Abstract
We consider several classes of dynamical systems on manifolds, like Hamiltonian systems, volume preserving systems, etc. On each of these classes there is a more or less natural topology. The word generic is used for properties of dynamical systems which hold for almost all elements, in the topological sense, of a given class. Formal definitions will be given below. Generic properties are known which imply a certain local simplicity of the system, e.g. that fixed points or equilibria are isolated. Still for such a system the global dynamics can be very complicated.
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References
V. I. Arnol’d, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer-Verlag, 1982. (Russian original: Moscow, 1977.)
V. I. Arnol’d and A. Avez, Ergodic Problems of Classical Mechanics. Benjamin 1978. (French original: Gauthier-Villars, 1967.)
G. D. Birkhoff, Dynamical Systems. AMS Publications, 1927.
G. D. Birkhoff, Nouvelles recherches sur les systèmes dynamiques. Mem. Pont. Acad. Sci. Novi Lyncaei 1 (1935), 85–216.
E. Borel, Sur quelques points de la théorie des fonctions (Thèse). Ann. Sci. École Norm. Sup., 3 (1895), 9–55.
B. L. J. Braaksma, H. W. Broer and G. B. Huitema, Towards a Quasi-periodic Bifurcation Theory. Preprint, Groningen, 1988.
H. W. Broer, Bifurcations of Singularities in Volume Preserving Vector Fields. PhD thesis, Groningen, 1979.
H. W. Broer, Formal normal forms for vector fields and some consequences for bifurcations in the volume preserving case. In Dynamical Systems and Turbulence, Warwick, 1980, Springer LNM 898.
H. W. Broer, Quasi-periodic flow near a codimension one singularity of a divergence free vector field in dimension three. In Dynamical Systems and Turbulence, Warwick, 1980, Springer LNM 898.
H. W. Broer and S. J. van Strien, Infinitely many moduli of strong stability in divergence free unfoldings of singularities of vector fields. In Geometric Dynamics, Proceedings, Rio de Janeiro, 1981, Springer LNM 1007.
H. W. Broer and Tangerman, From a differentiable to a real analytic perturbation theory, applications to the Kupka-Smale theorems. Ergod. Th. and Dynam. Sys., 6 (1986), 345–62.
H. W. Broer and G. Vegter, Subordinate Sil’nikov bifurcations near some singularities of vector fields having low codimension. Ergod. Th and Dynam. Sys., 4 (1984), 509–525.
J. Carr, Applications of Center Manifold Theory. Appl. Math. Sci. 35, Springer-Verlag, 1981.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 1983.
M. R. Herman, Mesure de Lebesque et nombre de rotation. In Geometry and Topology, Rio de Janerior, July 1976, Springer LNM 597.
M. W. Hirsch, Differential Topology. Springer-Verlag, 1976.
I. Kupka, Contribution à la théorie des champs génériques. Contr. Diff. Eqs., 2 (1963), 457–84; ibid., 3 (1964), 411–420.
J. K. Moser, Nonexistence of integrals for canonical systems of differential equations. Comm. Pure Appl. Math., 8 (1955), 409–436.
J. K. Moser, Lectures on Hamiltonian systems. Mem. of the AMS, 81 (1968), 1–60.
J. Palis and W. C. de Melo, Geometric Theory of Dynamical Systems. Springer-Verlag, 1982.
H. Poincaré, Thèse. In Oeuvres I, pp. LIX-CXXIX, 1879. Gauthier-Villars, 1928.
R. C. Robinson, Generic properties of conservative systems I, II. Amer. J. Math., 92 (1972), 562–603, 897–906.
L. P. Sil’nikov, A case of the existence of a denumerable set of periodic motions. Soviet Math. Dokl., 6 (1965), 163–6.
L. P. Sil’nikov, A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle focus type. Math. USSR Sbornik, 10 (1970), 91–102.
S. Smale, Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Normale Superiore Pisa, 18 (1963), 97–116.
S. Smale, Differentiable dynamical systems. Bull. Am. Math. Soc., 73 (1967), 747–817.
J. Sotomayor, Generic bifurcations in dynamical systems. In Dynamical Systems, Academic Press, 1973.
S. J. van Strien, Center manifolds are not C. Math. Z., 166 (1979), 143–5.
F. Takens, A nonstabilisable jet of a singularity of a vector field. In Dynamical Systems, Academic Press, 1973.
F. Takens, Singularities of vector fields. Publ. Math. IHES, 43 (1974), 48–100.
F. Takens, Forced oscillations and bifurcations. In Applications of Global Analysis I, Comm. Math. Inst. Rijksuniversiteit Utrecht, 3 (1974), 1–59.
J.-C. Yoccoz, C1-conjugaison des difféomorphismes du cercle. In: Geometric Dynamics, Proceedings, Rio de Janeiro 1981, Springer LNM 1007.
E. Zehnder, Homoclinic points near elliptic fixed points. Comm. Pure Appl. Math., 26 (1973), 131–82.
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© 1989 John Wiley & Sons Ltd and B. G. Teubner
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Broer, H.W., Takens, F. (1989). Formally Symmetric Normal Forms and Genericity. In: Kirchgraber, U., Walther, H.O. (eds) Dynamics Reported. Dynamics Reported, vol 2. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-96657-5_2
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DOI: https://doi.org/10.1007/978-3-322-96657-5_2
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
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