High-order study of the canard explosion in an aircraft ground dynamics model. (English) Zbl 1459.34133
Summary: A planar system has been proposed in the paper [J. Rankin et al., “Canard cycles in aircraft ground dynamics”, Nonlinear Dyn. 66, No. 4, 681–688 (2011; doi:10.1007/s11071-010-9940-y)] to understand the canard explosion detected in a 6D aircraft ground dynamics model. A specific feature of this minimal 2D system is a critical manifold with a single fold and an asymptote. In this paper, we provide a high-order analytical prediction (in fact, up to any wanted order) of the canard explosion in this system. Using a nonlinear time transformation method, we are able to approximate not only the critical parameter value, but also the critical manifold in the phase space. The comparison of our theoretical results with the corresponding numerical continuations shows a very good agreement.
MSC:
| 34E17 | Canard solutions to ordinary differential equations |
| 70K70 | Systems with slow and fast motions for nonlinear problems in mechanics |
References:
| [1] | Benoit, E.; Callot, JF; Diener, F.; Diener, M., Chasse au canard, Collect. Math., 31-32, 37-119 (1981) · Zbl 0529.34046 |
| [2] | Zvonkin, AK; Shubin, MA, Non-standard analysis and singular perturbations of ordinary differential equations, Russ. Math. Surv., 39, 69-131 (1984) · Zbl 0549.34055 |
| [3] | Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. In: Asymptotic Analysis II. Lecture Notes in Mathematics, vol. 985, pp. 449-494. Springer, Berlin (1983) · Zbl 0509.34037 |
| [4] | Mishchenko, EF; Kolesov, YS; Kolesov, AY; Rozov, NK, Asymptotic Methods in Singularly Perturbed Systems (1994), New York: Consultants Bureau, New York · Zbl 0947.34046 |
| [5] | Dumortier, F., Roussarie, R.: Canard Cycles and Center Manifolds, vol. 577. Memoirs of the American Mathematical Society (1996) · Zbl 0851.34057 |
| [6] | Dumortier, F.; Roussarie, R., Multiple canard cycles in generalized Liénard equations, J. Differ. Equ., 174, 1-29 (2001) · Zbl 1024.34038 |
| [7] | Krupa, M.; Szmolyan, P., Relaxation oscillation and canard explosion, J. Differ. Equ., 174, 312-368 (2001) · Zbl 0994.34032 |
| [8] | Brøns, M., Bifurcations and instabilities in the Greitzer model for compressor system surge, Math. Eng. Ind., 2, 51-63 (1988) · Zbl 0836.34009 |
| [9] | Peng, B.; Gáspár, V.; Showalter, K., False bifurcations in chemical systems: canards, Philos. Trans. R. Soc. Lond. A, 337, 275-289 (1991) · Zbl 0744.92037 |
| [10] | Brøns, M.; Bar-Eli, K., Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction, J. Phys. Chem., 95, 8706-8713 (1991) |
| [11] | Freire, E.; Gamero, E.; Rodríguez-Luis, AJ, First-order approximation for canard periodic orbits in a van der Pol electronic oscillator, Appl. Math. Lett., 12, 73-78 (1999) · Zbl 1066.34510 |
| [12] | Schuster, S.; Marhl, M., Bifurcation analysis of calcium oscillations: time-scale separation, canards, and frequency lowering, J. Biol. Syst., 9, 291-314 (2001) |
| [13] | Moehlis, J., Canards in a surface oxidation reaction, J. Nonlinear Sci., 12, 319-345 (2002) · Zbl 1095.34527 |
| [14] | Brøns, M., Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures, Proc. R. Soc. A, 461, 2289-2302 (2005) · Zbl 1206.74005 |
| [15] | Moehlis, J., Canards for a reduction of the Hodgkin-Huxley equations, J. Math. Biol., 52, 141-153 (2006) · Zbl 1091.92019 |
| [16] | Rankin, J.; Desroches, M.; Krauskopf, B.; Lowenberg, M., Canard cycles in aircraft ground dynamics, Nonlinear Dyn., 66, 681-688 (2011) |
| [17] | Ambrosio, B.; Aziz-Alaoui, MA; Yafia, R., Canard phenomenon in a slow-fast modified Leslie-Gower model, Math. Biosci., 295, 48-54 (2018) · Zbl 1380.92050 |
| [18] | Doedel, EJ; Freire, E.; Gamero, E.; Rodríguez-Luis, AJ, An analytical and numerical study of a modified van der Pol oscillator, J. Sound Vib., 256, 755-771 (2002) · Zbl 1237.34059 |
| [19] | Rotstein, HG; Kopell, N.; Zhabotinsky, AM; Epstein, IR, Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback, J. Chem. Phys., 119, 8824-8832 (2003) |
| [20] | Shchepakina, E., Black swans and canards in self-ignition problem, Nonlinear Anal. Real, 4, 45-50 (2003) · Zbl 1082.34523 |
| [21] | Ginoux, JM; Llibre, J.; Chua, LO, Canards from Chua’s circuit, Int. J. Bifurcat. Chaos, 23, 1330010 (2013) · Zbl 1270.34124 |
| [22] | Ginoux, JM; Llibre, J., Canards in memristor’s circuits, Qual. Theory Dyn. Syst., 15, 383-431 (2016) · Zbl 1370.34111 |
| [23] | Desroches, M.; Krupa, M.; Rodrigues, S., Spike-adding in parabolic bursters: the role of folded-saddle canards, Physica D, 331, 58-70 (2016) · Zbl 1365.92019 |
| [24] | Steindl, A.; Edelmann, J.; Plöchl, M., Limit cycles at oversteer vehicle, Nonlinear Dyn., 99, 313-321 (2020) · Zbl 1430.70082 |
| [25] | Köksal Ersöz, E.; Desroches, M.; Mirasso, CR; Rodrigues, S., Anticipation via canards in excitable systems, Chaos, 29, 013111 (2019) · Zbl 1406.92084 |
| [26] | Rotstein, HG; Coombes, S.; Gheorghe, AM, Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of FitzHugh-Nagumo type, SIAM J. Appl. Dyn. Syst., 11, 135-180 (2012) · Zbl 1266.92043 |
| [27] | Perc, M.; Marhl, M., Different types of bursting calcium oscillations in non-excitable cells, Chaos Solitons Fractals, 18, 759-773 (2003) · Zbl 1068.92017 |
| [28] | Perc, M.; Marhl, M., Synchronization of regular and chaotic oscillations: the role of local divergence and the slow passage effect, Int. J. Bifurcat. Chaos, 14, 2735-2751 (2004) · Zbl 1065.92015 |
| [29] | Desroches, M.; Guckenheimer, J.; Krauskopf, B.; Kuehn, C.; Osinga, HM; Wechselberger, M., Mixed-mode oscillations with multiple time scales, SIAM Rev., 54, 211-288 (2012) · Zbl 1250.34001 |
| [30] | Han, X.; Bi, Q., Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation, Nonlinear Dyn., 68, 275-283 (2012) · Zbl 1245.34058 |
| [31] | Desroches, M.; Kaper, TJ; Krupa, M., Mixed-mode bursting oscillations: dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos, 23, 046106 (2013) · Zbl 1375.92014 |
| [32] | Zheng, Y.; Bao, L., Time-delay effects on mixed-mode oscillations of modified Chua’s system, Nonlinear Dyn., 80, 1521-1529 (2015) |
| [33] | Kristiansen, KU, Blowup for flat slow manifolds, Nonlinearity, 30, 2138-2184 (2017) · Zbl 1371.34085 |
| [34] | Chan, HSY; Chung, KW; Xu, Z., A perturbation-incremental method for strongly non-linear oscillators, Int. J. Nonlinear Mech., 31, 59-72 (1996) · Zbl 0864.70015 |
| [35] | Lau, SL; Cheung, YK, Amplitude incremental variational principle for nonlinear vibration of elastic systems, ASME J. Appl. Mech., 48, 959-964 (1981) · Zbl 0468.73066 |
| [36] | Lau, SL; Yuen, SW, Solution diagram of nonlinear dynamics systems by the IHB method, J. Sound Vib., 167, 303-316 (1993) · Zbl 0925.70218 |
| [37] | Cao, YY; Chung, KW; Xu, J., A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dyn., 64, 221-236 (2011) |
| [38] | Fahsi, A.; Belhaq, M., Analytical approximation of heteroclinic bifurcation in a 1:4 resonance, Int. J. Bifurcat. Chaos, 22, 1250294 (2012) · Zbl 1258.34090 |
| [39] | Chung, KW; Cao, YY; Fahsi, A.; Belhaq, M., Analytical approximation of heteroclinic bifurcations in 1:4 resonance using a nonlinear transformation method, Nonlinear Dyn., 78, 2479-2486 (2014) |
| [40] | Qin, BW; Chung, KW; Fahsi, A.; Belhaq, M., On the heteroclinic connections in the 1:3 resonance problem, Int. J. Bifurcat. Chaos, 26, 1650143 (2016) · Zbl 1345.34077 |
| [41] | Qin, BW; Chung, KW; Rodríguez-Luis, AJ; Belhaq, M., Homoclinic-doubling and homoclinic-gluing bifurcations in the Takens-Bogdanov normal form with \(D_4\) symmetry, Chaos, 28, 093107 (2018) · Zbl 1397.37022 |
| [42] | Rucklidge, AM, Global bifurcations in the Takens-Bogdanov normal form with \(D_4\) symmetry near the \(O(2)\) limit, Phys. Lett. A, 284, 99-111 (2001) · Zbl 0983.37055 |
| [43] | Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X., Zhang, C.: AUTO-07P: continuation and bifurcation software for ordinary differential equations (with HomCont). Technical report, Concordia University (2012) |
| [44] | Dhooge, A.; Govaerts, W.; Kuznetsov, YA; Meijer, HGE; Sautois, B., New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn., 14, 147-175 (2008) · Zbl 1158.34302 |
| [45] | Algaba, A.; Chung, KW; Qin, BW; Rodríguez-Luis, AJ, A nonlinear time transformation method to compute all the coefficients for the homoclinic bifurcation in the quadratic Takens-Bogdanov normal form, Nonlinear Dyn., 97, 979-990 (2019) · Zbl 1430.37054 |
| [46] | Algaba, A.; Chung, KW; Qin, BW; Rodríguez-Luis, AJ, Computation of all the coefficients for the global connections in the \(\mathbb{Z}_2\)-symmetric Takens-Bogdanov normal forms, Commun. Nonlinear Sci. Numer. Simul., 81, 105012 (2020) · Zbl 1453.37045 |
| [47] | Qin, BW; Chung, KW; Algaba, A.; Rodríguez-Luis, AJ, High-order analysis of global bifurcations in a codimension-three Takens-Bogdanov singularity in reversible systems, Int. J. Bifurcat. Chaos, 30, 2050017 (2020) · Zbl 1436.34035 |
| [48] | Qin, BW; Chung, KW; Algaba, A.; Rodríguez-Luis, AJ, Analytical approximation of cuspidal loops using a nonlinear time transformation method, Appl. Math. Comput., 373, 125042 (2020) · Zbl 1433.37019 |
| [49] | Algaba, A.; Chung, KW; Qin, BW; Rodríguez-Luis, AJ, Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method, Physica D (2020) · Zbl 1482.34138 · doi:10.1016/j.physd.2020.132384 |
| [50] | Qin, B.W., Chung, K.W., Algaba, A., Rodríguez-Luis, A.J.: High-order analysis of canard explosion in the Brusselator equations. Int. J. Bifurcat. Chaos (2020) (accepted) · Zbl 1447.34023 |
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