Abstract
We review essential techniques in the study of families of periodic orbits of slow-fast systems in the plane. The techniques are demonstrated by treating orbits passing through unfoldings of transcritical intersections of curves of singular points in the most generic setting. We show that such transcritical intersections can generate canard type orbits. The stability of limit cycles of canard type containing that pass near transcritical intersections is examined by means of the slow divergence integral.
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Benoit, E.: Équations différentielles: relation entrée–sortie. C. R. Acad. Sci. Paris Sér. I Math. 293(5), 293–296 (1981)
Benoît, É., Callot, J.L., Diener, F., Diener, M.: Chasse au canard. I–IV. Collect. Math. 32(1–2), 37–119 (1981)
Canalis-Durand, M.: Formal expansion of van der Pol equation canard solutions are Gevrey. In: Dynamic Bifurcations (Luminy, 1990). Lecture Notes in Math., vol. 1493, pp. 29–39. Springer, Berlin (1991)
Canalis-Durand, M., Ramis, J.P., Schäfke, R., Sibuya, Y.: Gevrey solutions of singularly perturbed differential equations. J. Reine Angew. Math. 518, 95–129 (2000)
De Maesschalck, P.: Gevrey properties of real planar singularly perturbed systems. J. Differ. Equ. 238(2), 338–365 (2007)
De Maesschalck, P., Desroches, M.: Numerical continuation techniques for planar slow-fast systems. SIAM J. Appl. Dyn. Syst. 12(3), 1159–1180 (2013)
De Maesschalck, P., Dumortier, F.: Time analysis and entry-exit relation near planar turning points. J. Differ. Equ. 215(2), 225–267 (2005)
De Maesschalck, P., Dumortier, F.: Canard solutions at non-generic turning points. Trans. Am. Math. Soc. 358(5), 2291–2334 (2006) (electronic)
De Maesschalck, P., Dumortier, F.: Bifurcations of multiple relaxation oscillations in polynomial Liénard equations. Proc. Am. Math. Soc. 139(6), 2073–2085 (2011)
De Maesschalck, P., Dumortier, F., Roussarie, R.: Transitory canard cycles. In preparation
De Maesschalck, P., Dumortier, F., Roussarie, R.: Cyclicity of common slow-fast cycles. Indag. Math. 22(3-4), 165–206 (2011)
De Maesschalck, P., Huzak, R.: Slow divergence integrals in classical liénard equations near centers. J. Dyn. Differ. Equ. (2014), 9 pp. http://dx.doi.org/10.1007/s10884-014-9358-1. doi:10.1007/s10884-014-9358-1,
Dumortier, F.: Slow divergence integral and balanced canard solutions. Qual. Theory Dyn. Syst. 10(1), 65–85 (2011)
Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100 (1996). With an appendix by Cheng Zhi Li
Dumortier, F., Roussarie, R.: Bifurcation of relaxation oscillations in dimension two. Discrete Contin. Dyn. Syst. 19(4), 631–674 (2007)
Dumortier, F., Roussarie, R.: Birth of canard cycles. Discrete Contin. Dyn. Syst. Ser. S 2(4), 723–781 (2009)
Eckhaus, W.: Asymptotic Analysis of Singular Perturbations. Studies in Mathematics and Its Applications, vol. 9. North-Holland Publishing Co., Amsterdam-New York (1979)
Eckhaus, W.: Relaxation oscillations including a standard chase on French ducks. In: Asymptotic Analysis, II. Lecture Notes in Math., vol. 985, pp. 449–494. Springer, Berlin (1983)
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
Françoise, J.P., Piquet, C., Vidal, A.: Enhanced delay to bifurcation. Bull. Belg. Math. Soc. Simon Stevin 15(5), 825–831 (2008). http://projecteuclid.org/euclid.bbms/1228486410
van Gils, S., Krupa, M., Szmolyan, P.: Asymptotic expansions using blow-up. Z. Angew. Math. Phys. 56(3), 369–397 (2005)
Grasman, J.: Asymptotic Methods for Relaxation Oscillations and Applications. Applied Mathematical Sciences, vol. 63. Springer, New York (1987)
Huzak, R., Maesschalck, P., Dumortier, F.: Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. Commun. Pure Appl. Anal. 13(6), 2641–2673 (2014)
Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, vol. 114. Springer, New York (1996)
Krupa, M., Szmolyan, P.: Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14(6), 1473–1491 (2001)
Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)
Kuznetsov, Y.A., Muratori, S., Rinaldi, S.: Homoclinic bifurcations in slow-fast second order systems. Nonlinear Anal. 25(7), 747–762 (1995)
Li, C., Zhu, H.: Canard cycles for predator-prey systems with Holling types of functional response. J. Differ. Equ. 254(2), 879–910 (2013)
Mamouhdi, L., Roussarie, R.: Canard cycles of finite codimension with two breaking parameters. Qual. Theory Dyn. Syst. 11(1), 167–198 (2012)
Mishchenko, E.F., Kolesov, Y.S., Kolesov, A.Y., Rozov, N.K.: Asymptotic Methods in Singularly Perturbed Systems. Monographs in Contemporary Mathematics. Consultants Bureau, New York (1994). Translated from the Russian by Irene Aleksanova
Panazzolo, D.: Desingularization of nilpotent singularities in families of planar vector fields. Mem. Am. Math. Soc. 158(753), viii+108 (2002)
Rosenzweig, M.L., MacArthur, R.H.: Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 47(895), 209–223 (1963)
Schecter, S.: Persistent unstable equilibria and closed orbits of a singularly perturbed equation. J. Differ. Equ. 60(1), 131–141 (1985)
Shen, J., Han, M.: Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete Contin. Dyn. Syst. 33(7), 3085–3108 (2013)
Verhulst, F.: Methods and Applications of Singular Perturbations. Boundary Layers and Multiple Timescale Dynamics. Texts in Applied Mathematics, vol. 50. Springer, New York (2005)
Wasow, W.R.: Singular perturbation methods for nonlinear oscillations. In: Proceedings of the Symposium on Nonlinear Circuit Analysis, pp. 75–98. Polytechnic Institute of Brooklyn, New York (1953)
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The author thanks the anonymous reviewers for their many useful remarks and suggestions. The author acknowledges support from FWO Vlaanderen.
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De Maesschalck, P. Planar Canards with Transcritical Intersections. Acta Appl Math 137, 159–184 (2015). https://doi.org/10.1007/s10440-014-9994-9
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DOI: https://doi.org/10.1007/s10440-014-9994-9