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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 [Mar84, Mar85] and Guillemin and Sternberg [GS84] for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the… (More)

The presence of symmetries in a Hamiltonian system usually implies 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… (More)

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, 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… (More)

- Jerrold E. Marsden, Gerard Misio, Juan-Pablo Ortega, Matthew Perlmutter, Tudor S. Ratiu
- 2007

- Joan-Andreu Lázaro-Camı, Juan-Pablo Ortega
- 2007

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with… (More)

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a… (More)

- Lyudmila Grigoryeva, Julie Henriques, Laurent Larger, Juan-Pablo Ortega
- Neural Networks
- 2014

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… (More)

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie– Dirac reduction. This procedure simultaneously includes Lagrangian, Hamil-tonian, and a variational view of reduction. The goal of the present paper is to generalize Lie–Dirac reduction to the case of a general… (More)

- Lyudmila Grigoryeva, Julie Henriques, Laurent Larger, Juan-Pablo Ortega
- Scientific reports
- 2015

Reservoir computing is a recently introduced brain-inspired machine learning paradigm capable of excellent performances in the processing of empirical data. We focus in a particular kind of time-delay based reservoir computers that have been physically implemented using optical and electronic systems and have shown unprecedented data processing rates.… (More)

- Stéphane Chrétien, Juan-Pablo Ortega
- Computational Statistics & Data Analysis
- 2014

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the " curse of dimensionality ". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the… (More)