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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
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References:
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