Explicit non-algebraic limit cycles for polynomial systems

Abstract

We consider a system of the form x ˙ = P n ( x , y ) + xR m ( x , y ) , y ˙ = Q n ( x , y ) + yR m ( x , y ) , where P n ( x , y ) , Q n ( x , y ) and R m ( x , y ) are homogeneous polynomials of degrees n, n and m, respectively, with n ⩽ m . We prove that this system has at most one limit cycle and that when it exists it can be explicitly found and given by quadratures. Then we study a particular case, with n = 3 and m = 4 . We prove that this quintic polynomial system has an explicit limit cycle which is not algebraic. To our knowledge, there are no such type of examples in the literature. The method that we introduce to prove that this limit cycle is not algebraic can be also used to detect algebraic solutions for other families of polynomial vector fields or for probing the absence of such type of solutions.