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Non-autonomous equations related to polynomial two-dimensional systems. (English) Zbl 0618.34026

As is well known, Hilbert’s celebrated 16th problem is the question of the number of limit cycles of plane polynomial systems. It is a long- standing fascinating problem and continues to attract much interest.
Some good results for plane homogeneous polynomial systems (1) \(\dot x=\lambda x+y+p(x,y)\), \(\dot y=-x+\lambda y+q(x,y)\), have already been obtained. If p and q are quadratic, then (1) can have at most 3 small amplitude limit cycles around any one singularity; if p and q are cubic, then (1) have at most 5 such limit cycles; examples of systems with exactly 3 and 5 cycles were constructed, respectively. Suppose p and q are of degree n (n\(\geq 2)\). In polar coordinates, (1) can be transformed into the form (2) \(dp/d\theta =A(\theta)p^ 3-B(\theta)p^ 2-\lambda (n-1)p.\) Limit cycles of (1) correspond to positive \(2\pi\)-periodic solutions of (2). It is certainly useful to consider the appropriate complexified form of (2): (3) \(dZ/d\theta =\alpha (\theta)Z^ 3+\beta (\theta)Z^ 2+\gamma (\theta)Z\) where Z is complex-valued and \(\alpha\),\(\beta\),\(\gamma\) are real functions of real independent variable \(\theta\). Specify a real number \(\omega\) and seek solutions of (4) with \(Z(0)=Z(\omega)\). Clearly, (4) has at least as many periodic solutions as (2). In the particular case arising from Hilbert’s 16th problem, \(\omega =2\pi\), \(\alpha\) and \(\beta\) are polynomials of cos t and sin t, \(\gamma\) is independent of t.
In this paper the maximum number of periodic solutions of the one- dimensional non-autonomous equations (3) are estimated when the coefficients are polynomials of t or cos t, sin t. It answers some questions of A. Lins Neto [Invent. Math. 59, 67-76 (1980; Zbl 0448.34012)]. Finally the author returns to Hilbert’s problem, the paper’s original motivation. He gives a condition for the origin of (1) to be a center which is a generalisation of the classical divergence criterion.
Reviewer: J.H.Tian

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations

Citations:

Zbl 0448.34012
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References:

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