CRITICAL SETS AND PROPERTIES OF ENDOMORPHISMS BUILT BY COUPLING OF TWO IDENTICAL QUADRATIC MAPPINGS

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Abstract

ABSTRACT

In this paper, it is shown that there are parameter values such that endomorphisms built by coupling of two identical 1-dimensional quadratic mappings (a) have two kinds of trapping regions in the phase space: a large simply-connected domain inside of which there is a smaller trapping subregion consisting of two disjoint domains; (b) restrictions of the main diagonal y = x of their nonwandering sets are invariant subsets, which may not belong to attractors of the given endomorphisms, but in any way, can be nonisolated in their nonwandering sets. Numerical investigation results represented in the Appendix display the existence of a couple of bifurcation cascades and this leads to a couple of nontrivial symmetrically disposed chaotic strange attractors each of which consists of four disjoint simply connected regions. As parameters vary, these attractors merge into the one consisting at first of two and then of one such region as mentioned above.

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