1 / 20

SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY

Communicación en el III Congreso Latinoamericano de Matemáticos August 31th.-September 4th. 2009. SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY. Eleonora Catsigeras Marcelo Cerminara Heber Enrich. Instituto de Matemática. Facultad de Ingeniería.

zenia
Download Presentation

SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Communicación en el III Congreso Latinoamericano de Matemáticos August 31th.-September 4th. 2009 SIMULTANEOUS CONTINUATION OF INFINITELY MANY SINKS NEAR A QUADRATIC HOMOCLINIC TANGENCY Eleonora Catsigeras Marcelo Cerminara Heber Enrich Instituto de Matemática. Facultad de Ingeniería. Universidad de la República URUGUAY eleonora@fing.edu.uy

  2. ASSUMPTIONS: • M is a 2-dimensional compact manifold • has a dissipative saddle periodic point with a homoclinic quadratic tangency • is a one-parameter family generically unfolding the homoclinic tangency Theorems of Newhouse 1970, 1974, 1979 and Robinson 1983:

  3. Theorems of Newhouse 1970, 1974, 1979 and Robinson 1983: OPEN QUESTION: HAVE THE INFINITELY MANY COEXISTING SINKS SIMULTANEOUS CONTINUATIONS IN SOMEOPEN NEIGHBORHOOD IN THE FUNCTIONAL SPACE? ?

  4. THEOREM 1: There exists an open interval of parameter values ,and a residual subset such that: For all 1. exhibits infinitely many coexisting sinks (Newhouse-Robinson) 2. There exists a infinite-dimensional manifold such that:

  5. THEOREM 2:

  6. Route of the proofs of Theorems 1 and 2 In SIX STEPS: STEP 1: Apply Newhouse Theorem:

  7. STEP 2: • Consider TRIVIALIZING COORDINATES OF THE INVARIANT FOLIATIONS in the neighborhood • Use dissipative hypothesis of the given hyperbolic periodic point exhibiting a quadratic homoclinic tangency in f sub0, to obtain STRONG DISSIPATIVE CONDITION of the hyperbolic set for all • Apply r-normality to conclude that • The STABLE LOCAL FOLIATION is • while the unstable foliation is not necessarily more than

  8. STEP 3: Compute the iteration of f in the trivializing coordinates as in [PT 1993] (up to non trivial adaptations, using C3 differentiability), to conclude the following : LEMMA (Uniform approximation to the quadraticfamily):

  9. LEMMA (Uniform approximation to the quadraticfamily):

  10. STEP 4 Consequences of Lemma:

  11. STEP 4 Consequences of Lemma:

  12. STEP 5:

  13. STEP 5:

  14. STEP 6 (CONCLUSIONS): End of the proof of Theorem 1:

  15. STEP 6 (CONCLUSIONS): End of the proof of Theorem 1:

  16. STEP 6 (CONCLUSIONS):

  17. STEP 6 (CONCLUSIONS): End of the proof of Theorem 2:

More Related