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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 |
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. |
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. |
[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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