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

José Mujica; Bernd Krauskopf; Hinke M. Osinga

doi:10.3934/jcd.2017005

Article Contents

2017, Volume 4: 143-165. Doi: 10.3934/jcd.2017005

This issue Previous Article Set-oriented numerical computation of rotation sets Next Article Addendum to "Optimal control of multiscale systems using reduced-order models"

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

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand

* Corresponding author: b.krauskopf@auckland.ac.nz
* Corresponding author: b.krauskopf@auckland.ac.nz

Published: March 2018

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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.

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Mathematics Subject Classification: Primary: 34A26, 65L10; Secondary: 37C10, 65L11.

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Figure 1. Lin's method setup for finding canard orbits in system (16) with $\mu = 8.5$. Panel (a1) shows $S^a_{\varepsilon}$ (red surface) and $S^r_{\varepsilon}$ (blue surface) computed from $L^a$ and $L^r$, respectively, up to $\Sigma$. The initial orbit segments $u_a$ (red curve) and $u_r$ (blue curve) are each other's symmetric counterparts and define the Lin space $Z = span\{(0, 0, 1)\}$ (dark-gray line) that defines the Lin gap $\eta$. Panel (b1) shows the situation when the Lin gap is closed and the canard orbit $\xi_1$ (orange) is detected. The relevant objects in $\Sigma$ are shown in panels (a2) and (b2), respectively.

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Figure 2. Illustration of the Lin's method approach to detect canard orbits for (16) with $\mu = 8.5$ in the section $\Sigma$, which is the $(x, z)$-plane. Shown are the intersection sets $\widehat{S}^a_{\varepsilon}$ (red curve) and $\widehat{S}^r_{\varepsilon}$ (blue curve), together with the Lin space $Z$ (vertical dark-gray line). Panels (a1) and (a2) show the detection of the canard orbit $\xi_0$ (cyan), and panels (b1) and (b2) show the detection of $\xi_1$ (orange).

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Figure 3. Three-dimensional view of the slow manifolds $S^a_{\varepsilon}$ and $S^r_{\varepsilon}$, and all canard orbits $\xi_0$-$\xi_4$ of the normal form (16) for $\mu = 8.5$.

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Figure 4. Illustration of the Lin's method approach for (16) with $\mu = 8.5$ and a Lin space $Z$ in general position. Panel (a1) shows when $Z$ becomes tangent to $\widehat{S}^a_{\varepsilon}$ as the end points tracing $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$ move to the right, and panel (a2) shows that it is not possible to detect a canard orbit by keeping $Z$ fixed. Panels (b1) and (b2) show a similar situation when the end points tracing $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$ move to the left.

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Figure 5. Three-dimensional view of the slow manifolds computed up to section $\Sigma \subset \{z = -0.8\}$ of the Koper model (19) for the parameters values given by (20).

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Figure 6. Illustration of the Lin's method approach to detect canard orbits of (19) with parameter values as (20) in section $\Sigma$, represented by the $(x, y)$-plane. Shown are the intersection sets $\widehat{S}^a_{\varepsilon}$ (red curve) and $\widehat{S}^r_{\varepsilon}$ (blue curve), together with the corresponding Lin space (dark-gray line). Panel (a1) shows the detection of the canard orbit $\xi_0$ (cyan) and panel (a2) shows a tangency of the Lin space $Z_0$ with $\widehat{S}^r_{\varepsilon}$. Panels (b1) and (b2) show the detection of $\xi_1$ (orange) and $\xi_2$ (green), respectively.

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Figure 7. Intermediate step (Ⅲ) for the detection of a simultaneous tangency of the Lin space with $\widehat{S}^a_{\varepsilon}$ and $\widehat{S}^r_{\varepsilon}$, for the Koper model (19) with parameters as in (20). Panel (a1) shows the detection of the points defining the Lin space $Z_1$ and panel (a2) shows the corresponding fold of $\beta_a$. Panels (b1) and (b2) show step (Ⅲ) for the detection of the points defining the Lin space $Z_2$.

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Figure 8. Slow manifolds and the canard orbits $\xi_0$-$\xi_5$ of the Koper model (19) for the parameter values (20).

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