We consider a family of one-dimensional discontinuous invertible maps from an application in engineering. It is defined by a linear function and by a hyperbolic function with real exponent. The presence of vertical and horizontal asymptotes of the hyperbolic branch leads to particular codimension-two border collision bifurcation (BCB) such that if the parameter point approaches the bifurcation value from one side then the related cycle undergoes a regular BCB, while if the same bifurcation value is approached from the other side then a nonregular BCB occurs, involving periodic points at infinity, related to the asymptotes of the map. We investigate the bifurcation structure in the parameter space. Depending on the exponent of the hyperbolic branch, different period incrementing structures can be observed, where the boundaries of a periodicity region are related either to subcritical, or supercritical, or degenerate flip bifurcations of the related cycle, as well as to a regular or nonregular BCB. In particular, if the exponent is positive and smaller than one, then the period incrementing structure with bistability regions is observed and the corresponding flip bifurcations are subcritical, while if the exponent is larger than one, then the related flip bifurcations are supercritical and, thus, also the regions associated with cycles of double period are involved into the incrementing structure.
In this paper we prove the existence of full measure unbounded chaotic attractors which are persistent under parameter perturbation (also called robust). We show that this occurs in a discontinuous piecewise smooth one-dimensional map f , belonging to the family known as Nordmark's map. To prove the result we extend the properties of a full shift on a …nite or in…nite number of symbols to a map, here called Baker-like map with in…nitely many branches, de…ned as a map of the interval I = [0; 1] into itself with in…nitely branches due to expanding functions with range I except at most the rightmost one. The proposed example is studied by using the …rst return map in I, which we prove to be chaotic in I making use of the border collision bifurcations curves of basic cycles. This leads to a robust unbounded chaotic attractor, the interval (1; 1], for the map f. Kyewords. Unbounded chaotic attractors, Robust full measure chaotic attractors, Piecewise smooth systems, Full shift maps, Border collision bifurcations
We investigate the dynamics of a family of one-dimensional linearpower maps. This family has been studied by many authors mainly in the continuous case, associated with Nordmark systems. In the discontinuous case, which is much less studied, the map has vertical and horizontal asymptotes giving rise to new kinds of border collision bifurcations. We explain a mechanism of the interplay between smooth bifurcations and border collision bifurcations with singularity, leading to peculiar sequences of attracting cycles of periods n, 2n, 4n−1, 2(4n−1),..., n ≥ 3. We show also that the transition from invertible to noninvertible map may lead abruptly to chaos, and the role of organizing center in the parameter space is played by a particular bifurcation point related to this transition and to a flip bifurcation. Robust unbounded chaotic attractors characteristic for certain parameter ranges are also described. We provide proofs of some properties of the considered map. However, the complete description of its rich bifurcation structure is still an open problem.
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