Canard cycles in the presence of slow dynamics with singularities

Abstract

We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve. All existing results deal with a non-zero slow movement, permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular, our study includes the cyclicity of the slow–fast two-saddle cycle, formed by a regular saddle connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow-divergence integral.

Multi-layer canard cycles and translated power functions

“…This property was supposed to hold for all contact points in M but from now on, we will restrict to an arbitrary neighborhood U of Γ and we just need the quadratic contact property for the contact points on Γ . We also suppose that the slow dynamics has no zeros on the slow curves contained in Γ (occurrence of such zeros are studied in [7]). …”

Bifurcations of multiple relaxation oscillations in polynomial Liénard equations

“…For a geometrical explanation of this notion, we refer to [DMD08]. Observe that I(Y, λ) does not depend on ε, nor on B 0 , and so the required condition formulated in Proposition 1 is a condition depending solely on f (x, λ) and G(x, λ).…”

Cyclicity of a fake saddle inside the quadratic vector fields

“…It follows, for a = 0, from results in [8] and [10] that the number of canard cycles around the contact point (x, y) = (+1, 0) is one, i.e. we cannot have more than one canard cycle in each nest.…”

“…Then, see Proposition 21, system (27) has four singularities, two of them on the line {x = 0}, which are (0, 1) and (0, −1), and the other two singularities are the intersections of the line {ax + by = 0} and the circle {x 2 + y 2 = 1}, which we denote by p 1 and p 2 . A straightforward computation of the trace and the determinant at these singularities, by using (8) and (9), and the Hartman-Grobmann Theorem imply that (0, 1) and (0, −1) are nodes, and that p 1 and p 2 are saddles.…”