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An improvement of the non-existence region for limit cycles of the Bogdanov-Takens system. (English) Zbl 1524.34084

For the universal unfolding (also known as the Bogdanov-Takens system) \[ \dot{x}=y,\;\;\dot{y}=\mu_1+\mu_2y+xy+x^2,\tag{1} \] the author’s aim is to enlarge the domain in the parameter space \((\mu_1,\mu_2)\subset\mathbb{R}^2\) where System (1) surely has no limit cycles. The new result here is that in the parameter set given by \(F=\{(\mu_1,\mu_2):\mu_1\leq -(\mu_2+\epsilon)^2,\,\epsilon\geq\frac13\}\), there are no limit cycles in (1). This extends to \(E\cup F\) the similar previous author’s result about \(E\) [M. Hayashi, Far East J. Math. Sci. (FJMS) 14, No. 1, 127–136 (2004; Zbl 1105.34013)]. A comparison with some other earlier results is also provided in the paper.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations

Citations:

Zbl 1105.34013
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References:

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