Juan-Pablo Ortega

Momentum maps and Hamiltonian reduction

Introduction.- Manifolds and smooth structures.- Lie group actions.- Pseudogroups and groupoids.- The standard momentum map.- Generalizations of the momentum map.- Regular symplectic reduction theory.- The Symplectic Slice Theorem.- Singular reduction and the stratification theorem.- Optimal reduction.- Poisson reduction.- Dual Pairs.- Bibliography.- Index.

Dynamics on Leibniz manifolds

Abstract This paper shows that various well-known dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some control and dissipative systems, and non-holonomically constrained simple mechanical systems, among other dynamical behaviors, can be described using this mathematical construction that generalizes the standard Poisson bracket currently used in Hamiltonian mechanics. The symmetries of these systems and the associated reduction procedures are described in detail. A number of examples illustrate the theoretical developments in the paper.

Stochastic hamiltonian dynamical systems

We use the global stochastic analysis tools introduced by R A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, are characterized by a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

A Symplectic Slice Theorem

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamiltons equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.

Forecasting Growth During the Great Recession: Is Financial Volatility the Missing Ingredient?

The Great Recession endured by the main industrialized countries during the period 2008–2009, in the wake of the financial and banking crisis, has pointed out the major role of the financial sector on macroeconomic fluctuations. In this respect, many researchers have started to reconsider the linkages between financial and macroeconomic areas. In this paper, we evaluate the leading role of the daily volatility of two major financial variables, namely commodity and stock prices, in their ability to anticipate the output growth. For this purpose, we propose an extended MIDAS model that allows the forecasting of the quarterly output growth rate using exogenous variables sampled at various higher frequencies. Empirical results on three industrialized countries (US, France, and UK) show that mixing daily financial volatilities and monthly industrial production is useful at the time of predicting gross domestic product growth over the Great Recession period.

The optimal momentum map

The presence of symmetries in a Hamiltonian system usually simplies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden-Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian system with fewer degrees of freedom. However, in a large number of situations, this standard approach does not work or is not e cient enough, in the sense that it does not use all the information encoded in the symmetry of the system. In this work, a new momentum map will be defined that is capable of overcoming most of the problems encountered in the traditional approach.

Bifurcation of relative equilibria in mechanical systems with symmetry

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [Nonlinearity 11 (1998) 1637-1649] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example illustrates the use of this approach.

Hamiltonian Hopf bifurcation with symmetry

Abstract In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.

REDUCTION, RECONSTRUCTION, AND SKEW-PRODUCT DECOMPOSITION OF SYMMETRIC STOCHASTIC DIFFERENTIAL EQUATIONS

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. In addition, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any point in the open and dense principal orbit type. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples. The Hamiltonian category deserves a separate study since in that situation the presence of symmetries implies in most cases the existence of conservation laws, mathematically described via momentum maps, that should be taken into account in the analysis.

Stochastic Nonlinear Time Series Forecasting Using Time-Delay Reservoir Computers: Performance and Universality

Reservoir computing is a recently introduced machine learning paradigm that has already shown excellent performances in the processing of empirical data. We study a particular kind of reservoir computers called time-delay reservoirs that are constructed out of the sampling of the solution of a time-delay differential equation and show their good performance in the forecasting of the conditional covariances associated to multivariate discrete-time nonlinear stochastic processes of VEC-GARCH type as well as in the prediction of factual daily market realized volatilities computed with intraday quotes, using as training input daily log-return series of moderate size. We tackle some problems associated to the lack of task-universality for individually operating reservoirs and propose a solution based on the use of parallel arrays of time-delay reservoirs.