# American Institute of Mathematical Sciences

March  2013, 18(2): 467-482. doi: 10.3934/dcdsb.2013.18.467

## Periodic canard trajectories with multiple segments following the unstable part of critical manifold

 1 Institute for Information Transmission Problems, 19 Bolshoi Karetny, Moscow 127994, Russian Federation, National Research University Higher School of Economics, 20 Myasnitskaya Street, Moscow 101000 2 Department of Applied Mathematics, University College Cork, Ireland 3 Department of Applied Mathematics, University College, Cork

Received  October 2011 Revised  March 2012 Published  November 2012

We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.
Citation: Alexander M. Krasnosel'skii, Edward O'Grady, Alexei Pokrovskii, Dmitrii I. Rachinskii. Periodic canard trajectories with multiple segments following the unstable part of critical manifold. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 467-482. doi: 10.3934/dcdsb.2013.18.467
##### References:
 [1] B. Appelbe, D. Rachinskii and A. Zhezherun, Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis, Physica B, 40 (2008), 301-304. [2] V. I. Arnold, V. S. Afraimovich, Yu. S. Il'yashenko and L. P. Shil'nikov, Theory of Bifurcations, in "Dynamical Systems'' (ed. V. Arnold), 5, Encyclopedia of Mathematical Sciences, New York, Springer, 1994. [3] S. Aubry, Exact models with a complete Devil's staircase, J. Phys. C: Solid State Phys., 16 (1983), 2497-2508. [4] Z. Balanov, W. Krawcewicz, D. Rachinskii and A. Zhezherun, "Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis," J. Dyn. Diff. Equat., (2012). doi: 10.1007/s10884-012-9271-4. [5] E. Benoit, Chasse au canard. II. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 37-74. [6] P.-A. Bliman, A. M. Krasnosel'skii and D. I. Rachinskii, Sector estimates of nonlinearities and the existence of self-oscillations in control systems, Automat. Remote Control, 61 (2000), 889-903. [7] E. Bouse, A. M. Krasnosel'skii, A. V. Pokrovskii and D. I. Rachinskii, Nonlocal branches of cycles, bistability, and topologically persistent mixed mode oscillations, Chaos, 18 (2008), 015109. [8] M. Brφns, Canard explosion of limit cycles in templator models of self-replication mechanisms, J. Chem. Phys., 134 (2011), 144105. [9] J. L. Callot, F. Diener and M. Diener, Le probleme de la "chasse au canard'', C. R. Acad. Sci. Paris Ser. A-B, 286 (1978), A1059-A1061. [10] K. Deimling, "Nonlinear Functional Analysis,'' Springer, 1980. [11] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765. doi: 10.1088/0951-7715/23/3/017. [12] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Redwood City, CA: Addison-Wesley, 1987. [13] P. Diamond, N. A. Kuznetsov and D. I. Rachinskii, On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity, J. Differential Equations, 175 (2001), 1-26. [14] P. Diamond, D. Rachinskii and M. Yumagulov, Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity, Nonlinear Anal. Theory, Methods, Appl., 42 (2000), 1017-1031. [15] , "Dynamic Bifurcations,", Lecture Notes in Math. 1493, (1493). [16] V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas, Proc. London Roy. Soc. A, 452 (1996), 2103-2119. doi: 10.1098/rspa.1996.0111. [17] J. Guckenheimer, K. Hoffman and W. Weckesser, Numerical computation of canards, Int. J. Bifurcation Chaos Appl. Sci. Eng., 10 (2000), 2669-2687. [18] J. Guckenheimer and M. D. Lamar, Periodic orbit continuation in multiple time scale systems, in "Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems" (eds B. Krauskopf, H. M. Osinga and J. Galan-Vioque), Berlin, Springer, (2007), 253-268. [19] A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications to differential equations, Differential Equations, 33 (1997), 59-66. [20] A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, On resonant differential equations with unbounded non-linearities, Zeitshrift für Analysis und ihre Anwendungen, 21 (2002), 639-668. [21] A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, Resonant equations with unbounded nonlinearities, Doklady Mathematics, 62 (2000), 44-48. [22] A. M. Krasnosel'skii, R. Mennicken and D. I. Rachinskii, Small periodic solutions generated by sublinear terms, J. Differential Equations, 179 (2002), 97-132. [23] A. M. Krasnosel'skii and D. I. Rachinskii, Continua of cycles of higher-order equations, Differential Equations, 39 (2003), 1690-1702. [24] A. M. Krasnosel'skii and D. I. Rachinskii, Existence of continua of cycles in hamiltonian control systems, Aut. Remote Control, 62 (2001), 227-235. doi: 10.1023/A:1002842206198. [25] A. M. Krasnosel'skii and D. I. Rachinskii, Continuous branches of cycles in systems with nonlinearizable nonlinearities, Doklady Mathematics, 67 (2003), 153-157. [26] A. M. Krasnosel'skii and D. I. Rachinskii, Nonlinear Hopf bifurcations, Doklady Mathematics, 61 (2000), 389-392. [27] A. M. Krasnosel'skii and D. I. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Mathematics, 63 3 (2001), 339-344. [28] A. M. Krasnosel'skii and D. I. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA: Nonlinear Differential Equations Appl., 9 (2002), 93-115. [29] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer, 1984. [30] A. Pokrovskii and D. Rachinskii, Effect of positive feedback on Devil's staircase input-output relationship,, DCDS-S, (). [31] A. Pokrovskii, D. Rachinskii, V. Sobolev and A. Zhezherun, Topological degree in analysis of canard-type trajectories in 3-D systems, Applicable Analysis: Int. J., 90 (2011), 1123-1139. [32] A. Pokrovskii, E. Shchepakina and V. Sobolev, Canard doublet in Lotka-Volterra type model, J. Phys. Conf. Ser., 138 (2008), 012019. [33] A. Pokrovskii and A. Zhezherun, Topological method for analysis of periodic canards, Automat. Remote Control, 70 (2009), 967-981. [34] A. Pokrovskii and A. Zhezherun, Topological degree in analysis of chaotic behavior in singularly perturbed systems, Chaos, 18 (2008), 023130. [35] M. Sekikawa, N. Inaba and T. Tsubouchi, Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation, Physica D, 194 (2004), 227-249. [36] , "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005). [37] V. A. Sobolev, Geometry of singular perturbations: Critical cases, in "Singular Perturbations and Hysteresis" (eds. R. E. O'Malley, M. P. Mortell, A.V. Pokrovskii and V. A. Sobolev), SIAM, (2005), 153-206.

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##### References:
 [1] B. Appelbe, D. Rachinskii and A. Zhezherun, Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis, Physica B, 40 (2008), 301-304. [2] V. I. Arnold, V. S. Afraimovich, Yu. S. Il'yashenko and L. P. Shil'nikov, Theory of Bifurcations, in "Dynamical Systems'' (ed. V. Arnold), 5, Encyclopedia of Mathematical Sciences, New York, Springer, 1994. [3] S. Aubry, Exact models with a complete Devil's staircase, J. Phys. C: Solid State Phys., 16 (1983), 2497-2508. [4] Z. Balanov, W. Krawcewicz, D. Rachinskii and A. Zhezherun, "Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis," J. Dyn. Diff. Equat., (2012). doi: 10.1007/s10884-012-9271-4. [5] E. Benoit, Chasse au canard. II. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 37-74. [6] P.-A. Bliman, A. M. Krasnosel'skii and D. I. Rachinskii, Sector estimates of nonlinearities and the existence of self-oscillations in control systems, Automat. Remote Control, 61 (2000), 889-903. [7] E. Bouse, A. M. Krasnosel'skii, A. V. Pokrovskii and D. I. Rachinskii, Nonlocal branches of cycles, bistability, and topologically persistent mixed mode oscillations, Chaos, 18 (2008), 015109. [8] M. Brφns, Canard explosion of limit cycles in templator models of self-replication mechanisms, J. Chem. Phys., 134 (2011), 144105. [9] J. L. Callot, F. Diener and M. Diener, Le probleme de la "chasse au canard'', C. R. Acad. Sci. Paris Ser. A-B, 286 (1978), A1059-A1061. [10] K. Deimling, "Nonlinear Functional Analysis,'' Springer, 1980. [11] M. Desroches, B. Krauskopf and H. M. Osinga, Numerical continuation of canard orbits in slow-fast dynamical systems, Nonlinearity, 23 (2010), 739-765. doi: 10.1088/0951-7715/23/3/017. [12] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Redwood City, CA: Addison-Wesley, 1987. [13] P. Diamond, N. A. Kuznetsov and D. I. Rachinskii, On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity, J. Differential Equations, 175 (2001), 1-26. [14] P. Diamond, D. Rachinskii and M. Yumagulov, Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity, Nonlinear Anal. Theory, Methods, Appl., 42 (2000), 1017-1031. [15] , "Dynamic Bifurcations,", Lecture Notes in Math. 1493, (1493). [16] V. Gol'dshtein, A. Zinoviev, V. Sobolev and E. Shchepakina, Criterion for thermal explosion with reactant consumption in a dusty gas, Proc. London Roy. Soc. A, 452 (1996), 2103-2119. doi: 10.1098/rspa.1996.0111. [17] J. Guckenheimer, K. Hoffman and W. Weckesser, Numerical computation of canards, Int. J. Bifurcation Chaos Appl. Sci. Eng., 10 (2000), 2669-2687. [18] J. Guckenheimer and M. D. Lamar, Periodic orbit continuation in multiple time scale systems, in "Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems" (eds B. Krauskopf, H. M. Osinga and J. Galan-Vioque), Berlin, Springer, (2007), 253-268. [19] A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications to differential equations, Differential Equations, 33 (1997), 59-66. [20] A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, On resonant differential equations with unbounded non-linearities, Zeitshrift für Analysis und ihre Anwendungen, 21 (2002), 639-668. [21] A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, Resonant equations with unbounded nonlinearities, Doklady Mathematics, 62 (2000), 44-48. [22] A. M. Krasnosel'skii, R. Mennicken and D. I. Rachinskii, Small periodic solutions generated by sublinear terms, J. Differential Equations, 179 (2002), 97-132. [23] A. M. Krasnosel'skii and D. I. Rachinskii, Continua of cycles of higher-order equations, Differential Equations, 39 (2003), 1690-1702. [24] A. M. Krasnosel'skii and D. I. Rachinskii, Existence of continua of cycles in hamiltonian control systems, Aut. Remote Control, 62 (2001), 227-235. doi: 10.1023/A:1002842206198. [25] A. M. Krasnosel'skii and D. I. Rachinskii, Continuous branches of cycles in systems with nonlinearizable nonlinearities, Doklady Mathematics, 67 (2003), 153-157. [26] A. M. Krasnosel'skii and D. I. Rachinskii, Nonlinear Hopf bifurcations, Doklady Mathematics, 61 (2000), 389-392. [27] A. M. Krasnosel'skii and D. I. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Mathematics, 63 3 (2001), 339-344. [28] A. M. Krasnosel'skii and D. I. Rachinskii, On a bifurcation governed by hysteresis nonlinearity, NoDEA: Nonlinear Differential Equations Appl., 9 (2002), 93-115. [29] M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer, 1984. [30] A. Pokrovskii and D. Rachinskii, Effect of positive feedback on Devil's staircase input-output relationship,, DCDS-S, (). [31] A. Pokrovskii, D. Rachinskii, V. Sobolev and A. Zhezherun, Topological degree in analysis of canard-type trajectories in 3-D systems, Applicable Analysis: Int. J., 90 (2011), 1123-1139. [32] A. Pokrovskii, E. Shchepakina and V. Sobolev, Canard doublet in Lotka-Volterra type model, J. Phys. Conf. Ser., 138 (2008), 012019. [33] A. Pokrovskii and A. Zhezherun, Topological method for analysis of periodic canards, Automat. Remote Control, 70 (2009), 967-981. [34] A. Pokrovskii and A. Zhezherun, Topological degree in analysis of chaotic behavior in singularly perturbed systems, Chaos, 18 (2008), 023130. [35] M. Sekikawa, N. Inaba and T. Tsubouchi, Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation, Physica D, 194 (2004), 227-249. [36] , "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005). [37] V. A. Sobolev, Geometry of singular perturbations: Critical cases, in "Singular Perturbations and Hysteresis" (eds. R. E. O'Malley, M. P. Mortell, A.V. Pokrovskii and V. A. Sobolev), SIAM, (2005), 153-206.
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