A Lin's method approach for detecting all canard orbits arising from a folded node

José Mujica,Hinke M Osinga,Bernd Krauskopf

doi:10.3934/jcd.2017005

Abstract

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

Highlights

Slow-fast systems arise as mathematical models in various applications and they describe physical phenomena, such as chemical reactions, nonharmonic oscillations, spiking and bursting [8, 21, 29, 30, 34, 42, 44, 51]; their study has been an active area of research
Note that canard orbits cannot be found analytically and must be computed numerically, which explains the keen interest in detecting them. While they may arise more widely, canard orbits in R3 are closely related with folded singularities, which are singularities of the slow flow that are located on fold curves of the critical manifold
We describe the numerical techniques we use for the computation of slow manifolds via the continuation of solutions to a 2PBVP implemented in AUTO [10], as well as the numerical setup for the detection of canard orbits with Lin’s method

Summary

Introduction

Slow-fast systems arise as mathematical models in various applications and they describe physical phenomena, such as chemical reactions, nonharmonic oscillations, spiking and bursting [8, 21, 29, 30, 34, 42, 44, 51]; their study has been an active area of research. The intersection points of the two intersection curves of the attracting and repelling slow manifolds are found in the cross section by inspection When this computation is performed with a continuation and boundary-value-problem setup, a correction step can be used to find canard orbits accurately [7, 8]. We simultaneously compute and continue two orbit segments — one on the attracting slow manifold and one on the repelling slow manifold — up to a section Σ through the folded node These two orbit segments are coupled by requiring that their end points lie along a fixed direction. The approach has been successfully applied in the slow-fast context; for detecting so-called connecting canard orbits arising as codimension-zero intersections between the twodimensional unstable manifold of a saddle-focus equilibrium and a two-dimensional repelling slow manifold in a model with a singular Hopf bifurcation [45].

Background

Discussion