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Cubic systems with four real line invariants

Published online by Cambridge University Press:  24 October 2008

Robert E. Kooij
Robert E. Kooij
Affiliation:
University of Technology, Delft, Mekelweg 4, 2628 CD Delft, Netherlands
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Extract

A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:

with aij, bij ∈ ℝ and where x = x(t) and y = y(t) are real-valued functions.

The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[16]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[9] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for nth degree polynomial systems, in terms of n. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[8] or Il'yashenko[10]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of n. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 7 - 19
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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