DOI QR코드

DOI QR Code

PERSISTENCE OF HOMOCLINIC ORBITS AFTER DISCRETIZATION OF A TWO DIMENSIONAL DEGENERATE DIFFERENTIAL SYSTEM

  • Mehidi, Noureddine (Departement de mathematiques Laboratoire de Mathematiques Appliquees Universite A. Mira de Bejaia) ;
  • Mohdeb, Nadia (Departement de mathematiques Laboratoire de Mathematiques Appliquees Universite A. Mira de Bejaia)
  • Received : 2013.08.01
  • Published : 2014.09.30

Abstract

The aim of this work is to construct a general family of two dimensional differential systems which admits homoclinic solutions near a non-hyperbolic fixed point, such that a Jacobian matrix at this point is zero. We then discretize it by using Euler's method and look after the persistence of the homoclinic solutions in the obtained discrete system.

References

  1. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second Order Dynamic Systems, John Wiley and Sons, New York, Toronto, 1973.
  2. W. J. Beyn, The effect of discretization on homoclinic orbits, Bifurcation, Analysis, Algorithms, Applications, Birkhauser, Basel, 1-8, 1987.
  3. W. J. Beyn, On invariant closed curves for one-step methods, Numer. Math. 51 (1987), no. 1, 103-122. https://doi.org/10.1007/BF01399697
  4. W. J. Beyn and B. M. Garay, Estimates of variable stepsize Runge-Kutta methods for sectorial evolution equations with nonsmooth data, Appl. Numer. Math. 41 (2002), no. 3, 369-400. https://doi.org/10.1016/S0168-9274(01)00126-X
  5. W. J. Beyn and J. M. Kleinhauf, The numerical computation of homoclinic orbits for maps, SIAM J. Numer. Anal. 34 (1997), no. 3, 1207-1236. https://doi.org/10.1137/S0036142995281693
  6. W. J. Beyn and M. Stiefenhofer, A direct approach to homoclinic orbits in the fast dynamics of singularly perturbed systems, J. Dynam. Differential Equations 11 (1999), no. 4, 671-709. https://doi.org/10.1023/A:1022663512855
  7. N. Chen, J. Sun, Y. Sun, and M. Tang, Visualizing the complex dynamics of families of polynomials with symmetric critical points, Chaos Solitons Fractals 42 (2009), no. 3, 1611-1622. https://doi.org/10.1016/j.chaos.2009.03.042
  8. E. J. Doedel, M. J. Friedman, and B. I. Kunin, Successive continuation for locating connecting orbits, Numer. Algorithms 14 (1997), no. 1-3, 103-124. https://doi.org/10.1023/A:1019152611342
  9. E. J. Doedel, M. J. Friedman, and A. C. Monteiro, On locating connecting orbits, Appl. Math. Comput. 65 (1994), no. 1-3, 231-239. https://doi.org/10.1016/0096-3003(94)90179-1
  10. B. Fiedler and J. Sheurle, Discretization of homoclinic orbits, rapid forcing and "invis-ible" chaos, Mem. Amer. Math. Soc. 119 (1996), no. 570, viii+79 pp.
  11. S. J. Greenfield and R. D. Nussbaum, Dynamics of a quadratic map in two complex variables, J. Differential Equations 169 (2001), no. 1, 57-141. https://doi.org/10.1006/jdeq.2000.3895
  12. S. Lefschetz, Differential Equations: Geometric Theory, Interscience publishers, 1957.
  13. N. Mehidi and N. Mohdeb, Homoclinic solutions in a quadratic differential system under discretization, J. Difference Equ. Appl. 19 (2013), no. 4, 538-542. https://doi.org/10.1080/10236198.2012.658383
  14. Y. K. Zou and W. J. Beyn, On manifolds of connecting orbits in discretizations of dynamical systems, Nonlinear Anal. 52 (2003), no. 5, 1499-1520. https://doi.org/10.1016/S0362-546X(02)00269-9