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Discretization of second-order variational systems. (English) Zbl 0770.58008
Let us consider the mapping \(\varphi_ h(x,y)=(2\cdot x-y+h^ 2\cdot f(x),x)\), where \((x,y)\in R^ m\times R^ m\), \(f:R^ m\to R^ m\), \(f=\text{grad }g\), \(f(0)=0\), \(h>0\) is a small parameter. The author proves that \(\varphi_ h\) possesses periodic orbits for a dense set of parameter values \(h\). If the mapping \(f\) is asymptotically linear, then for fixed \(h > 0\) sufficiently small there is an increasing sequence of natural numbers \(\{n_ i\}^ \infty_{i=0}\), \(\lim_{i \to \infty}n_ i = \infty\) and \(\varphi_ h\) has a periodic orbit with the minimal period \(n_ i\) for each \(i\).
An interesting conjecture and discussion about the number of families of invariant circles of the mapping \(\varphi_ h\) is presented.
Reviewer: A.Klíč (Praha)

58E30 Variational principles in infinite-dimensional spaces
37B99 Topological dynamics
37G99 Local and nonlocal bifurcation theory for dynamical systems
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