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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)

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