# Quadratic Planar Differential Systems with Algebraic Limit Cycles via Quadratic Plane Cremona Maps

## TL;DR

This paper introduces birational transformations via quadratic plane Cremona maps to generate and analyze quadratic differential systems with algebraic limit cycles, including new families with degree 5 cycles and phase portraits.

## Contribution

It presents a novel method to produce quadratic systems with algebraic limit cycles using quadratic plane Cremona maps, expanding known families and providing detailed phase portraits.

## Key findings

- New family of quadratic systems with algebraic limit cycle of degree 5
- Known families with algebraic limit cycles of degree > 4 derived using transformations
- Phase portraits of all families on the Poincaré disk

## Abstract

In this paper we show how we can transform quadratic systems into new quadratic systems after some kind of birational transformations, the quadratic plane Cremona maps. We afterwards apply these transformations to the families of quadratic differential systems having an algebraic limit cycle. As a consequence, we provide a new family of quadratic systems having an algebraic limit cycle of degree 5. Moreover we show how the known families of quadratic differential systems having an algebraic limit cycle of degree greater than four are obtained using these transformations. We also provide the phase portraits on the Poincar\'e disk of all the families of quadratic differential systems having algebraic limit cycles.