[1] V. 1. Arnold, Geometrical Method in the Theory of Ordinary Dijjerential Equations, Springer-Verlag, Berlin (1983).
[2] R. I. Bogdanov, Bifurcation of the limit cycle of a family of plane vector field, Sel. Math.Sov., 1 (1981), 373-387.
[3] R. I. Bogdanov, Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues, Sel. Math. Sov., 1 (1981), 389-421.
[4] A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin(1989).
[5] S. N. Chow and J. K. Hale, Methods of B办rcation Theory, Springer-Verlag, Berlin(1982).
[6] S. N. Chow and D. Wang, Normal form of bifurcating periodic orbits, Multi-parameter bifurcation theory, M. Golubitsky and J. Guckenheimer (eds), Contemporary Math., 56(1986), 9-18.
[7] R. Cushman and J. Sanders, Nilpotent normal forms and representation theory of s1(2, R), Multi-parameter bifurcation theory, M. Golubitsky and J. Gguckenheimer (eds), Contemporary. Math., 56 (1986), 31-51.
[8] R. Cushman and J. Sanders;Splitting algorithm for nilpotent normal forms, Dynamics and Stabilitv oJSystems. 2 (1988), 235-246.
[9] R. Cushman, A. Deprit and R. Mosad, Normal forms and representation theory, J.Math. Phys., 24 (1983), 2103-2116.
[10] C. Elphick, E. Tirapegui, M. E. Bracher, P. Coullet and G. Iooss, A simple global characterization for normal forms of singular vector fields. Phys. D., 29 (1987), 95-117.
[11] J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin (1983).
[12] R. Rand and D. Armbruster, Perturbation Method, B如rcation Theory and Computer Algebra, Springer-Verlag, Berlin (1987).
[13] F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier, 23(1973), 163-195.
[14] F. Takens, Singularities of vector fields, Publ. Math. IHES, 43 (1974), 47-100.
[15] P. Holmes and D. A. Rand, Phase portraits and bifurcations of the nonlinear oscillator x+(a+yx2)x+βx+δ3=0, Int. J. Non-Linear Mech., 15 (1980), 449-458.
[16] P. Holmes, Center manifolds, normal forms and bifurcations of vector Gelds with application to coupling between periodic and steady motions, Phy. D., 2 (1981), 449-481.
[17] A. K. Bajaj, Bifurcations in a parametrically excited nonlinear oscillator, Int. J.Nonlinear Mech., 22 (1987), 47-59.
[18] A. K. Bajaj, Nonlinear dynamics of tubes carrying a pulsatile folw, Dynamics and Stability of Systems, 2 (1987), 19-41.
[19] J. Shaw and S. W. Shaw, The effects of unbalance on oil whirl, Nonlinear Dynamic's, 1(1990), 293-311.
[20] N. Sri. Namachchivaya, Co-dimension two bifurcations in the presence of noise, ASME, J. Appl. Mech., 58 (1991), 259-265.
[21] W. Zhang and Q. Z. Huo, Degenerate bifurcations of codimension two in nonlinear oscillator under combined parametric and forcing excitation, Acta Mechanica Sinica, 24(1992), 717-727.
[22] W. Zhang and Q. Z. 1-Iuo, Degenerate bifurcations of codimension two in nonlinear oscillator for 1/2 subharmonic resonance-primary parametric resonance, Theory, Method and Application of Nonlinear Mechanics, C. J. Cheng and Z. H. Guo (eds), Modern Mathematics and Mechanics (MMM) IV (1991), 431- 437.
[23] W. Zhang and Q. Z. Huo, Bifurcations of the cusp singularity in a nonlinear oscillator under combined parametric and forcing excitation, J. Vibration Engineering, 6 (1992), 355-366.
[24] Y. S. Chen and J. Xu, Periodic respones and bifurcation theory of nonlinear Hill system, J. Nonlinear Dynamics in Science and Technology, 1 (1993), 1-14.
[25] F. Dumortier, R. Roussarie aid J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part the cusp case, Ergodic Theory and Dynamical, Systems, 7 (1987), 375-413.
[26] F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of planar vector fields, unfolding of saddle, focus and elliptic singularities with nilpotent linear parts, Preprint .(1990).
[27] F. Dumortier and P. Fiddelares, Quadratic models for generic local 3 parameter bifurcations on the plane, Trans. Amer. Math. Soc., 326 (1991), 101-126.
[28] D. Wang, An introduction to the normal form theory of ordinary differentital equations, Advances in Mathematics, 19 (1990), 38-71.
[29] W. Zhang, Computation of the higher order normal form and codimension three degenerate bifurcation in a nonlinear dynamical system with Z2-symmetry, Acta Mechanica Sinica, 25 (1993), 548-559.
[30] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, John Wiley and Sons, Interscience (1980). |
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