[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Relaxation Oscillations and Chaos in a Duffing Type Equation: A Case Study

Discontinuity, Nonlinearity, and Complexity 5(4) (2016) 457--474 | DOI:10.5890/DNC.2016.12.010

L. Lerman; A. Kazakov; N.Kulagin

L. Lerman1, A. Kazakov2,1, N.Kulagin3

Download Full Text PDF

 

Abstract

Results of numerical simulations of a Duffing type Hamiltonian system with a slow periodically varying parameter are presented. Using theory of adiabatic invariants, reversibility of the system and theory of symplectic maps, along with thorough numerical experiments, we present many details of the orbit behavior for the system. In particular, we found many symmetric mixed mode periodic orbits, both being hyperbolic and elliptic, the regions with a perpetual adiabatic invariant and chaotic regions. For the latter region we present details of chaotic behavior: calculation of homoclinic tangles and Lyapunov exponents.

Acknowledgments

1 Institute of Information Technology, Mathematics & Mechanics, Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia 2 National Research University Higher School of Economics, Nizhny Novgorod, 603155, Russia 3 Moscow Aviation Institute (MAI), Moscow, Russia

References

  1. [1]  Mischenko, E.F. and Rozov, N.Kh. (1980), Differential Equations with Small Parameters and Relaxation Oscillations, Plenum Press, New York and London.
  2. [2]  Desroches, M., Guckenheimer, J., Krauskopf, B., Kuehn, C., Osinga, H.M., and Wechselberger, M. (2012), Mixedmode oscillations with multiple time scales, SIAM Reviews, 54, 211-288.
  3. [3]  Arnold, V.I. (1963), Small denominators and problems of stability of motion in classical and celestial mechanics, Russ. Math. Surveys, 18(6), 85-191.
  4. [4]  Neishtadt, A.I. (2014), Averaging, passage through resonance, and capture into resonance in two-frequency systems, Russ. Math. Surveys, 69(5), 771-843.
  5. [5]  Sanders, J.A., Verhulst, F., and Murdock, J. (2007), Averaging Methods in Nonlinear Dynamical Systems, Appl. Math. Sci., v.59 (Second edition), Springer-Verlag, Berlin-New York-London.
  6. [6]  Fenichel, N. (1979), Geometric singular perturbation theory for ordinary differential equations, J. Diff. Equat., 31, 53-98.
  7. [7]  Dumortier, F. and Roussarie, R. (1996) Canard cycles and center manifolds, Memoirs of the AMS, 557.
  8. [8]  M. Krupa, M. and Szmolyan, P. (2001), Geometric analysis of the singularly perturbed planar fold, in Multiple-Time- Scale Dynamical Systems, C.K.T.R. Jones et al (eds.), Springer Science + Business Media, New York.
  9. [9]  Jardon-Kojakhmetov,H.J., Broer, H.W., and Roussarie, R. (2015)Analysis of a slow-fast system near a cusp singularity, J. Diff. Equations, 260, 3785-3843.
  10. [10]  Zaslavsky G.M. and Filonenko, N.N. (1968), Stochastic instability of trapped particles and conditions of applicability of the quasi-linear approximation, Sov. Phys. JETP, 27, 851-857.
  11. [11]  Chirikov, B.V. (1979), A universal instability of many-dimensional oscillatory system, Phys. Reports, 264-379.
  12. [12]  Elskence, Y. and Escande, D.F. (1991), Slow Pulsating Separatrices Sweep Homoclinic Tangles where Islands must be Small: An Extension of Classical Adiabatical Theory Nonlinearity, 4, 615-667.
  13. [13]  Meiss, J.D. (1992), Symplectic maps, variational principles, and transport, Rev. Modern Phys., 64(3), 795-848.
  14. [14]  White, R.B., Zaslavsky G.M. (1998) Near threshold anomalous transport in the standard map, Chaos, 8(4), 758-767.
  15. [15]  Simo, C. and Vieiro, A. (2011), Dynamics in chaotic zones of area-preserving maps: Close to separatrix and global instability zones, Physica D, 240, 732-753.
  16. [16]  Luo, A.C.J. (2004) Chaotic motion in the resonant separatrix bands of a Mathieu-Duffing oscillator with a twin-well potential, J. Sound and Vibration, 273, 653-666.
  17. [17]  Lerman, L.M. and Yakovlev, E.I. (2015) Geometry of slow-fast Hamiltonian systems and Painlevé equations, Indagationes Mathematicae 27(5).
  18. [18]  Haberman, R. (1979), Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems, SIAM J. Appl. Math., 37(1), 69-106.
  19. [19]  Diminnie, D.C. and Haberman, R. (2000) Slow passage through a saddle-center bifurcation, J. Nonlinear Science, 10 (2), 197-221.
  20. [20]  H. Chiba, H. (2011), Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points, J. Diff. Equat., 250, 112-160.
  21. [21]  Neishtadt, A.I. (1975), Passage through a separatrix in the resonance problem with slow varying parameter, J. Appl. Math. Mech., 39(4), 594-605 (transl. from Russian: Prikl. Matem. Mekhan., v.39 (1975), No.4, 621-632).
  22. [22]  Cary, J.R., Escande, D.F., and Tennyson, J.L. (1986), Adiabatic-invariant change due to separatrix crossing, Phys. Rev. A, 34(5), 4356-4275.
  23. [23]  Neishtadt, A.I., Treschev, D.V. and Sidorenko, V.V. (1997), Stable periodic motions in the problem of passage through a separatrix, Chaos, 7, 2-11.
  24. [24]  Devaney, R. (1976) Reversible diffeomorphisms and flows, Trans. AMS, 218, 89-113.
  25. [25]  Vanderbauwhede,A. and Fiedler B. (1992), Homoclinic period blow-up in reversible and conservative systems, ZAMP, 43, 292-318.
  26. [26]  Morozov, A.D. (1990), Resonances, Cycles, and Chaos in Quasiconservative Systems, Nonlinear Dynamics, 1, 401- 420.
  27. [27]  Gelfreich, V. and Lerman, L. (2002), Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system, Nonlinearity, 15, 447-457.
  28. [28]  Brüning, J., Dobrokhotov, S.Yu. and Poteryakhin, M.A. (2001), Averaging for Hamiltonian Systems with One Fast Phase and Small Amplitudes, Matem. Notes, 70(5-6), 599-607.
  29. [29]  Neishtadt, A.I. (1984) The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48, 133-139 (transl. from Russian Prikl. Mat. Mekh., 48 (1984), 197-204).
  30. [30]  Fenichel, N. (1974) Asymptotic stability with rate conditions. Indiana Univ. Math. J., 23(12), 1109-1137.
  31. [31]  Fenichel, N. (1977) Asymptotic stability with rate conditions, II. Indiana Univ. Math. J., 26(1), 81-93.
  32. [32]  Lerman, L.M. and Meiss, J.D. (2016),Mixed dynamics in a parabolic standard map, Physica D, 315, 58-71.
  33. [33]  Bochner, S. and Montgomery, D. (1946), Locally Compact Groups of Differentiable Transformations, Ann. Math., 47(4), 639-653.
  34. [34]  Arnold, V.I. (1989),Mathematical Methods of Classical Mechanics, Second Edition, Springer-Verlag.
  35. [35]  Bateman, H. and Erdelyi, A. (1955), Higher transcendental functions, 3, New York, Toronto, London, MC Graw-Hill Book Company, Inc..
  36. [36]  Akhiezer, N.I. (1990) Elements of the Theory of Elliptic Functions (Translations of Mathematical Monographs), 79, New Edition, American Mathematical Society, Providence, R.I.
  37. [37]  Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, NBS, 10th Edition.
  38. [38]  Lyapunov, A.M. (1893), Investigation of one special case of the problem of the stability motion,Matem. Sbornik, 17(2), 253-333 (in Russian).
  39. [39]  Kiselev, O.M. and Glebov, S.G. (2003), An asymptotic solution slowly crossing the separatrix near a saddle-centre bifurcation point, Nonlinearity, 16, 327-362.
  40. [40]  P. Painlevé (1900), Mémoire sur les équationes differentielles dont l'intégrale générale est uniforme, Bull. Soc. Math., 28, 201-261. P. Painlevé, Sur les équationes differentielles du second ordre et d'ordre supérieur don't l'intégrale générale est uniforme, Acta Math., 21, 1-85.
  41. [41]  Clarkson, P.A. (2003), Painlevé equations - nonlinear special functions, J. Comput. Applied Math., 153, 127-140.
  42. [42]  A.R. Its, V.Yu. Novokshenov (1986), The Isomonodromic Deformation Method in the Theory of Painleve equations. Lecture Notes in Mathematics, 1191, Springer-Verlag, 313 pp.
  43. [43]  Boutroux, P. (1913), Recherches sur les transcendantes de M. Painlevé et l'étude asymptotique des équations differentielles du second ordre, Ann. École Norm., 30, 255-375.
  44. [44]  Joshi, N. and Kitaev, A.V. (2001), On Boutroux's Tritronquée Solution of the First Painlevé Equation, Studies Appl. Math., 107, 253-291.
  45. [45]  Clarkson, Peter A. (2010), Numerics and Asymptotics for the Painleve Equations, the talk given at the Conference "Numerical solution of the Painleve equations", ICMS, Edinburgh, May 2010.
  46. [46]  Miguel N., Simo C., Vieiro A. (2013), From the Hénon Conservative Map to the Chirikov Standard Map for Large Parameter Values, Regul. Chaot. Dynam., 18(5), 469-489.
  47. [47]  Neishtadt A.I. and Vasiliev, A.V. (2008) Phase change between separatrix crossings in slow-fast Hamiltonian systems Nonlinearity, 18, No.3, 1393-1406.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates