Finite cyclicity of slow-fast Darboux systems with a two-saddle loop

Abstract

We prove that the cyclicity of a quadratic slow-fast integrable system of Darboux type with a double heteroclinic loop, is finite and uniformly bounded.

The number of limit cycles for regularized piecewise polynomial systems is unbounded

“…Hilbert's 16th problem remains unsolved to this day. Whereas general progress has been made on N = 2 [1,4,5,20,24,25,31,53] and on Smale's version of the problem where (1.5) is restricted to classical Liénard type: P N (x, y) = y − p N (x), Q N = −x, see [11,46,50,54], there has been an emphasis on obtaining good lower bounds on the number of limit cycles (see [12,29,30,51] and references therein). Following the work of De Maesschalck, Dumortier and Roussarie, see [14,15,19,22,23], a key tool in this effort has been the slow divergence-integral from slow-fast systems and canard theory; in particular, the roots of the slow divergence-integral provide candidates for limit cycles.…”

The number of limit cycles for regularized piecewise polynomial systems is unbounded

“…Hilbert's 16th problem remains unsolved to this day. Whereas general progress has been made on N = 2 [1,4,5,19,23,24,30,51] and on Smale's version of the problem where (1.4) is restricted to classical Liénard type: Q N (x, y) = y −q N (x), P N = −x, see [10,44,48,52], there has been an emphasis on obtaining good lower bounds on the number of limit cycles (see [11,28,29,49] and references therein). Following the work of De Maesschalck, Dumortier and Roussarie, see [13,14,18,21,22], a key tool in this effort has been the slow divergence-integral from slow-fast systems and canard theory; in particular, the roots of the slow divergence-integral provide candidates for limit cycles.…”

Finite cyclicity of the contact point in slow-fast integrable systems of Darboux type

“…The fast subsystem X 0 of X , has the curve of singularities {y − x 2 = 0}, often called the critical or slow curve, and (fast) regular horizontal orbits (see Figure 2). The goal in [1,3] was to prove -uniform finiteness of the number of limit cycles bifurcating from the compact set Ō := O in a polynomial deformation X ,δ of X , with δ = (δ 1 , . .…”

“…The deformations are smooth or analytic depending on the region in the parameter space. This article is a natural continuation of [1,3], where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the "integrable" direction in the parameter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the "slow-fast" direction in the parameter space.…”