The 16th Hilbert problem restricted to circular algebraic limit cycles

Abstract

Abstract. We prove the following two results.First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles.Second a planar polynomial vector field of degree S, admits at most S − 1 invariant circles which are algebraic limit cycles.In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S − 1 algebraic limit cycles given by circles, and this number is reached.

Limit Cycles in the Equation of Whirling Pendulum With Piecewise Smooth Perturbations

“…Later, this problem was posed again by Smale [30] in 1998 for the 21st Century. Although there are plenty of excellent articles corresponding to it, see for instance [8,11,22] and the references quoted there, this problem is still open.…”

Differential equations with a given set of solutions

“…The inverse problem for determining the ordinary differential equations with given partial and first integrals was studied in [10]. The obtained results were applied in particular (i) to construct Lagrangian mechanical systems with a given set of linear constraints with respect to the velocity and to obtain Hamiltonian systems with a given set of first integrals (see [11,20]), (ii) to solve the 16th Hilbert problem for algebraic limit cycles (see [11,12,13,14,21]), and (iii) to study the center-focus problem (see [15,16,17]).…”

“…Proposition 5. Let w j = w j (t) for j = 1, 2, 3 be solutions of the Abel differential equation of second kind (14). Then equation (14) has the invariant…”

“…As we prove in section 3, the Abel equation of first kind (27) under the change z = z 4 (t)+ 1 w , and introducing the respectively notations,becomes into the Abel equation of a second kind (28) or equivalent (14).…”

The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

“…In [4] the authors proved, first that every planar polynomial vector field of degree n with n invariant circles is Darboux integrable without limit cycles, and second that a planar polynomial vector field of degree n has at most n − 1 invariant circles as algebraic limit cycles. So, in particular, cubic systems have at most two circles as algebraic limit cycles.…”