Reversible equivariant Hamiltonian bifurcation theory
The main aim is to develop systematic methods for studying local and
global bifurcations in Hamiltonian dynamical systems with symmetries
(time-preserving [equivariance] and time-reversing [reversibility]).
It is well known that symmetries affect such bifurcations, and the aim is
to classify such bifurcations as well as study (local) dynamical consequences.
Subtheme is the comparison between reversible-equivariant and Hamiltonian
reversible-equivariant systems.
Current projects include:
- Linear systems
Development of Arnol'd normal
form and unfolding theory for linear systems depending on parameters for
the RE and Hamiltonian RE categories. Work in progress with I. Hoveijn
(Groningen) and Mark Roberts (Surrey)
- Steady-state bifurcation
Understanding families of equilibria in generic (parameter families of)
RE and RE Hamiltonian systems.
Work in progress with Pietro-Luciano Buono (CRM Montreal) and Mark Roberts
(Surrey).
- Hopf bifurcation
Studying branching and bifurcation of families of periodic solutions near
equilibria, including Liapunov Center families.
Work in progress with Claudio Buzzi (UNESP Sao Jose do Rio Preto) and
Takeo Fujihira (Imperial).
- Homoclinic bifurcation
Studying dynamics near homoclinic and heteroclinic connections of RE and
RE Hamiltonian systems. Work in progress with Ale Jan Homburg (Amsterdam)
and Alice Jukes (Imperial).
- Elliptic points
Studying the dynamics near elliptic points in reversible systems, in
particular in comparison to dynamics near elliptic points in Hamiltonian
systems. Work in progress with Isabel Rios (UFF Niteroi).