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Two-parameter chaos. (English) Zbl 0799.58051

Summary: Let \(I\) be a real compact interval and \(0 \leq \alpha \leq \beta\) be real numbers smaller than the length of \(I\). A continuous function \(f\) from \(I\) into itself is said to be generically or densely \((\alpha,\beta)\)- chaotic if the set of all points \([x,y]\), for which \(\liminf_{n \to \infty}| f^{(n)}(x) - f^ n(y)| \leq \alpha\) and \(\limsup_{n \to \infty}| f^ n(x) - f^ n(y)| > \beta\), is residual or dense in \(I\times I\), respectively. In the paper such functions are characterized in terms of behaviour of subintervals of \(I\) under iterates of \(f\) provided \(\alpha > 0\) [for \(\alpha = 0\) see the author, Commentat. Math. Univ. Carol. 31, No. 4, 793-810 (1990; Zbl 0724.58044) and ibid. 33, No. 4, 747-752 (1992; Zbl 0784.58043)].

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E99 Low-dimensional dynamical systems
54H20 Topological dynamics (MSC2010)
26A18 Iteration of real functions in one variable

Keywords:

chaos; interval; iterates
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