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Discrete geometric singular perturbation theory

  • *Corresponding author: Samuel Jelbart

    *Corresponding author: Samuel Jelbart 

SJ and CK acknowledge funding from the SFB/TRR 109 Discretization and Geometry in Dynamics grant. CK thanks the VolkswagenStiftung for support via a Lichtenberg Professorship

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  • We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established geometric singular perturbation theory for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold $ S $. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of $ S $. The persistence of the critical manifold $ S $, local stable/unstable manifolds $ W^{s/u}_{loc}(S) $ and foliations of $ W^{s/u}_{loc}(S) $ by stable/unstable fibers is described in detail. The practical utility of the resulting discrete geometric singular perturbation theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincaré maps, the geometry and dynamics of which can be described in detail using DGSPT.

    Mathematics Subject Classification: Primary: 37C05, 37D10, 37C86; Secondary: 34D15, 37C15.

    Citation:

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  • Figure 1.  Examples of normally hyperbolic critical manifolds $ S $ in dimension $ n = 3 $. In (a)-(d) $ \dim S = 2 $ and there exists a single non-trivial multiplier $ |\mu| \neq 1 $. There are four generic possibilities depending on $ \mu $: (a) attracting and orientation preserving; (b) attracting and orientation reversing; (c) repelling and orientation preserving; (d) repelling and orientation reversing. In (e)-(f) $ S $ is saddle-type with $ \dim S = 1 $ and two non-trivial multipliers $ |\mu_a| < 1 $ and $ |\mu_r| > 1 $. Two of four generic possibilities are shown: (e) orientation preserving along stable/unstable fibers $ w^{s/u}_{loc}(z) $; (f) orientation reversing along $ w^s_{loc}(z) $

    Figure 2.  Stable/unstable manifolds $ W^{s/u}_{loc}(S_n) $ associated to a normally hyperbolic critical manifold $ S_n $ are foliated by lower dimensional stable/unstable manifolds $ w^{s/u}_{loc}(z) $ associated to base points $ z \in S_n $. The case of a $ 1- $dimensional saddle-type critical manifold in dimension $ n = 3 $ is shown here ($ S_n $ is shown in green). In this case the manifolds $ W^{s/u}_{loc}(S_n) $ (shown in blue/red) are $ 2- $dimensional and foliated by the $ 1- $dimensional stable/unstable manifolds $ w^{s/u}_{loc}(z) $ associated to base points $ z \in S_n $. Representative stable/unstable fibers $ w^{s/u}(z_i) $ with base points $ z_i \in S_n $ for $ i = 1, 2 $ are also shown

    Figure 3.  Action of the oblique projection operator $ \Pi^{S_n}_{\mathcal N} $ along $ \mathcal N $ onto $ TS_n $ in the case of a $ 1- $dimensional attracting critical manifold $ S_n $ (green). The splitting (18) implies that the leading order perturbation $ G(z, 0) $ at each point $ z \in S_n $ can be decomposed into components in the tangent bundle $ T_z S_n $, and the (transverse) linear fast fiber $ \mathcal N_z $ tangent to the nonlinear fast fiber $ w^s_{loc}(z) $ shown in blue in the figure. The component in $ T_zS_n $ is given by $ \Pi_{\mathcal N}^{S_n}G(z, 0) \in T_zS_n $, and sufficient to determine the reduced map (24), see also Proposition 2.16

    Figure 4.  Reduced and $ m $'th iterate maps on $ S_{n, \varepsilon} $ defined in equations (24) and (26) respectively. The normally hyperbolic critical manifold and the corresponding perturbed slow manifold are shown again in green and shaded green respectively. The dynamics of the map (10) within $ S_{n, \varepsilon} $ are governed to leading order in $ \varepsilon $ by the reduced map (24). Iterates are generically separated by distances of $ O( \varepsilon) $, as shown in (a). Iterates of the $ m $'th iterate reduced map (26) shown in (b) can be separated by $ O(1) $ distances if the number of iterates $ m $ is comparable to $ \varepsilon^{-1} $. For both reduced and $ m $'th iterate maps, non-trivial dynamics is only possible for $ 0 < \varepsilon \ll 1 $. See however Remark 2.22, which describes a possible approach to identifying non-trivial dynamics on $ S $ for $ \varepsilon = 0 $ in the dual limit $ ( \varepsilon, m) \to (0, \infty) $

    Figure 5.  Fast-slow maps in the standard form (30) can be characterised geometrically as the subclass of fast-slow maps in general form (10) for which the fast foliation has been or can be globally rectified. Such a situation is sketched in (a). In (b) we show an example of a fast-slow map with an isolated fixed point $ Q $. Such a map cannot be written in standard form (30) since the fast foliation cannot be globally rectified

    Figure 6.  A normally hyperbolic critical manifold $ S $ for the map (36) perturbs to an $ O( \varepsilon)- $close locally invariant slow manifold $ S_ \varepsilon $ for $ 0 < \varepsilon \ll 1 $. This is described in Theorem 3.1. Dynamics within $ S_ \varepsilon $ are governed by the map (21), i.e. approximated to leading order in $ \varepsilon $ by the reduced map (24). Several iterates $ \bar z_ \varepsilon, \bar z_ \varepsilon^2, \ldots , \bar z_ \varepsilon^6 \in S_ \varepsilon $ starting from a point $ z_ \varepsilon \in S_ \varepsilon $ are shown for illustrative purposes. Theorem 3.3 provides further details for the case in which $ S $ is has a graph representation $ y = \varphi_0(x) $, in which case $ S_ \varepsilon $ can be calculated up to $ O( \varepsilon^2) $ using the formula (38)

    Figure 7.  The stable/unstable manifolds $ W^{u/s}_{loc}(S) $ (here in shaded blue/red) of a normally hyperbolic critical manifold $ S $ (shaded green) perturb to $ O( \varepsilon)- $close positively/negatively invariant manifolds $ W^{s/u}_{loc}(S_ \varepsilon) $ (blue/red) which intersect along the slow manifold $ S_ \varepsilon $ (green), see Theorem 3.4

    Figure 8.  Positive invariance of the perturbed stable fibers $ \{w_{loc}^s(z_ \varepsilon)\}_{z_ \varepsilon \in S_ \varepsilon} $ of a $ 1- $dimensional critical manifold in dimension $ n = 2 $. By the invariance property in Theorem 3.5 (i), initial conditions $ q_1, q_2 $ in the stable fiber $ w^s_{loc}(z_{ \varepsilon, 1}) $ with base point $ z_{ \varepsilon, 1} \in S_ \varepsilon $ are mapped by $ H $ into the stable fiber $ w^s_{loc}(z_{ \varepsilon, 2}) $, then $ w^s_{loc}(z_{ \varepsilon, 3}) $, with base points $ z_{ \varepsilon, 2} $ and $ z_{ \varepsilon, 3} $ corresponding to iterates of $ z_{ \varepsilon, 1} $. Moreover, iterates are exponentially contracted towards their base points on $ S_ \varepsilon $ by Theorem 3.5 (iii). In (a): the orientation preserving case in which $ S $ has a non-trivial multiplier $ \mu \in (0, 1) $. In (b): the orientation reversing case with $ \mu \in (-1, 0) $

    Figure 9.  Geometry and dynamics of the fast-slow Chialvo map (77) in the singular limit $ \varepsilon = 0 $, with parameter values $ \tilde a = 1 $, $ \tilde b = 5 $, $ \tilde c = 3.5 $ and $ k = 0.035 $. The critical manifold $ S $ has a cubic-like profile, with normally hyperbolic and attracting (repelling) critical manifolds $ S^a_{\pm} $ ($ S^r_{\pm} $) shown in blue (red) separated by non-hyperbolic fold points $ F_\pm $ (orange) and a supercritical flip point $ Q $ (green); see Lemma 5.4. Iterates of the layer map (79) along vertical fast fibers (shaded grey) are sketched along with the direction of iteration, in order to illustrate the fast dynamics. The reduced map, i.e. (82) truncated at $ O( \varepsilon^2) $, has a unique unstable fixed point $ p_\ast \in S_-^r $, see Proposition 5.10. The direction of 'slow iteration' under the reduced map is indicated with single arrows. Also shown in cyan is the (numerically computed) curve of period-2 points $ S^2 $ bifurcating from the flip point $ Q $, as well as numerically identified flip bifurcations $ Q_{\pm}^2 $ in the second iterate map $ v \mapsto \bar v^2 $. A period-doubling route to chaos occurs with increasing $ w $; see [71] and Remark 5.6

    Figure 10.  Global dynamics of the fast-slow Chialvo map (77) for $ 0 < \varepsilon \ll 1 $. Singular structure for $ \varepsilon = 0 $ including critical manifolds $ S_{\pm}^a $, $ S_{\pm}^r $, the fold, flip and equilibrium points $ F_{\pm} $, $ Q $ and $ p_\ast $ respectively are shown overlaid with numerically computed iterates of (77) with initial condition $ (w, v) = (1/4, 2) $, $ \varepsilon = 10^{-3} $, $ \tilde a = 1 $, $ \tilde b = 5 $ and varying values of $ \tilde c, k $. The 'slow nullcline' $ \{g(w, v, 0) = 0\} $ is also shown in shaded grey. Dynamics away from the non-normally hyperbolic points $ F_\pm $ and $ Q $ is described by the theory in Sections 2 and 3. We identify four distinct behaviours depending on $ (\tilde c, k) $, which are described in the text. (a) Case I: Excitability, with $ (\tilde c, k) = (7, 0.07) $. (b) Case II: Relaxation, with $ (\tilde c, k) = (3.5, 0.07) $. (c) Case III: Non-chaotic bursting, with $ (\tilde c, k) = (3.5, 0.035) $. (d) Case IV: Chaotic bursting, with $ (\tilde c, k) = (3.5, 0.02) $

    Figure 11.  Setup for $ \varepsilon = 0 $. The layer problem (99) has a manifold of hyperbolic limit cycles $ \mathcal M $, shown here in shaded purple, which contains the limit cycle $ \Gamma_{\alpha_\ast} $ and extends over the open $ \alpha- $interval $ V_\alpha \ni \alpha_\ast $. A Poincaré map $ P_\Delta : \Delta \to \Delta $ on the cross-section $ \Delta $ shown in shaded green is induced by the flow of the layer problem (99). Hyperbolicity of the limit cycles $ \Gamma_\alpha $ implies that the 1-dimensional submanifold $ S_\Delta = \mathcal M \cap \Delta $ (bold green) is a normally hyperbolic critical manifold for $ P_\Delta $, see Lemma 5.22

    Figure 12.  Perturbation of the singular geometry and dynamics for $ \varepsilon = 0 $ (left) for $ 0 < \varepsilon \ll 1 $ (right). We sketch the 3-dimensional case of a normally hyperbolic and attracting limit cycle manifold $ \mathcal M $. Theorem 5.23 describes the persistence properties for the Poincaré map $ P_{\Delta} $ on an arbitrary cross-section $ \Delta $. Three representative stable fibers on $ \Delta $ for $ \varepsilon = 0 $ are shown in blue on the left, which perturb to invariant stable fibers for $ 0 < \varepsilon \ll 1 $ (also in blue) on the right, e.g. $ w_{loc}^s(z) $ perturbs to $ w_{loc}^s(z_ \varepsilon) $. A number of example iterates are shown on $ \Delta $ for both $ \varepsilon = 0 $ and $ 0 < \varepsilon \ll 1 $. Note that the arrows connecting points on e.g. $ w_{loc}^s(z_ \varepsilon) $ and $ w_{loc}^s({\bar z_ \varepsilon}^5) $ only show the direction of iteration (they do not indicate a single iteration). The persistence of the manifold of limit cycles $ \mathcal M $ as a nearby locally invariant 'slow manifold' $ \mathcal M_ \varepsilon $ as described by Theorem 5.23 is also shown ($ \mathcal M_ \varepsilon $ is shown in a darker shade of purple)

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