Year-end Sale % OFF 
Abstract
In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.
Highlights
Limit cycles of planar polynomial vector fields have long been a focus of extensive research
In this way a polynomial complex vector field is obtained and the holomorphic curves tangent to it form a partition of the complex plane by Riemann surfaces, called a polynomial complex foliation with singularities, or in short polynomial complex foliation [12], [13]
Since having all zero resonant terms is an extremely special property for maps with root-of-unity multiplier, we can expect that the Poincare transformations for most foliations of the form (1) will have a lot of isolated periodic orbits and the foliations themselves will have many multi-fold limit cycles
Summary
Limit cycles of planar polynomial vector fields have long been a focus of extensive research. A central problem in the study of multi-fold limit cycles is their existence in families of polynomial foliations of the form (1). Using Pontryagin’s theorem, we could find a family of 1-fold cycles which gives a family of isolated fixed points for the corresponding Poincare map Pε. For such ε a local continuous family of m-periodic isolated orbits could bifurcate from the fixed point This will happen as long as some of the resonant terms of the map’s normal form do not vanish, i.e. the map is not analytically equivalent to a rotation. Since having all zero resonant terms is an extremely special property for maps with root-of-unity multiplier, we can expect that the Poincare transformations for most foliations of the form (1) will have a lot of isolated periodic orbits and the foliations themselves will have many multi-fold limit cycles. This fact imposes a challenge since the connection between the polynomial foliation and its Poincare transformation is implicit and indirect
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
