The Geometrical Demonstration of the Order of Resonant Saddle Points in C2

• Wu, Hao ^[1] ; Xu Shoujun ^[1]
  1. [1] Southeast University

     Southeast University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00370-9
• Enlaces
  • Texto completo
• Resumen

  • Here we investigate the Minkowski box dimension of complex integral curves of the vector fields near resonant saddles in C2. The results provide the geometrical explanation of the order of the saddle points and a quantitative description for the non-integrability via monodromy.

• Referencias bibliográficas
  • 1. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Theory of Bifurcations of Dynamic Systems on a Plane. Wiley, New York (1973)
  • 2. Arnold, V.I., Ilyashenko, Y.S.: Dynamical systems-I, ordinary differential equations. In: Encyclopaedia of Mathematical Sciences, Vol....
  • 3. Broer, H.W.: Normal forms in perturbation theory. In: Mathematics of Complexity and Dynamical Systems, pp. 1152–1171 (2011)
  • 4. Li, W., Llibre, J., Zhang, X.: Planar analytic vector fields with generalized rational first integrals. Bull. Sci. Math. 125, 341–361 (2001)
  • 5. Mandelbrot, B.B.: The Fractal Geometry of Nature. W. H. Freeman and co., San Francisco (1982)
  • 6. Martynchuk, N., Broer, H.W., Efstathiou, K.: Hamiltonian monodromy and Morse theory. arXiv:1901.00705
  • 7. Poincaré, H.: Sur lintgration des équations différentielles du premier ordre et du premier degr I. Rendiconti del Circolo Matematico di...
  • 8. Poincaré, H.: Sur lintgration des équations différentielles du premier ordre et du premier degr II. Rendiconti del Circolo Matematico di...
  • 9. Roussarie, R.: On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Math....
  • 10. Stolovitch, L.: Smooth Gevrey normal forms of vector fields near a fixed point. Ann. Inst. Fourier 63, 241–267 (2013)
  • 11. Wu, H.: Gevrey smooth topology is proper to detect normalization under Siegel type small divisor conditions. Math. Z. 284, 1223–1243 (2016)
  • 12. Zoladek, H.: The Monodromy Group. Birkhäuser, Basel (2006)
  • 13. Zubrini´ ˘ c, D., Zubrini´ ˘ c, V.: Box dimension of spiral trajectories of some vector fields in R3. Qual. Theory Dyn. Syst. 6, 251–272...
  • 14. Zubrini´ ˘ c, D., Zubrini´ ˘ c, V.: Fractal analysis of spiral trajectories of some planar vector fields. Bull. Sci. Math. 129, 457–485...