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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

Alexander M. Krasnosel'skii 1, , Edward O'Grady 2, , Alexei Pokrovskii 3, and  Dmitrii I. Rachinskii 2,

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.
Keywords: stability., canard trajectory, degree theory, Periodic solution.
Mathematics Subject Classification: Primary: 34E17, 34C25; Secondary: 34C5.
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 & 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. Google Scholar

[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. Google Scholar

[3]

S. Aubry, Exact models with a complete Devil's staircase, J. Phys. C: Solid State Phys., 16 (1983), 2497-2508. Google Scholar

[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.  Google Scholar

[5]

E. Benoit, Chasse au canard. II. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 37-74.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Brφns, Canard explosion of limit cycles in templator models of self-replication mechanisms, J. Chem. Phys., 134 (2011), 144105. Google Scholar

[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.  Google Scholar

[10]

K. Deimling, "Nonlinear Functional Analysis,'' Springer, 1980.  Google Scholar

[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.  Google Scholar

[12]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Redwood City, CA: Addison-Wesley, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

, "Dynamic Bifurcations,", Lecture Notes in Math. 1493, (1493).   Google Scholar

[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.  Google Scholar

[17]

J. Guckenheimer, K. Hoffman and W. Weckesser, Numerical computation of canards, Int. J. Bifurcation Chaos Appl. Sci. Eng., 10 (2000), 2669-2687.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, Resonant equations with unbounded nonlinearities, Doklady Mathematics, 62 (2000), 44-48.  Google Scholar

[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.  Google Scholar

[23]

A. M. Krasnosel'skii and D. I. Rachinskii, Continua of cycles of higher-order equations, Differential Equations, 39 (2003), 1690-1702.  Google Scholar

[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.  Google Scholar

[25]

A. M. Krasnosel'skii and D. I. Rachinskii, Continuous branches of cycles in systems with nonlinearizable nonlinearities, Doklady Mathematics, 67 (2003), 153-157.  Google Scholar

[26]

A. M. Krasnosel'skii and D. I. Rachinskii, Nonlinear Hopf bifurcations, Doklady Mathematics, 61 (2000), 389-392.  Google Scholar

[27]

A. M. Krasnosel'skii and D. I. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Mathematics, 63 3 (2001), 339-344.  Google Scholar

[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.  Google Scholar

[29]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer, 1984.  Google Scholar

[30]

A. Pokrovskii and D. Rachinskii, Effect of positive feedback on Devil's staircase input-output relationship,, DCDS-S, ().   Google Scholar

[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.  Google Scholar

[32]

A. Pokrovskii, E. Shchepakina and V. Sobolev, Canard doublet in Lotka-Volterra type model, J. Phys. Conf. Ser., 138 (2008), 012019. Google Scholar

[33]

A. Pokrovskii and A. Zhezherun, Topological method for analysis of periodic canards, Automat. Remote Control, 70 (2009), 967-981.  Google Scholar

[34]

A. Pokrovskii and A. Zhezherun, Topological degree in analysis of chaotic behavior in singularly perturbed systems, Chaos, 18 (2008), 023130.  Google Scholar

[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.  Google Scholar

[36]

, "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005).   Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[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. Google Scholar

[3]

S. Aubry, Exact models with a complete Devil's staircase, J. Phys. C: Solid State Phys., 16 (1983), 2497-2508. Google Scholar

[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.  Google Scholar

[5]

E. Benoit, Chasse au canard. II. Tunnels-entonnoirs-peignes, Collect. Math., 32 (1981), 37-74.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Brφns, Canard explosion of limit cycles in templator models of self-replication mechanisms, J. Chem. Phys., 134 (2011), 144105. Google Scholar

[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.  Google Scholar

[10]

K. Deimling, "Nonlinear Functional Analysis,'' Springer, 1980.  Google Scholar

[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.  Google Scholar

[12]

R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Redwood City, CA: Addison-Wesley, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

, "Dynamic Bifurcations,", Lecture Notes in Math. 1493, (1493).   Google Scholar

[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.  Google Scholar

[17]

J. Guckenheimer, K. Hoffman and W. Weckesser, Numerical computation of canards, Int. J. Bifurcation Chaos Appl. Sci. Eng., 10 (2000), 2669-2687.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[21]

A. M. Krasnosel'skii, N. A. Kuznetsov and D. I. Rachinskii, Resonant equations with unbounded nonlinearities, Doklady Mathematics, 62 (2000), 44-48.  Google Scholar

[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.  Google Scholar

[23]

A. M. Krasnosel'skii and D. I. Rachinskii, Continua of cycles of higher-order equations, Differential Equations, 39 (2003), 1690-1702.  Google Scholar

[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.  Google Scholar

[25]

A. M. Krasnosel'skii and D. I. Rachinskii, Continuous branches of cycles in systems with nonlinearizable nonlinearities, Doklady Mathematics, 67 (2003), 153-157.  Google Scholar

[26]

A. M. Krasnosel'skii and D. I. Rachinskii, Nonlinear Hopf bifurcations, Doklady Mathematics, 61 (2000), 389-392.  Google Scholar

[27]

A. M. Krasnosel'skii and D. I. Rachinskii, On continua of cycles in systems with hysteresis, Doklady Mathematics, 63 3 (2001), 339-344.  Google Scholar

[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.  Google Scholar

[29]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer, 1984.  Google Scholar

[30]

A. Pokrovskii and D. Rachinskii, Effect of positive feedback on Devil's staircase input-output relationship,, DCDS-S, ().   Google Scholar

[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.  Google Scholar

[32]

A. Pokrovskii, E. Shchepakina and V. Sobolev, Canard doublet in Lotka-Volterra type model, J. Phys. Conf. Ser., 138 (2008), 012019. Google Scholar

[33]

A. Pokrovskii and A. Zhezherun, Topological method for analysis of periodic canards, Automat. Remote Control, 70 (2009), 967-981.  Google Scholar

[34]

A. Pokrovskii and A. Zhezherun, Topological degree in analysis of chaotic behavior in singularly perturbed systems, Chaos, 18 (2008), 023130.  Google Scholar

[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.  Google Scholar

[36]

, "Singular Perturbations and Hysteresis,", (eds. M. P. Mortell, (2005).   Google Scholar

[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. Google Scholar

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