Seminário de Teoria Qualitativa dos Sistemas Dinâmicos - "On invariant tori and the averaging method"

Murilo Cândido (recém doutor pela Universitat Autònoma de Barcelona e PósDoc do IMECC.)
Data do Evento: 
terça-feira, 13 de Novembro de 2018 - 10:00
Sala 321

The averaging method is widely used for nding periodic solutions of
di erential systems in the form
_ x = "F1(t; x; ) + "2eF
(t; x; "; ); (1)
where F1 : R 
 ! Rn and eF
: R  !  (􀀀"0; "0) ! Rn are Ck+1 functions,
T-periodic at the rst variable, being
 an open and bounded subset of Rn,
 2 R and " a small positive parameter.
The classical averaging method for detecting periodic solutions of system
(1) consists basically of nding some xed points of the autonomous
di erential system (often called averaged system)
y_ = g1(y; ); (2)
here g1(y; ) =
0 F1(t; y; )dt.
For planar di erential systems (i.e. n = 2), some authors also relate the
occurrence of a Hopf bifurcation in system (2) with the birth of an invariant
torus in the original system (1). In many works, only heuristic arguments
are given in order to justify such relation.
In this talk, we provide results that can be used for proving the above
relation. In fact, using these results we can show that a co-dimension 1
Hopf bifurcation in system (2), is a sucient condition for the existence of a
Neimark-Sacker bifurcation in the rst return map of the original system (1)
at time T. Consequently system (1) shall have a periodic solution surrounded
by an invariant torus.
Finally, we show how to use our results for studying the existence of invariant
tori in 3D autonomous systems. This method will be use for providing
sucient conditions for the existence of an invariant torus in the Rossler system.
As far as we know, this is the rst time that such invariant torus was
analytically detected.