Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model. Publié en ligne: 10 nov. 2020 Pages: 239 - 256 Reçu: 19 janv. 2020 Accepté: 24 févr. 2020 © 2020 René Lozi et al., published by Sciendo This work is licensed under the Creative Commons Attribution 4.0 International License.
Figure 1 Canard explosion of the van der Pol oscillator for ɛ = 0.02 happens in an exponentially small parameter interval near a = a* ≈ 0.997461066999 where the transition from relaxation oscillations for (a) a < 0.997461066999, to small-amplitude limit cycles for (c) a ≥ 0.997461066999, comes via (b) canard cycles. Figure 2 Electrical circuit equivalent to the Fitzhugh–Nagumo model. Figure 3 HLB curve in the (b, α) parameter space. Figure 4 Fractional-order fast and slow subsystems of system (10). Single arrows indicate slow motions along the slow curve S0. Double arrows indicate fast motions outside S0, which possesses two attracting branches, Sa, and one repelling branch, Sr, separated by fold points (red) of the slow curve, corresponding to saddle-node bifurcation points of the fast subsystem. Figure 5 HLB curve in the (b, α) parameter space. Figure 6 Canard solutions observed from the fractional-order system (10): (a) Phase portrait for (b, α) = (0.7974389863166, 0.9512805068416). (b) Time evolution of x for (b, α) = (0.7974389863166, 0.9512805068416). (c) Phase portrait for (b, α) = (0.7974389863168, 0.9512805068415). (d) Time evolution of x for (b, α) = (0.7974389863168, 0.9512805068415). Figure 7 Repetitive patterns of MMO 15 − 15 − 16 and 15 − 16 for α = 0.9490476218825, b = 0.8019047562350 Figure 8 Repetitive patterns of MMO 14 − 13 and 14 − 14 for α = 0.9512573981986, b = 0.7974852036028 Figure 9 Nonidentical MMO 41, 31, 21, 11 for α = 0.9608101829227, b = 0.7783796341546, red (x0, y0) = (−0.94, −0.27), blue (x0, y0) = (−0.94, −0.26). Some canard explosion parameter sub-segments:
CEPS=[(b¯i,α¯i) (b¯i+2×10−13,α¯i+10−13)]CEPS = [({\bar b.i},{\bar \alpha .i})\;({\bar b.i} + 2 \times {10^{- 13}},{\bar \alpha .i} + {10^{- 13}})]
. i = 1, 2,..., 13, with their corresponding NSAO, and tf, determined using GLCESA as both parameters b and α are varied. NSAO (α , b )t f (α , b )α ¯ i {\bar \alpha _i} b ¯ i {\bar b_i} 14 702.59 0.9460520040915 0.8078959918168 13 669.43 0.9462002574928 0.8075994850142 12 637.49 0.9463712718829 0.8072574562340 11 609.12 0.9465701755015 0.8068596489968 10 955.08 0.9468114549302 0.8063770901394 9 500.67 0.9470807229186 0.8058385541626 8 469.37 0.9474164241791 0.8051671516416 7 436.78 0.9478315727039 0.8043368545920 6 365.56 0.9483577759722 0.8032844480554 5 332.4 0.9490323331705 0.8019353336588 4 300.35 0.9499488046301 0.8001023907396 3 227.11 0.9512805068416 0.7974389863166 2 189.64 0.9532285469108 0.7935429061782 1 156.56 0.9564716090280 0.7870567819438 0 204.82 0.9608101829226 0.7783796341546