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

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

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

    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.

    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).

    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 $\mu = 8.5$.

    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.

    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).

    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.

    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 $Z_2$.

    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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