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Shift spaces and attractors in noninvertible horseshoes

100%

Bothe H. G.

Fundamenta Mathematicae

1997

tom 152

nr 3

267-289

As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square $I^2$ in $ℝ^2$ (or more generally, of the cube $I^m$ in $ℝ^m$) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of $I^2$ (or $I^m$). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.

Ergodic theory approach to chaos: Remarks and computational aspects

88%

Mitkowski P., Mitkowski W.

International Journal of Applied Mathematics and Computer Science

2012

tom 22

nr 2

259-267

We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 International Journal of Applied Mathematics and Computer Science

1 Bothe H. G.

1 Mitkowski P.

1 Mitkowski W.

1 2012

1 1997

Wyniki wyszukiwania

Shift spaces and attractors in noninvertible horseshoes

Ergodic theory approach to chaos: Remarks and computational aspects