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Periodic orbits of certain Hénon-like maps. (English) Zbl 0810.58016
For the mappings $$r_ p : \mathbb{R}^ 2 \to \mathbb{R}^ 2$$ given by $$r_ p (x,y) = (bx + dy - f(p,x), cy)$$ $$[b, d, c \in \mathbb{R}$$, $$cd = -1$$, $$p \in \mathbb{R}$$ a parameter, $$f \in C^ 2(\mathbb{R}^ 2, \mathbb{R})$$, $$f(p, 0) = 0$$] conditions are given under which to any neighbourhood $$U$$ of $$(0,0)$$ in $$\mathbb{R}^ 2$$ and any $$n \geq 1$$ the set of all $$p$$ for which $$r_ p$$ has periodic orbits in $$U$$ with minimal period $$\geq n$$ is dense in a certain interval. Similar results concern mappings $$(x,y) \mapsto (bx + dy - q(x), cx)$$, when $$q : \mathbb{R} \to \mathbb{R}$$ has finitely many zeros and $$\lim_{| x | \to \infty} q(x)/x$$ exists. Here, of course, the cycles are not necessarily close to (0,0). The methods are related to those developed by S. B. Angenent [Commun. Math. Phys. 115, No. 3, 353-374 (1988; Zbl 0665.58034)].
Reviewer: H.G.Bothe (Berlin)
##### MSC:
 37B99 Topological dynamics 39B12 Iteration theory, iterative and composite equations
##### Keywords:
Hénon-like maps; periodic orbits
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##### References:
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