Alexandre Artur Pinho Rodrigues's Annual Report

Alexandre Artur Pinho Rodrigues's Annual Report

Year: 

2015

Brief description of the research activities: 

My main research interest is dynamical systems, specially the qualitative theory of differential equations, with emphasis on stability and bifurcations. My particular interests include dynamical properties near heteroclinic networks that become generic in special classes of equations: symmetric, reversible and conservative.

The dynamics near a network deserves a more specific description than the dichotomy: regular dynamics vs chaos; in the latter case, generically the distant future of solutions near the network is practically inaccessible and may only be described in probabilistic and ergodic terms. The aim of my research is to verify the convergence of spatial averages associated to trajectories that remain near the networks for all time. I would like to estimate the size of the set of points persistently close to a given connection in the network. This will provide useful information about nearby dynamics and visibility of each cycle. The main question in my research is the Takens' Last Problem: whether there are persistent classes of smooth dynamical systems such that the set of initial conditions which give rise to orbits with historical behaviour has positive Lebesgue measure.

My research builds on recent results by Labouriau, Kiriki and myself, on the dynamics near symmetric heteroclinic networks.

Seminars and courses given at CMUP or conferences: 

Talks / Seminars / Courses : 

Seminars

Title:
"Dynamics near a network involving a bifocus", Centro de Matemática da Universidade do Porto
Speakers
Alexandre Artur Pinho Rodrigues
Date:
19.06.2015
Country:
Portugal
Location / City:
Porto
Title:
Intermitência Heteroclínica no Campo Geomagnético,
Speakers
Alexandre Artur Pinho Rodrigues
Talk:
Invited talk
Date:
06.10.2015
Host institution:
Universidade de Évora
Country:
Portugal
Location / City:
Évora
Title:
Bifocal homoclinic orbits: next challenges
Speakers
Alexandre Artur Pinho Rodrigues
Talk:
Invited talk
Date:
19.01.2015
Host institution:
Universidade de Oviedo
Country:
Espanha
Location / City:
Oviedo

Communications in international conferences

Title:
Dynamics near a network involving a bifocus
Name of the event:
Shilnikov Workshop 2015
Speakers
Alexandre Artur Pinho Rodrigues
Talk:
Invited talk
Date:
19.07.2015 to 24.07.2015
Host institution:
Niznhy Novgorod University
Country:
Russia
Location / City:
Niznhy Novgorod
Title:
Dynamics near a network involving a bifocus
Name of the event:
DMV - Encontro Nacional da Sociedade Alemã de Matemática,
Speakers
Alexandre Artur Pinho Rodrigues
Talk:
Invited talk
Date:
21.09.2015 to 25.09.2015
Host institution:
Hamburg University
Country:
Germay
Location / City:
Hamburg

Organization of scientific meetings: 

Role: 

Co-Organizer of the Thematic Session "Heteroclinic cycles and networks: Stability and Dynamics",AMS-EMS-SPM International Meeting 2015

Name of the event: 

Heteroclinic cycles and networks: Stability and Dynamics

start and end date: 

June 10, 2015 to June 13, 2015

Host institution: 

University og Porto

Country: 

Portugal

Location / City: 

Porto

Organization of regular seminars: 

Organizer of the DYNAMICAL SYSTEMS seminars at CMUP

Work visits: 

Imperial College of London, joint work with Mike Field (July 2015)