In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval begin{document}$ (0,frac{pi}{2}] $end{document} with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on begin{document}$ (0,delta] $end{document} (with begin{document}$ delta $end{document} small) while a Chebyshev series expansion is used to solve the problem on begin{document}$ [delta,frac{pi}{2}] $end{document}. The two setups are incorporated in a larger zero-finding problem of the form begin{document}$ F(a) = 0 $end{document} with begin{document}$ a $end{document} containing the coefficients of the Taylor and Chebyshev series. The problem begin{document}$ F = 0 $end{document} is solved rigorously using a Newton-Kantorovich argument.
In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval begin{document}$ (0,frac{pi}{2}] $end{document} with a removable singularity at zero. The singularity is removed by solving the equation with Taylor series on begin{document}$ (0,delta] $end{document} (with begin{document}$ delta $end{document} small) while a Chebyshev series expansion is used to solve the problem on begin{document}$ [delta,frac{pi}{2}] $end{document}. The two setups are incorporated in a larger zero-finding problem of the form begin{document}$ F(a) = 0 $end{document} with begin{document}$ a $end{document} containing the coefficients of the Taylor and Chebyshev series. The problem begin{document}$ F = 0 $end{document} is solved rigorously using a Newton-Kantorovich argument.
{"title":"Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach","authors":"J. B. Berg, Gabriel William Duchesne, J. Lessard","doi":"10.3934/jcd.2022005","DOIUrl":"https://doi.org/10.3934/jcd.2022005","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval <inline-formula><tex-math id=\"M1\">begin{document}$ (0,frac{pi}{2}] $end{document}</tex-math></inline-formula> with a <i>removable</i> singularity at zero. The singularity is removed by solving the equation with Taylor series on <inline-formula><tex-math id=\"M2\">begin{document}$ (0,delta] $end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id=\"M3\">begin{document}$ delta $end{document}</tex-math></inline-formula> small) while a Chebyshev series expansion is used to solve the problem on <inline-formula><tex-math id=\"M4\">begin{document}$ [delta,frac{pi}{2}] $end{document}</tex-math></inline-formula>. The two setups are incorporated in a larger zero-finding problem of the form <inline-formula><tex-math id=\"M5\">begin{document}$ F(a) = 0 $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M6\">begin{document}$ a $end{document}</tex-math></inline-formula> containing the coefficients of the Taylor and Chebyshev series. The problem <inline-formula><tex-math id=\"M7\">begin{document}$ F = 0 $end{document}</tex-math></inline-formula> is solved rigorously using a Newton-Kantorovich argument.</p>","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89109291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in begin{document}$ mathbb{R}^3 $end{document} resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.
Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in begin{document}$ mathbb{R}^3 $end{document} resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.
{"title":"Continuation methods for principal foliations of embedded surfaces","authors":"J. Guckenheimer","doi":"10.3934/jcd.2022007","DOIUrl":"https://doi.org/10.3934/jcd.2022007","url":null,"abstract":"Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in begin{document}$ mathbb{R}^3 $end{document} resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in \"Monge coordinates.\" Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81636158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. 200 words.
{"title":"Sensitivity analysis for periodic orbits and quasiperiodic invariant tori using the adjoint method","authors":"H. Dankowicz, J. Sieber","doi":"10.3934/jcd.2022006","DOIUrl":"https://doi.org/10.3934/jcd.2022006","url":null,"abstract":"This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is shown to be directly amenable to analysis using the COCO software package, specifically its paradigm for staged problem construction. The general theory is illustrated in the context of algebraic equations and boundary-value problems, with emphasis on periodic orbits in smooth and hybrid dynamical systems, and quasiperiodic invariant tori of flows. In the latter case, normal hyperbolicity is used to prove the existence of continuous solutions to the adjoint conditions associated with the sensitivities of the orbital periods to parameter perturbations and constraint violations, even though the linearization of the governing boundary-value problem lacks a bounded inverse, as required by the general theory. An assumption of transversal stability then implies that these solutions predict the asymptotic phases of trajectories based at initial conditions perturbed away from the torus. Example COCO code is used to illustrate the minimal additional investment in setup costs required to append sensitivity analysis to regular parameter continuation. 200 words.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85373020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose begin{document}$ 1 $end{document}-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.
Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose begin{document}$ 1 $end{document}-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.
{"title":"A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems","authors":"Morgan Jones, M. Peet","doi":"10.3934/jcd.2022019","DOIUrl":"https://doi.org/10.3934/jcd.2022019","url":null,"abstract":"Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose begin{document}$ 1 $end{document}-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81101460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we combine an approach based on Runge-Kutta Nets considered in [Benning et al., J. Comput. Dynamics, 9, 2019] and a technique on augmenting the input space in [Dupont et al., NeurIPS, 2019] to obtain network architectures which show a better numerical performance for deep neural networks in point and image classification problems. The approach is illustrated with several examples implemented in PyTorch.
在本文中，我们结合了[Benning et al.， J. Comput]中考虑的基于龙格-库塔网的方法。[杜邦等人，NeurIPS, 2019]中的一种增强输入空间的技术，以获得在点和图像分类问题中表现出更好数值性能的深度神经网络网络架构。通过在PyTorch中实现的几个示例说明了该方法。
{"title":"Classification with Runge-Kutta networks and feature space augmentation","authors":"E. Giesecke, Axel Kroner","doi":"10.3934/jcd.2021018","DOIUrl":"https://doi.org/10.3934/jcd.2021018","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we combine an approach based on Runge-Kutta Nets considered in [<i>Benning et al., J. Comput. Dynamics, 9, 2019</i>] and a technique on augmenting the input space in [<i>Dupont et al., NeurIPS</i>, 2019] to obtain network architectures which show a better numerical performance for deep neural networks in point and image classification problems. The approach is illustrated with several examples implemented in PyTorch.</p>","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82476325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.
{"title":"An algebraic approach to the spontaneous formation of spherical jets","authors":"M. Viviani","doi":"10.3934/jcd.2021028","DOIUrl":"https://doi.org/10.3934/jcd.2021028","url":null,"abstract":"The global structure of the atmosphere and the oceans is a continuous source of intriguing challenges in geophysical fluid dynamics (GFD). Among these, jets are determinant in the air and water circulation around the Earth. In the last fifty years, thanks to the development of more and more precise and extensive observations, it has been possible to study in detail the atmospheric formations of the giant-gas planets in the solar system. For those planets, jets are the dominant large scale structure. Starting from the 70s, various theories combining observations and mathematical models have been proposed in order to describe their formation and stability. In this paper, we propose a purely algebraic approach to describe the spontaneous formation of jets on a spherical domain. Analysing the algebraic properties of the 2D Euler equations, we give a characterization of the different jets' structures. The calculations are performed starting from the discrete Zeitlin model of the Euler equations. For this model, the classification of the jets' structures can be precisely described in terms of reductive Lie algebras decomposition. The discrete framework provides a simple tool for analysing both from a theoretical and and a numerical perspective the jets' formation. Furthermore, it allows to extend the results to the original Euler equations.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85358098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Henon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.
{"title":"Computer-assisted estimates for birkhoff normal forms","authors":"Chiara Caracciolo, U. Locatelli","doi":"10.3934/jcd.2020017","DOIUrl":"https://doi.org/10.3934/jcd.2020017","url":null,"abstract":"Birkhoff normal forms are commonly used in order to ensure the so called \"effective stability\" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Henon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89127094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the Mendeley Data repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.
{"title":"Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers","authors":"L. Valvo, U. Locatelli","doi":"10.3934/jcd.2022002","DOIUrl":"https://doi.org/10.3934/jcd.2022002","url":null,"abstract":"We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the Mendeley Data repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84492498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The shuffle algorithm is applied to analysis of the fractional descriptor discrete-time linear systems. Using the shuffle algorithm the singularity of the fractional descriptor linear system is eliminated and the system is decomposed into dynamic and static parts. Procedures for computation of the solution and dynamic and static parts of the system are proposed. Sufficient conditions for the positivity of the fractional descriptor discrete-time linear systems are established.
{"title":"Analysis of the fractional descriptor discrete-time linear systems by the use of the shuffle algorithm","authors":"T. Kaczorek, A. Ruszewski","doi":"10.3934/JCD.2021007","DOIUrl":"https://doi.org/10.3934/JCD.2021007","url":null,"abstract":"The shuffle algorithm is applied to analysis of the fractional descriptor discrete-time linear systems. Using the shuffle algorithm the singularity of the fractional descriptor linear system is eliminated and the system is decomposed into dynamic and static parts. Procedures for computation of the solution and dynamic and static parts of the system are proposed. Sufficient conditions for the positivity of the fractional descriptor discrete-time linear systems are established.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73177752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper analyzes conservation issues in the discretization of certain stochastic dynamical systems by means of stochastic begin{document}$ vartheta $end{document}-mehods. The analysis also takes into account the effects of the estimation of the expected values by means of Monte Carlo simulations. The theoretical analysis is supported by a numerical evidence on a given stochastic oscillator, inspired by the Duffing oscillator.
This paper analyzes conservation issues in the discretization of certain stochastic dynamical systems by means of stochastic begin{document}$ vartheta $end{document}-mehods. The analysis also takes into account the effects of the estimation of the expected values by means of Monte Carlo simulations. The theoretical analysis is supported by a numerical evidence on a given stochastic oscillator, inspired by the Duffing oscillator.
{"title":"Numerical preservation issues in stochastic dynamical systems by $ vartheta $-methods","authors":"R. D'Ambrosio, S. Di Giovacchino","doi":"10.3934/jcd.2021023","DOIUrl":"https://doi.org/10.3934/jcd.2021023","url":null,"abstract":"<p style='text-indent:20px;'>This paper analyzes conservation issues in the discretization of certain stochastic dynamical systems by means of stochastic <inline-formula><tex-math id=\"M2\">begin{document}$ vartheta $end{document}</tex-math></inline-formula>-mehods. The analysis also takes into account the effects of the estimation of the expected values by means of Monte Carlo simulations. The theoretical analysis is supported by a numerical evidence on a given stochastic oscillator, inspired by the Duffing oscillator.</p>","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79739189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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