6533b82efe1ef96bd1293ca2

RESEARCH PRODUCT

Limit Periodic Sets

Robert Roussarie

subject

Surface (mathematics)PhysicsMathematical analysisOrbit (dynamics)Periodic sequenceAnnulus (mathematics)Vector fieldAstrophysics::Earth and Planetary AstrophysicsSingular point of a curvePerturbation theoryLimit superior and limit inferior

description

As explained at the end of the previous chapter, the most difficult problem in the study of bifurcations in a family of vector fields on a surface of genus 0 is the control of the periodic orbits. In fact, in generic smooth families the periodic orbits will be isolated for each value of the parameter. For analytic families we have two possibilities for each orbit: it may be isolated or belong to a whole annulus of periodic orbits. In this last case and for the parameter values for which the system has infinitely many periodic orbits, the vector field has a local analytic first integral and the nearby vector fields in the family may be studied by the perturbation theory introduced in Chapter 4. They have in general isolated periodic orbits. The interest in the study of isolated periodic orbits is also justified by tradition and by applications.

yearjournalcountryeditionlanguage
1998-01-01
https://doi.org/10.1007/978-3-0348-8798-4_2