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Mixed-Mode Oscillations Based on Complex Canard Explosion in a Fractional-Order Fitzhugh-Nagumo Model.

René Lozi

René Lozi

Laboratoire de Mathématiques J.A. Dieudonné, Université Côte d’AzurFrance
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Lozi, René
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Mohammed-Salah Abdelouahab

Mohammed-Salah Abdelouahab

Laboratory of Mathematics and Their Interactions, University Abdelhafid Boussouf, MilaAlgeria
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Abdelouahab, Mohammed-Salah
 und 
Guanrong Chen

Guanrong Chen

Department of Electrical Engineering, City University of Hong KongP.R. China
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Chen, Guanrong
  
10. Nov. 2020
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Online veröffentlicht: 10. Nov. 2020
Seitenbereich: 239 - 256
Eingereicht: 19. Jan. 2020
Akzeptiert: 24. Feb. 2020
DOI: https://doi.org/10.2478/amns.2020.2.00047
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© 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.
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.
Electrical circuit equivalent to the Fitzhugh–Nagumo model.

Figure 3

HLB curve in the (b, α) parameter space.
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.
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.
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).
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
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
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).
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}
14702.590.94605200409150.8078959918168
13669.430.94620025749280.8075994850142
12637.490.94637127188290.8072574562340
11609.120.94657017550150.8068596489968
10955.080.94681145493020.8063770901394
9500.670.94708072291860.8058385541626
8469.370.94741642417910.8051671516416
7436.780.94783157270390.8043368545920
6365.560.94835777597220.8032844480554
5332.40.94903233317050.8019353336588
4300.350.94994880463010.8001023907396
3227.110.95128050684160.7974389863166
2189.640.95322854691080.7935429061782
1156.560.95647160902800.7870567819438
0204.820.96081018292260.7783796341546
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