Time analysis and entry–exit relation near planar turning points

Abstract

The paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions. We follow a geometric approach based on the use of C k -normal forms, centre manifolds and (family) blow up, as we did in (Trans. Amer. Math. Soc., to appear). In (Trans. Amer. Math. Soc., to appear) we considered the existence of manifolds of canard solutions for given appropriate boundary conditions. These manifolds need not be smooth at the turning point. In this paper we essentially study the transition time along such manifolds, as well as the divergence integral, providing a structure theorem for these integrals. As a consequence we get a nice structure theorem for the transition equation, governing the canard solutions. It permits to compare different control manifolds and to obtain a precise description of the entry-exit relation of different canard solutions. Attention is also given to the special case in which the canard manifolds are smooth, i.e. when "formal" canard solutions exist.

Existence of canards under non-generic conditions

“…In [18], Li studied the existence of multiple canard cycles for a class of planar fast-slow systems under non-generic conditions by using classical asymptotic analysis. Very recently, Maesschalck and Dumortier [20] used a geometric approach to consider canards in the following fast-slow system:…”

Cyclicity of a fake saddle inside the quadratic vector fields

“…Let X be the vector field associated to system (6). Then its linearization matrix at a point (x, y) is…”

“…The origin of this system corresponds to the point (x 0 , y 0 ) of (6). Finally, the expressions of the Lyapunov quantities V 3 and V 5 given in (10) and (11) follows after rewriting system (12) as…”

“…They consist of (a perturbation of) a fast vertical path connecting (x, y) = (cos θ 0 , sin θ 0 ) and (x, y) = (cos θ 0 , − sin θ 0 ), (for θ 0 ≈ 0 or θ 0 ≈ π) together with a small arc on the circle. The stability of the obtained periodic orbits can be computed using a slow divergence integral, see [6]. This integral is the divergence of the fast vector field, integrated along the slow arcs, and it is well-known that it can be computed in arbitrary coordinate systems.…”

Delayed Hopf bifurcation in time-delayed slow-fast systems

“…For example, with the slow state variable regarded as the bifurcation parameter, the bifurcation does not happen immediately at the bifurcation point as the bifurcation parameter passes through the bifurcation point, but at a point which is above the bifurcation point by obvious distance. Such a bifurcation, firstly reported in the study of Hopf bifurcation of a slow-fast system by Shishkova [13], is called delayed bifurcation, and has been studied by many authors [14][15][16]. A general theory of delayed bifurcation was developed by Neishtadt [17,18] on the basis of analytic continuation of solutions in the plane of complex time, and the so-called exit-point and entry-exit functions were introduced to characterize the delayed bifurcation.…”