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HamInstab SIGNED

Instabilities and homoclinic phenomena in Hamiltonian systems

Total Cost €

EC-Contrib. €

Partnership

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Outcomes and
results

HamInstab project word cloud

Explore the words cloud of the HamInstab project. It provides you a very rough idea of what is the project "HamInstab" about.

ouml averages breathers nonlinear equations showing modes infinite setting homoclinic interaction dynamics pdes manifolds place global arbitrarily phenomena drifting stability klein nearly proper oscillatory solutions puntual heteroclinic motion physically mechanics construct accumulates analyze solution nevertheless leads body schr energy gordon partial mathematical transversal dinger diffusion theory celestial astronomers instabilities deeply framework scarce objects orbits techniques dimensional wave dynamical localized models usually motions time fundamental hyperbolic masses intersection arnold effect connections push perturbation plan differential instability hamiltonian mean force invariant prove gravitational periodic integrable transfer secular existence model resonances stationary relying ascertain

Project "HamInstab" data sheet

The following table provides information about the project.

Coordinator	UNIVERSITAT POLITECNICA DE CATALUNYA Organization address address: CALLE JORDI GIRONA 31 city: BARCELONA postcode: 8034 website: www.upc.edu contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.
Coordinator Country	Spain [ES]
Project website	https://haminstab.barcelonatech-upc.eu/
Total cost	1˙100˙347 €
EC max contribution	1˙100˙347 € (100%)
Programme	1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
Code Call	ERC-2017-STG
Funding Scheme	ERC-STG
Starting year	2018
Duration (year-month-day)	from 2018-01-01 to 2022-12-31

Partnership

Take a look of project's partnership.

#	participants	country	role	EC contrib. [€]
1	UNIVERSITAT POLITECNICA DE CATALUNYA UNIVERSITAT POLITECNICA DE CATALUNYA Organization address address: CALLE JORDI GIRONA 31 city: BARCELONA postcode: 8034 website: www.upc.edu contact info title: n.a. name: n.a. surname: n.a. function: n.a. email: n.a. telephone: n.a. fax: n.a.	ES (BARCELONA)	coordinator	1˙100˙347.00

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Project objective

A fundamental problem in the study of dynamical systems is to ascertain whether the effect of a perturbation on an integrable Hamiltonian system accumulates over time and leads to a large effect (instability) or it averages out (stability). Instabilities in nearly integrable systems, usually called Arnold diffusion, take place along resonances and by means of a framework of partially hyperbolic invariant objects and their homoclinic and heteroclinic connections.

The goal of this project is to develop new techniques, relying on the role of invariant manifolds in the global dynamics, to prove the existence of physically relevant instabilities and homoclinic phenomena in several problems in celestial mechanics and Hamiltonian Partial Differential Equations.

The N body problem models the interaction of N puntual masses under gravitational force. Astronomers have deeply analyzed the role of resonances in this model. Nevertheless, mathematical results showing instabilities along them are rather scarce. I plan to develop a new theory to analyze the transversal intersection between invariant manifolds along mean motion and secular resonances to prove the existence of Arnold diffusion. I will also apply this theory to construct oscillatory motions.

Several Partial Differential Equations such as the nonlinear Schrödinger, the Klein-Gordon and the wave equations can be seen as infinite dimensional Hamiltonian systems. Using dynamical systems techniques and understanding the role of invariant manifolds in these Hamiltonian PDEs, I will study two type of solutions: transfer of energy solutions, namely solutions that push energy to arbitrarily high modes as time evolves by drifting along resonances; and breathers, spatially localized and periodic in time solutions, which in a proper setting can be seen as homoclinic orbits to a stationary solution.

Publications

List of publications.
year	authors and title	journal	last update
2019	Marcel Guardia, Vadim Kaloshin, Jianlu Zhang Asymptotic Density of Collision Orbits in the Restricted Circular Planar 3 Body Problem published pages: 799-836, ISSN: 0003-9527, DOI: 10.1007/s00205-019-01368-7	Archive for Rational Mechanics and Analysis 233/2	2019-08-05

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