Phase response to arbitrary perturbations: Geometric insights and resetting surfaces

Figure 1.  Phase resets with $ \boldsymbol{d} = [1, 0] $ for $ A = 0.5 $, $ A = 1 $, and $ A = 1.5 $ in Winfree's model (7) with $ \varepsilon = 0 $. Panel (a) shows $ \Gamma $ (black) and the three perturbation sets $ \Gamma_{0.5} $ (orange), $ \Gamma_{1} $ (red), and $ \Gamma_{1.5} $ (purple) together with 20 isochrons uniformly distributed in phase, which are coloured in increasingly darker shades from 0 (cyan) to 1 (dark blue). The resulting PRCs and PTCs are shown in matching colours in panels (b) and (c), respectively; the vertical line (grey) at $ \vartheta_{{\rm{o}}} = 0.5 $ represents the discontinuity for $ A = 1 $, and the shaded unit square (green) in panel (c) represents $ \mathbb{S}^1 \times \mathbb{S}^1 $

Figure 2.  Directional resets of system (7) with $ \varepsilon = 0 $ at $ \gamma_{0.125} $ for $ A = 0.5 $, $ A = 1 $, and $ A = 1.5 $ with $ \boldsymbol{d} = [\cos{(2 \pi\, \varphi_{{\rm{d}}})}, \, \sin{(2 \pi\, \varphi_{{\rm{d}}})}] $ and $ \varphi_{{\rm{d}}} \in \mathbb{S}^1 $. Panel (a) shows $ \Gamma $ (black) and the three perturbation sets $ C_{0.5} $ (orange), $ C_{1} $ (red), and $ C_{1.5} $ (purple) centred at $ C_0 = \gamma_{0.125} $ (black dot) together with 20 isochrons uniformly distributed in phase, coloured from 0 (cyan) to 1 (dark blue). The resulting DTCs in matching colours are shown in the $ ( \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-plane in panel (b) and on the torus in panel (c); the discontinuity for $ A = 1 $ occurs at $ \varphi_{{\rm{d}}} = 0.625 $ (grey line). Compare with Fig. 1

Figure 3.  Resetting surface $ \mathrm{graph}{( \mathcal{P}_A)} $ of system (7) with $ \varepsilon = 0 $ for $ A = 0.5 $ in panel (a), $ A = 1 $ in panel (b), and $ A = 1.5 $ in panel (c), shown in $ ( \vartheta_{{\rm{o}}}, \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-space for $ \vartheta_{{\rm{n}}} \in [-1.5, 1.5] $ with the corresponding DTCs for $ \vartheta_{{\rm{o}}} = 0.125 $, labelled $ C_{0.5} $ (orange), $ C_{1} $ (red), and $ C_{1.5} $ (purple), respectively

Figure 4.  Resetting surface of system (7) with $ \varepsilon = 0 $ for fixed $ \vartheta_{{\rm{o}}} = 0.125 $. Shown in panel (a) are two copies of the surface in $ ( \varphi_{{\rm{d}}}, A, \vartheta_{{\rm{n}}}) $-space over the range $ \vartheta_{{\rm{n}}} \in [-0.5, 1.5] $, as well as the corresponding three DTCs for $ A = 0.5 $, $ A = 1 $ and $ A = 1.5 $. The surface is discontinuous at the vertical line $ \{ \varphi_{{\rm{d}}} = 0.625, \, A = 1 \} $ (grey); compare with Fig. 2(b). Panel (b) shows the associated isochron surface in $ (x, y, \vartheta) $-space near $ \boldsymbol{0} $ over the range $ \vartheta \in [-0.5, 1.5] $, with the 20 straight isochrons from Fig. 2(a)

Figure 5.  Phase resets with $ \boldsymbol{d} = [1, 0] $ for $ A = 0.5 $, $ A = 1 $, and $ A = 1.5 $ in Winfree's model (7) with $ \varepsilon = -1 $. Panel (a) shows $ \Gamma $ (black), 20 isochrons uniformly distributed in phase, coloured from 0 (cyan) to 1 (dark blue), and $ \Gamma_{0.5} $ (orange), $ \Gamma_{1} $ (red), and $ \Gamma_{1.5} $ (purple). The resulting PRCs and PTCs are shown in matching colours in panels (b) and (c), respectively; the discontinuity for $ A = 1 $ occurs at $ \vartheta_{{\rm{o}}} = 0.5 $, and the shaded unit square (green) in panel (c) represents $ \mathbb{S}^1 \times \mathbb{S}^1 $. Compare with Fig. 1

Figure 6.  Directional resets of system (7) with $ \varepsilon = -1 $ at $ \gamma_{0.125} $ for $ A = 0.5 $, $ A = 1 $, and $ A = 1.5 $. Panel (a) shows $ \Gamma $ (black) with 20 isochrons uniformly distributed in phase, coloured from 0 (cyan) to 1 (dark blue), and $ C_0 = \gamma_{0.125} $ (black dot), $ C_{0.5} $ (orange), $ C_{1} $ (red), and $ C_{1.5} $ (purple). The resulting DTCs in matching colours are shown in the $ ( \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-plane in panel (b) and on the torus in panel (c); the discontinuity for $ A = 1 $ occurs at $ \varphi_{{\rm{d}}} = 0.625 $. Compare with Figs. 2 and 5

Figure 7.  Resetting surface $ \mathrm{graph}{( \mathcal{P}_A)} $ of system (7) with $ \varepsilon = -1 $ for $ A = 0.5 $ in panel (a), $ A = 1 $ in panel (b), and $ A = 1.5 $ in panel (c), shown in $ ( \vartheta_{{\rm{o}}}, \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-space for $ \vartheta_{{\rm{n}}} \in [-1.5, 1.5] $ with the corresponding DTCs for $ \vartheta_{{\rm{o}}} = 0.125 $; compare with Fig. 3

Figure 8.  Resetting surface of system (7) with $ \varepsilon = -1 $ for fixed $ \vartheta_{{\rm{o}}} = 0.125 $, shown in panel (a) in $ ( \varphi_{{\rm{d}}}, A, \vartheta_{{\rm{n}}}) $-space with the three DTCs for $ A = 0.5 $, $ A = 1 $ and $ A = 1.5 $ from Fig. 6. Panel (b) shows the associated isochron surface in $ (x, y, \vartheta) $-space near $ \boldsymbol{0} $, with the 20 spiralling isochrons from Fig. 6(a). Compare with Fig. 4

Figure 9.  First twin tangency for $ 0 < A < 1 $ of the DTC with $ \vartheta_{{\rm{o}}} = 0.125 $ of system (7) with $ \varepsilon = -1 $. Panel (a) shows $ \Gamma $ (black) with $ C_0 = \gamma_{0.125} $ (black dot) and the perturbation sets $ C_A $ for $ A = 0.7908 $ (orange), $ A = 0.8408 $ (magenta), and $ A = 0.8908 $ (purple), which have tangencies, respectively, with the isochrons $ I_{0.2658} $ (navy) and $ I_{0.5101} $ (brown), with $ I_{0.2738} $ (olive) twice, and with $ I_{0.2818} $ (dark green) and $ I_{0.8184} $ (light green); the tangency points are marked. Panel (b1) shows the corresponding DTCs for $ \vartheta_{{\rm{n}}} \in \mathbb{R} $, and the highlighted rectangle is enlarged in panel (b2); here the unit square (green shading) represents $ \mathbb{S}^1 \times \mathbb{S}^1 $, and the $ \vartheta_{{\rm{n}}} $-values of the extrema of the DTCs as indicated by horizontal lines (black)

Figure 10.  The second and third twin tangencies for $ 0 < A < 1 $ of the DTC with $ \vartheta_{{\rm{o}}} = 0.125 $ of system (7) with $ \varepsilon = -1 $. Panels (a) and (c) show $ C_A $ (magenta) and the twin tangency isochron (olive) with $ k = 2 $ at $ A = 0.9201 $, and with $ k = 3 $ at $ A = 0.9468 $, respectively; panels (b1) and (d1) and the enlargements in panels (b2) and (d2) show the corresponding DTCs on the unit square representing $ \mathbb{S}^1 \times \mathbb{S}^1 $

Figure 11.  Twin tangencies for $ A > 1 $ of the DTC with $ \vartheta_{{\rm{o}}} = 0.125 $ of system (7) with $ \varepsilon = -1 $, shown in the style of Fig. 10 for $ \ell = 3 $ at $ A = 1.0458 $ in panels (a) and (b), and for $ \ell = 2 $ at $ A = 1.0645 $ in panels (c) and (d)

Figure 12.  Last twin tangency for $ A > 1 $ of the DTC with $ \vartheta_{{\rm{o}}} = 0.125 $ of system (7) with $ \varepsilon = -1 $, where $ C_A = {1.1092} $ is tangent twice to $ I_{0.3175} $ (olive), which is shown with $ C_A $ for $ A = 1.1092 \pm 0.03 $ in the style of Fig. 9

Figure 13.  Transition through a cubic tangency at $ A = 1.1092 $ of the DTC with $ \vartheta_{{\rm{o}}} = 0.125 $ of system (7) with $ \varepsilon = -1 $. Shown in panel (a1) and the enlargement panel (a2) are $ \Gamma $ (black), 20 isochrons uniformly distributed in phase, $ C_0 = \gamma_{0.125} $, the perturbations sets $ C_A $ for $ A = 1.3 $ (orange), $ A = 1.4931 $ (magenta), and $ A = 1.7 $ (purple), and additional isochrons that have tangencies with them. Panel (b) shows the corresponding DTCs in matching colours for $ \vartheta_{{\rm{n}}} \in \mathbb{R} $, where the unit square (shaded green) represents $ \mathbb{S}^1 \times \mathbb{S}^1 $ and the horizontal lines (black) indicate extrema

Figure 14.  Periodic orbit $ \Gamma $ and isochron geometry of the Van der Pol system (14) with $ \mu = 1 $. Panel (a) shows $ \Gamma $ (black) with 12 isochrons evenly distributed in phase. Panel (b) shows the critical transition amplitude $ A_{{\rm{c}}} $ as a function of $ \vartheta_{{\rm{o}}} $, and panel (c) shows it as a function of the angle $ \varphi_{{\rm{d}}} $ of the associated direction $ \boldsymbol{d}( \varphi_{{\rm{d}}}) $

Figure 15.  Phase resets with $ \boldsymbol{d} = [1, 0] $ for $ A = 0.2 $, $ A = A_{{\rm{c}}} \approx 2.0086 $, and $ A = 3.8 $ in the Van der Pol system (14) with $ \mu = 1 $. Panel (a) shows $ \Gamma $ (black), 12 isochrons uniformly distributed in phase, coloured from 0 (cyan) to 1 (dark blue), and $ \Gamma_{0.2} $ (orange), $ \Gamma_{2.0086} $ (red), and $ \Gamma_{3.8} $ (purple). The resulting PTCs are shown in matching colours in panel (b); the discontinuity for $ A = A_{{\rm{c}}} $ is at $ \vartheta_{{\rm{o}}} = 0.5 $, and the shaded unit square (green) represents $ \mathbb{S}^1 \times \mathbb{S}^1 $

Figure 16.  Directional resets of system (14) with $ \mu = 1 $ at $ \gamma_{0} $ for $ A = 0.5 $, $ A = A_{{\rm{c}}} \approx 2.0086 $, and $ A = 3.5 $. Panel (a) shows $ \Gamma $ (black), 12 isochrons uniformly distributed in phase, coloured from 0 (cyan) to 1 (dark blue), and $ C_0 = \gamma_{0} $ (black dot) $ C_{0.5} $ (orange), $ C_{2.0086} $ (red), and $ C_{3.5} $ (purple). The resulting DTCs in matching colours are shown in the $ ( \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-plane in panel (b); the discontinuity for $ A = A_{{\rm{c}}} $ is at $ \varphi_{{\rm{d}}} = 0.625 $. Compare with Fig. 15

Figure 17.  Resetting surface $ \mathrm{graph}{( \mathcal{P}_A)} $ of system (14) with $ \mu = 1 $, shown in $ ( \vartheta_{{\rm{o}}}, \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-space for $ \vartheta_{{\rm{n}}} \in [-2, 1.5] $. Panel (a) is for $ A = 1.3 $, (b) for $ A = 1.5317 $, (c) for $ A = 2 $, (d) for $ A = 2.3 $, (e) for $ A = 2.8299 $, and (f) for $ A = 3 $. Vertical lines (grey) represent the singularities $ f_1 $ and $ f_1^* $ in (b), $ s_1 $, $ s_2 $, $ s_1^* $, $ s_2^* $ in (c) and (d), and $ f_2 $ and $ f_2^* $ in (e); see Fig. 14(b) and (c), and compare with Fig. 7

Figure 18.  Enlargement of $ {\rm{graph}}( \mathcal{P}_A) $ for $ A = 2 $ from Fig. 17(c) near the singularities $ s_1 $ and $ s_2 $ (grey vertical lines), shown for $ \vartheta_{{\rm{o}}} \in [-0.12, 0.48] $, $ \varphi_{{\rm{d}}} \in [0.05, 0.65] $ and $ \vartheta_{{\rm{n}}} \in [-1.9, 0.7] $. Anticlockwise loops around either $ s_1 $ or $ s_2 $ in the $ ( \vartheta_{{\rm{o}}}, \varphi_{{\rm{d}}}) $-plane lift, respectively, to downward and upward helices on $ {\rm{graph}}( \mathcal{P}_A) $ (red curves), while loops around both $ s_1 $ and $ s_2 $ lift to closed loops on $ {\rm{graph}}( \mathcal{P}_A) $ (blue curves)

Figure 19.  Resetting surface of system (14) with $ \mu = 1 $ for fixed $ \vartheta_{{\rm{o}}} = 0 $, shown in panel (a) in $ ( \varphi_{{\rm{d}}}, A, \vartheta_{{\rm{n}}}) $-space with the DTCs for $ A = 0.5 $, $ A = 2.0086 $ and $ A = 3.5 $ from Fig. 16. Panel (b) shows the associated isochron surface in $ (x, y, \vartheta) $-space near $ \boldsymbol{0} $, with the 12 spiralling isochrons from Fig. 14(a). Compare with Fig. 8

Figure 20.  Twin tangency at which the DTC for $ \vartheta_{{\rm{o}}} = 0 $ of system (14) with $ \mu = 1 $ becomes surjective, and nearby cubic tangency. Panel (a) shows in the $ (x, y) $-plane that $ C_{1.8444} $ (magenta) has two tangencies with $ I_{0.3021} $ (olive), and $ C_{1.623} $ (orange) has a cubic tangency with $ I_{0.9731} $ (brown). The corresponding DTCs are shown in the $ ( \varphi_{{\rm{d}}}, \vartheta_{{\rm{n}}}) $-plane in panel (b)

Figure 21.  DTCs for $ \vartheta_{{\rm{o}}} = 0 $ of system (14) with $ \mu = 1 $ extremely close to and on either side of the critical transition value $ A_{{\rm{c}}} $. Panel (a) shows the DTC in the $ ( \varphi_{{\rm{d}}} , \vartheta_{{\rm{n}}}) $-plane for $ A = 2.00861986 $ just before the first twin tangency with $ k = 1 $, and panel (b) shows it for $ A = 2.00861987 $ at the last twin tangency with $ \ell = 1 $. Compare with Fig. 9(b1) and Fig. 12(b1), respectively

Figure 22.  Schematic of the BVP setup with the periodic orbit $ \Gamma $ (black) of the Van der Pol system (14) with $ \mu = 1 $, showing the orbit segment $ \boldsymbol{u} $ (magenta) with begin point $ \boldsymbol{u}(0) = \gamma_{ \vartheta_{{\rm{o}}}} + A \, \boldsymbol{d}( \varphi_{{\rm{d}}}) $ (here for $ \vartheta_{{\rm{o}}} = 0.5 $, $ \varphi_{{\rm{d}}} = 0 $ and $ A \approx 1.8871 $) and end point $ \boldsymbol{u}(1) $ on the stable vector $ \mathbf{v}_{ \vartheta_{{\rm{n}}}} $ (blue) at $ \gamma_{ \vartheta_{{\rm{n}}}} $ (here for $ \vartheta_{{\rm{n}}} = 0.15 $), and the orbit segment $ \boldsymbol{w} $ (grey) with $ \boldsymbol{w}(0) = \gamma_{0} $ and $ \boldsymbol{w}(1) = \gamma_{ \vartheta_{{\rm{n}}}} $