Advertisement

Chaos in Ordinary Differential Equations

  • Michal Fečkan
Part of the Nonlinear Physical Science book series (NPS)

Abstract

Functional analytical methods are presented in this chapter to predict chaos for ODEs depending on parameters. Several types of ODEs are considered. We also study multivalued perturbations of ODEs, and coupled infinite-dimensional ODEs on the lattice ℂ as well. Moreover, the structure of bifurcation parameters for homoclinic orbits is investigated.

Keywords

Homoclinic Orbit Center Manifold Heteroclinic Orbit Exponential Dichotomy Homoclinic Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    F. BATTELLI & C. LAZZARI: Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations 86 (1990), 342–366.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    R. CHACÓN: Control of Homoclinic Chaos by Weak Periodic Perturbations, World Scientific Publishing Co., Singapore, 2005.zbMATHGoogle Scholar
  3. 3.
    J. GRUENDLER: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    J. GRUENDLER: The existence of transverse homoclinic solutions for higher order equations, J. Differential Equations 130 (1996), 307–320.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    J. GRUENDLER: Homoclinic solutions for autonomous dynamical systems in arbitrary dimensions, SIAM J. Math. Anal. 23 (1992), 702–721.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A.C.J. LUO: Global Transversality, Resonance and Chaotic Dynamics, World Scientific Publishing Co., Singapore, 2008.zbMATHCrossRefGoogle Scholar
  7. 7.
    K.J. PALMER: Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225–256.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    M. TABOR: Chaos and Integrability in Nonlinear Dynamics: An Introduction, Willey-Interscience, New York, 1989.zbMATHGoogle Scholar
  9. 9.
    M. FEČKAN: Topological Degree Approach to Bifurcation Problems, Springer, 2008.Google Scholar
  10. 10.
    C. CHICONE: Lyapunov-Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Differential Equations 112 (1994), 407–447.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    A. CODDINGTON & N. LEVINSON: Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.zbMATHGoogle Scholar
  12. 12.
    M. FARKAS: Periodic Motions, Springer-Verlag, New York, 1994.zbMATHCrossRefGoogle Scholar
  13. 13.
    J.K. HALE: Ordinary Differential Equations, 2nd ed., Robert E. Krieger, New York, 1980.zbMATHGoogle Scholar
  14. 14.
    F. BATTELLI: Perturbing diffeomorphisms which have heteroclinic orbits with semihyperbolic fixed points, Dyn. Sys. Appl. 3 (1994), 305–332.MathSciNetzbMATHGoogle Scholar
  15. 15.
    A. DOELMAN & G. HEK: Homoclinic saddle-node bifurcations in singularly perturbed systems, J. Dynamics Differential Equations 12 (2000), 169–216.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. 16.
    J.A. SANDERS, F. VERHULST & J. MURDOCK: Averaging Methods in Nonlinear Dynamical Systems, 2nd ed., Springer, 2007.Google Scholar
  17. 17.
    A.N. TIKHONOV: Systems of differential equations containing small parameters multiplying some of the derivatives, Mat. Sb. 31 (1952), 575–586.Google Scholar
  18. 18.
    F.C. HOPPENSTEADT: Singular perturbations on the infinite interval, Trans. Amer. Math. Soc. 123 (1966), 521–535.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    F.C. HOPPENSTEADT: Properties of solutions of ordinary differential equations with small parameters, Comm. Pure Appl. Math. 24 (1971), 807–840.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    N. FENICHEL: Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), 53–98.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    L. FLATTO & N. LEVINSON: Periodic solutions to singularly perturbed systems, J. Ration. Mech. Anal. 4 (1955), 943–950.MathSciNetzbMATHGoogle Scholar
  22. 22.
    P. SZMOLYAN: Heteroclinic Orbits in Singularly Perturbed Differential Equations, IMA preprint 576.Google Scholar
  23. 23.
    P. SZMOLYAN: Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations 92 (1991), 252–281.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    F. BATTELLI & C. LAZZARI: Bounded solutions to singularly perturbed systems of O.D.E., J. Differential Equations 100 (1992), 49–81.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M. FEČKAN: Melnikov functions for singularly perturbed ordinary differential equations, Nonlin. Anal., Th. Meth. Appl. 19 (1992), 393–401.zbMATHCrossRefGoogle Scholar
  26. 26.
    K. NIPP: Smooth attractive invariant manifolds of singularly perturbed ODE’s, SAM-ETH 92-13 (1992).Google Scholar
  27. 27.
    V.V. STRYGIN & V.A. SOBOLEV: Separation of Motions by the Method of Integral Manifolds, Nauka, Moscow, 1988, (in Russian).zbMATHGoogle Scholar
  28. 28.
    F. BATTELLI & M. FEČKAN: Global center manifolds in singular systems, NoDEA: Nonl. Diff. Eqns. Appl. 3 (1996), 19–34.CrossRefGoogle Scholar
  29. 29.
    M. FEČKAN & J. GRUENDLER: Transversal bounded solutions in systems with normal and slow variables, J. Differential Equation 165 (2000), 123–142.zbMATHCrossRefGoogle Scholar
  30. 30.
    M. FEČKAN: Transversal homoclinics in nonlinear systems of ordinary differential equations, in: “Proc. 6th Coll. Qual. Th. Differential Equations”, Szeged, 1999, Electr. J. Qual. Th. Differential Equations 9 (2000), 1–8.Google Scholar
  31. 31.
    M. FEČKAN: Bifurcation of multi-bump homoclinics in systems with normal and slow variables, Electr. J. Differential Equations 2000 (2000), 1–17.Google Scholar
  32. 32.
    F. BATTELLI: Heteroclinic orbits in singular systems: a unifying approach, J. Dyn. Diff. Eqns. 6 (1994), 147–173.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    G. KOVAČIČ: Singular perturbation theory for homoclinic orbits in a class of nearintegrable dissipative systems, SIAM J. Math. Anal. 26 (1995), 1611–1643.MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    X.-B. LIN: Homoclinic bifurcations with weakly expanding center manifolds, Dynamics Reported 5 (1995), 99–189.CrossRefGoogle Scholar
  35. 35.
    S. WIGGINS: Global Bifurcations and Chaos, Analytical Methods, Springer-Verlag, New York, 1988.zbMATHCrossRefGoogle Scholar
  36. 36.
    S. WIGGINS & P. HOLMES: Homoclinic orbits in slowly varying oscillators, SIAM J. Math. Anal. 18 (1987), 612–629. Erratum: SIAM J. Math. Anal. 19 (1988), 1254–1255.MathSciNetADSzbMATHCrossRefGoogle Scholar
  37. 37.
    D. ZHU: Exponential trichotomy and heteroclinic bifurcations, Nonlin. Anal. Th. Meth. Appl. 28 (1997), 547–557.zbMATHCrossRefGoogle Scholar
  38. 38.
    M. CARTMELL: Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman & Hall, London, 1990.zbMATHGoogle Scholar
  39. 39.
    M. RUIJGROK, A. TONDL & F. VERHULST: Resonance in a rigid rotor with elastic support, Z. Angew. Math. Mech. (ZAMM) 73 (1993), 255–263.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    M. RUIJGROK & F. VERHULST: Parametric and autoparametric resonance, Nonlinear Dynamical Systems and Chaos, Birkauser, Basel, (1996), 279–298.CrossRefGoogle Scholar
  41. 41.
    J.E. MARSDEN & M. MCCRACKEN: The Hopf Bifurcation and Its Applications, Springer, New York, 1976.zbMATHCrossRefGoogle Scholar
  42. 42.
    M. FEČKAN & J. GRUENDLER: Homoclinic-Hopf interaction: an autoparametric bifurcation, Proc. Royal Soc. Edinburgh A 130 A (2000), 999–1015.CrossRefGoogle Scholar
  43. 43.
    B. DENG & K. SAKAMATO: Šil’nikov-Hopf bifurcations, J. Differential Equations 119 (1995), 1–23.MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    A. VANDERBAUWHEDE: Bifurcation of degenerate homoclinics, Results in Mathematics 21 (1992), 211–223.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    M. GOLUBITSKY & V. GUILLEMIN: Stable Mappings and Their Singularities, Springer, New York, 1973.zbMATHCrossRefGoogle Scholar
  46. 46.
    J. KNOBLOCH & U. SCHALK: Homoclinic points near degenerate homoclinics, Nonlinearity 8 (1995), 1133–1141.MathSciNetADSzbMATHCrossRefGoogle Scholar
  47. 47.
    K.J. PALMER: Existence of a transversal homoclinic point in a degenerate case, Rocky Mountain J. Math. 20 (1990), 1099–1118.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    J. KNOBLOCH, B. MARX & M. EL MORSALANI: Characterization of Homoclinic Points Bifurcating from Degenerate Homoclinic Orbits, preprint, 1997.Google Scholar
  49. 49.
    B. MORIN: Formes canoniques des singularities d’une application différentiable, Comptes Rendus Acad. Sci., Paris 260 (1965), 5662–5665, 6503–6506.MathSciNetzbMATHGoogle Scholar
  50. 50.
    F. BATTELLI & K.J. PALMER: Tangencies between stable and unstable manifolds, Proc. Royal Soc. Edinburgh 121 A (1992), 73–90.MathSciNetCrossRefGoogle Scholar
  51. 51.
    B. AULBACH & T. WANNER: The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spacs, Nonlin. Anal., Th. Meth. Appl. 40 (2000), 91–104.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    B. AULBACH & T. WANNER: Integral manifolds for Carathéodory-type differential equations in Banach spacs, in: “Six Lectures on Dynamical System”, B. Aulbach, F. Colonius, Eds., World Scientific Publishing Co., Singapore, 1996, 45–119.CrossRefGoogle Scholar
  53. 53.
    G. COLOMBO, M. FEČKAN & B.M. GARAY: Multivalued perturbations of a saddle dynamics, Differential Equations & Dynamical Systems 18 (2010), 29–56.zbMATHCrossRefGoogle Scholar
  54. 54.
    F. BATTELLI & FEČKAN, Chaos arising near a topologically transversal homoclinic set, Top. Meth. Nonl. Anal. 20 (2002), 195–215.zbMATHGoogle Scholar
  55. 55.
    K.J. PALMER: Shadowing in Dynamical Systems, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000.zbMATHCrossRefGoogle Scholar
  56. 56.
    J.L. DALECKII & M.G. KREIN: Stability of Differential Equations, Nauka, Moscow, 1970, (in Russian).Google Scholar
  57. 57.
    A.M. SAMOILENKO & YU.V. TEPLINSKIJ: Countable Systems of Differential Equations, Brill Academic Publishers, Utrecht 2003.zbMATHCrossRefGoogle Scholar
  58. 58.
    M. PEYRARD, ST. PNEVMATIKOS & N. FLYTZANIS: Dynamics of two-component solitary waves in hydrogen-bounded chains, Phys. Rev. A 36 (1987), 903–914.ADSCrossRefGoogle Scholar
  59. 59.
    ST. PNEVMATIKOS, N. FLYTZANIS & M. REMOISSENET: Soliton dynamics of nonlinear diatomic lattices, Phys. Rev. B 33 (1986), 2308–2321.ADSCrossRefGoogle Scholar
  60. 60.
    R.B. TEW & J.A.D. WATTIS: Quasi-continuum approximations for travelling kinks in diatomic lattices, J. Phys. A: Math. Gen. 34, (2001), 7163–7180.MathSciNetADSzbMATHCrossRefGoogle Scholar
  61. 61.
    J.A.D. WATTIS: Solitary waves in a diatomic lattice: analytic approximations for a wide range of speeds by quasi-continuum methods, Phys. Lett. A 284 (2001), 16–22.ADSzbMATHCrossRefGoogle Scholar
  62. 62.
    S. AUBRY, G. KOPIDAKIS & V. KADELBURG: Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems, Discr. Cont. Dyn. Syst. B 1 (2001), 271–298.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    D. BAMBUSI & D. VELLA: Quasiperiodic breathers in Hamiltonian lattices with symmetries, Discr. Cont. Dyn. Syst. B 2 (2002), 389–399.MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    M. FEČKAN: Blue sky catastrophes in weakly coupled chains of reversible oscillators, Disc. Cont. Dyn. Syst. B 3, (2003), 193–200.zbMATHCrossRefGoogle Scholar
  65. 65.
    M. HASKINS & J.M. SPEIGHT: Breather initial profiles in chains of weakly coupled anharmonic oscillators, Phys. Lett. A 299 (2002), 549–557.MathSciNetADSzbMATHCrossRefGoogle Scholar
  66. 66.
    M. HASKINS & J.M. SPEIGHT: Breathers in the weakly coupled topological discrete sine-Gordon system, Nonlinearity 11 (1998), 1651–1671.MathSciNetADSzbMATHCrossRefGoogle Scholar
  67. 67.
    R.S. MACKAY & S. AUBRY: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity 7 (1994), 1623–1643.MathSciNetADSzbMATHCrossRefGoogle Scholar
  68. 68.
    J.A. SEPULCHRE & R.S. MACKAY: Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997), 679–713.MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Michal Fečkan
    • 1
  1. 1.Department of Mathematical Analysis and Numerical Mathematics Faculty of Mathematics, Physics and InformaticsComenius University Mlynská dolinaBratislavaSlovakia

Personalised recommendations