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Totally Bounded Cubic Systems in ℝ2

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1822)

Abstract

This work describes the behavior of trajectories of a planar cubic system such that

a) all the trajectories are bounded,

b) there exist just two singular points S, O,

c) the system is reversible about the line SO.

The exposition is divided into:

i) there is no heterocline. In particular O is a Poincaré center (Sec. 2.13), a degenerate center (Sec. 2.14), a right pseudo-center (Sec. 2.15), a left pseudo-center (Sec. 2.16);

ii) there exists just one pair of heteroclines. In particular O is a Poincaré saddle (Sec. 2.17), a degenerate saddle (Sec. 2.18), a tangential limit point with indexO=1 (Sec. 2.19);

iii) there exists a pair of bands of heteroclines (Sec. 2.23).

Maintaining the total boundedness, i.e., the boundedness of all the trajectories, the following topics

α ) O is the unique singular point (Sec. 1.5),

β ) invariancy of an ellipse (Sec. 2.20),

γ ) existence of limit cycles (Sec. 3.1)

are also considered.

Keywords

  • Singular Point
  • Phase Portrait
  • Regular System
  • Limit Line
  • Elliptic Sector

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Correspondence to Roberto Conti .

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© 2003 Springer-Verlag Berlin Heidelberg

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Conti, R., Galeotti, M. (2003). Totally Bounded Cubic Systems in ℝ2 . In: Macki, J.W., Zecca, P. (eds) Dynamical Systems. Lecture Notes in Mathematics, vol 1822. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45204-1_2

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  • DOI: https://doi.org/10.1007/978-3-540-45204-1_2

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40786-7

  • Online ISBN: 978-3-540-45204-1

  • eBook Packages: Springer Book Archive