\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Blue sky-like catastrophe for reversible nonlinear implicit ODEs

Abstract / Introduction Related Papers Cited by
  • We study for reversible implicit differential equations the bifurcation of bounded solutions connecting singularities in finite time and their approximation by shadowed periodic solutions. Melnikov like condition is derived. Application is given to planar nonlinear RLC system.
    Mathematics Subject Classification: Primary: 34A09, 34C23; Secondary: 37G40.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    F. Battelli and M. Fečkan, Melnikov theory for nonlinear implicit ODEs, J. Differential Equations, 256 (2014), 1157-1190.doi: 10.1016/j.jde.2013.10.012.

    [2]

    ________, Nonlinear RLC circuits and implicit ODEs, Differential Integral Equations, 27 (2014), 671-690.

    [3]

    ________, Melnikov theory for weakly coupled nonlinear RLC circuits}, Bound. Value Probl., 2014 (2014), 27pp.doi: 10.1186/1687-2770-2014-101.

    [4]

    A. W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Math., Vol. 629, Springer-Verlag, Berlin, 1978.

    [5]

    R. Devaney, Blue sky catastrophes in reversible and Hamiltonian systems, Indiana Univ. Math. J., 26 (1977), 247-263.doi: 10.1512/iumj.1977.26.26018.

    [6]

    M. C. Irwin, On the smoothness of the composition map, Quart. J. Math. Oxford Ser. (2), 23 (1971), 113-133.doi: 10.1093/qmath/23.2.113.

    [7]

    E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, Inc., New York, 1989.

    [8]

    X. B. Lin, Using Melnikov's method to solve Shilnikov's problems, Proc. Royal Soc. Edinburgh A, 116 (1990), 295-325.doi: 10.1017/S0308210500031528.

    [9]

    K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differential Equations, 65 (1986), 321-360.doi: 10.1016/0022-0396(86)90023-9.

    [10]

    P. J. Rabier and W. C. Rheinboldt, A general existence and uniqueness theorem for implicit differential algebraic equations, Differential Integral Equations, 4 (1991), 563-582.

    [11]

    ________, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations, 109 (1994), 110-146.doi: 10.1006/jdeq.1994.1046.

    [12]

    ________, On impasse points of quasilinear differential algebraic equations, J. Math. Anal. Appl., 181 (1994), 429-454.doi: 10.1006/jmaa.1994.1033.

    [13]

    ________, On the computation of impasse points of quasilinear differential algebraic equations, Math. Comp., 62 (1994), 133-154.doi: 10.2307/2153400.

    [14]

    R. Riaza, Differential-Algebraic Systems, Analytical Aspects and Circuit Applications, World Sci. Publ. Co. Pte. Ltd., Hackensack, NJ, 2008.

    [15]

    A. Vanderbauwhede, Heteroclinic cycles and periodic orbits in reversible systems, in Ordinary and Delay Differential Equations, (eds. J. Wiener and J.K. Hale), Pitman Res. Notes Math. Ser., 272, Longman Sci. Tech., Harlow, (1992), 250-253.

    [16]

    A. Vanderbauwhede and B. Fiedler, Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. (ZAMP), 43 (1992), 292-318.doi: 10.1007/BF00946632.

  • 加载中
SHARE

Article Metrics

HTML views(851) PDF downloads(147) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return