1 results
Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 1 / February 2023
- Published online by Cambridge University Press:
- 01 December 2021, pp. 104-114
- Print publication:
- February 2023
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In this paper we consider the unfolding of saddle-node
\[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \]parametrized by $(\varepsilon,\,a)$
with $\varepsilon \approx 0$
and $a$
in an open subset $A$
of $ {\mathbb {R}}^{\alpha },$
and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$
of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$
tends to $-\infty$
as $(s,\,\varepsilon )\to (0^{+},\,0)$
uniformly on compact subsets of $A.$
This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.
