre el e-portafolio en el marco de la Universidad Politècnica de Catalunya.Thu, 17 Jul 2014 07:51:51 GMThttp://hdl.handle.net/2117/235342014-07-17T07:51:51ZGarcía Planas, María Isabel; Taberna Torres, Juditnoe-portafolio, Moodle, Mahara, Google SitesEs bien conocido el potencial que tiene el uso del e-portfolio del estudiante para hacer visible tanto para el mismo como para los demás, de cómo y hasta que nivel ha logrado sus objetivos.
Para realizar un e-portafolio se pueden utilizar distintas aplicaciones, entre ellas se encuentran Google Sites y Mahara.
Google Sites permite de forma sencilla editando una plantilla previamente preparada por el profesor, crear una página web como muestra de sus trabajos desarrollados a lo largo de sus estudios.
Mahara es un sistema de e-portafolio que puede conectarse a Moodle y el cual puede ser controlado por el estudiante y puede ser visible por el grupo. Debido a la interoperabilidad de Mahara especialmente con Moodle, este permite que desde la plataforma de Atenea se puedan interconectar el profesorado con el e-portafolio del estudiante.
En este trabajo presentamos los resultados obtenidos desde el año 2011 hasta la actualidad en las investigaciones realizadas sobre el e-portafolio en el marco de la Universidad Politècnica de Catalunya.Optimal design of complex passive-damping systems for vibration control of large structures: an energy-to-peak approach
http://hdl.handle.net/2117/23527
Title: Optimal design of complex passive-damping systems for vibration control of large structures: an energy-to-peak approach
Authors: Palacios Quiñonero, Francisco; Rubió Massegú, Josep; Rossell Garriga, Josep Maria; Karimi, Hamid Reza
Abstract: We present a new design strategy that makes it possible to synthesize decentralized output-feedback controllers by solving two successive optimization problems with linear matrix inequality (LMI) constraints. In the initial LMI optimization problem, two auxiliary elements are computed: a standard state-feedback controller, which can be taken as a reference in the performance assessment, and a matrix that facilitates a proper definition of the main LMI optimization problem. Next, by solving the second optimization problem, the output-feedback controller is obtained. The proposed strategy extends recent results in static output-feedback control and can be applied to design complex passive-damping systems for vibrational control of large structures. More precisely, by taking advantages of the existing link between fully decentralized velocity-feedback controllers and passive linear dampers, advanced active feedback control strategies can be used to design complex passive-damping systems, which combine the simplicity and robustness of passive control systems with the efficiency of active feedback control. To demonstrate the effectiveness of the proposed approach, a passive-damping system for the seismic protection of a five-story building is designed with excellent results.Wed, 16 Jul 2014 09:47:48 GMThttp://hdl.handle.net/2117/235272014-07-16T09:47:48ZPalacios Quiñonero, Francisco; Rubió Massegú, Josep; Rossell Garriga, Josep Maria; Karimi, Hamid RezanoWe present a new design strategy that makes it possible to synthesize decentralized output-feedback controllers by solving two successive optimization problems with linear matrix inequality (LMI) constraints. In the initial LMI optimization problem, two auxiliary elements are computed: a standard state-feedback controller, which can be taken as a reference in the performance assessment, and a matrix that facilitates a proper definition of the main LMI optimization problem. Next, by solving the second optimization problem, the output-feedback controller is obtained. The proposed strategy extends recent results in static output-feedback control and can be applied to design complex passive-damping systems for vibrational control of large structures. More precisely, by taking advantages of the existing link between fully decentralized velocity-feedback controllers and passive linear dampers, advanced active feedback control strategies can be used to design complex passive-damping systems, which combine the simplicity and robustness of passive control systems with the efficiency of active feedback control. To demonstrate the effectiveness of the proposed approach, a passive-damping system for the seismic protection of a five-story building is designed with excellent results.Noise and adaptation in multistable perception: noise drives when to switch, adaptation determines percept choice.
http://hdl.handle.net/2117/23524
Title: Noise and adaptation in multistable perception: noise drives when to switch, adaptation determines percept choice.
Authors: Huguet Casades, Gemma; Rinzel, John; Hupé, Jean-Michel
Abstract: We study the dynamics of perceptual switching in ambiguous visual scenes that admit more than two interpretations/percepts to gain insight into the dynamics of perceptual multistability and its underlying neural mechanisms. We focus on visual plaids that are tristable and we present both experimental and computational results. We develop a firing-rate model based on mutual inhibition and adaptation that involves stochastic dynamics of multiple-attractor systems. The model can account for the dynamic properties (transition probabilities, distributions of percept durations, etc.) observed in the experiments. Noise and adaptation have both been shown to play roles in the dynamics of bistable perception. Here, tristable perception allows us to specify the roles of noise and adaptation in our model. Noise is critical in considering the time of a switch. On the other hand, adaptation mechanisms are critical in considering perceptual choice (in tristable perception, each time a percept ends, there is a possible choice between two new percepts).Wed, 16 Jul 2014 07:35:44 GMThttp://hdl.handle.net/2117/235242014-07-16T07:35:44ZHuguet Casades, Gemma; Rinzel, John; Hupé, Jean-MichelnoWe study the dynamics of perceptual switching in ambiguous visual scenes that admit more than two interpretations/percepts to gain insight into the dynamics of perceptual multistability and its underlying neural mechanisms. We focus on visual plaids that are tristable and we present both experimental and computational results. We develop a firing-rate model based on mutual inhibition and adaptation that involves stochastic dynamics of multiple-attractor systems. The model can account for the dynamic properties (transition probabilities, distributions of percept durations, etc.) observed in the experiments. Noise and adaptation have both been shown to play roles in the dynamics of bistable perception. Here, tristable perception allows us to specify the roles of noise and adaptation in our model. Noise is critical in considering the time of a switch. On the other hand, adaptation mechanisms are critical in considering perceptual choice (in tristable perception, each time a percept ends, there is a possible choice between two new percepts).R+aR2 loop quantum cosmology
http://hdl.handle.net/2117/23514
Title: R+aR2 loop quantum cosmology
Authors: Amorós Torrent, Jaume; De Haro, Jaume; Odintsov, Sergei D.
Abstract: Working in Einstein frame, we introduce, in order to avoid singularities, holonomy corrections to the f(R)=R+aR2 model. We perform a detailed analytical and numerical study when holonomy corrections are taken into account in both Jordan and Einstein frames, obtaining, in Jordan frame, a dynamics which differs qualitatively, at early times, from the one of the original model. More precisely, when holonomy corrections are taken into account, the Universe is not singular, starting at early times in the contracting phase and bouncing to enter the expanding one where, as in the original model, it inflates. This dynamics is completely different from the one obtained in the original R+aR2 model, where the Universe is singular at early times and never bounces. Moreover, we show that these holonomy corrections may lead to better predictions for the inflationary phase as compared with current observations.Tue, 15 Jul 2014 11:00:50 GMThttp://hdl.handle.net/2117/235142014-07-15T11:00:50ZAmorós Torrent, Jaume; De Haro, Jaume; Odintsov, Sergei D.noWorking in Einstein frame, we introduce, in order to avoid singularities, holonomy corrections to the f(R)=R+aR2 model. We perform a detailed analytical and numerical study when holonomy corrections are taken into account in both Jordan and Einstein frames, obtaining, in Jordan frame, a dynamics which differs qualitatively, at early times, from the one of the original model. More precisely, when holonomy corrections are taken into account, the Universe is not singular, starting at early times in the contracting phase and bouncing to enter the expanding one where, as in the original model, it inflates. This dynamics is completely different from the one obtained in the original R+aR2 model, where the Universe is singular at early times and never bounces. Moreover, we show that these holonomy corrections may lead to better predictions for the inflationary phase as compared with current observations.Auto-Backlund transformations and special integrals for differential-delay Painlevé hierarchies
http://hdl.handle.net/2117/23511
Title: Auto-Backlund transformations and special integrals for differential-delay Painlevé hierarchies
Authors: Fedorov, Yuri; Ruiz Gordoa, Maria Pilar; Pickering, Andrew
Abstract: The six Painleve equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painleve equations, i.e., analogues of the Painleve equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painleve hierarchy define solutions of higher-order members of a second differential-delay Painleve hierarchy. We also give an auto-Backlund transformation for a differential-delay Painleve hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies. (C) 2014 Elsevier B.V. All rights reserved.Tue, 15 Jul 2014 10:29:35 GMThttp://hdl.handle.net/2117/235112014-07-15T10:29:35ZFedorov, Yuri; Ruiz Gordoa, Maria Pilar; Pickering, AndrewnoDifferential-delay Painleve hierarchies, MONODROMY PRESERVING DEFORMATION, RATIONAL COEFFICIENTS, EQUATIONSThe six Painleve equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painleve equations, i.e., analogues of the Painleve equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painleve hierarchy define solutions of higher-order members of a second differential-delay Painleve hierarchy. We also give an auto-Backlund transformation for a differential-delay Painleve hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies. (C) 2014 Elsevier B.V. All rights reserved.Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies
http://hdl.handle.net/2117/23508
Title: Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies
Authors: Delshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Pere
Abstract: We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.Tue, 15 Jul 2014 08:21:36 GMThttp://hdl.handle.net/2117/235082014-07-15T08:21:36ZDelshams Valdés, Amadeu; Gonchenko, Marina; Gutiérrez Serrés, Perenosplitting of separatrices, Melnikov integrals, quadratic and cubic frequencies, INTEGRABLE HAMILTONIAN-SYSTEMS, ADIABATIC INVARIANTS, CONTINUED FRACTIONS, MELNIKOV METHOD, MCMILLAN MAP, UPPER-BOUNDS, RENORMALIZATION, APPROXIMATION, PERTURBATION, PENDULUMWe study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.