Periodic Behavior of Maps Obtained by Small Perturbations of Smooth Skew Products

2020 ◽  
Vol 9 (4) ◽  
pp. 519-523 ◽  
Author(s):  
L.S. Efremova
Keyword(s):  
Small Perturbations ◽  
Periodic Behavior ◽  
Skew Products

ATTRACTORS AND THEIR INVISIBLE PARTS FOR SKEW PRODUCTS WITH HIGH DIMENSIONAL FIBER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250182 ◽  
Author(s):  
F. H. GHANE ◽  
M. NAZARI ◽  
M. SALEH ◽  
Z. SHABANI
Keyword(s):  
Compact Manifold ◽  
High Dimensional ◽  
Skew Product ◽  
Small Perturbations ◽  
Skew Products ◽  
Lipschitz Constants ◽  
Structurally Stable

In this article, we study statistical attractors of skew products which have an m-dimensional compact manifold M as a fiber and their ε-invisible subsets. For any n ≥ 100 m2, m = dim (M), we construct a set [Formula: see text] in the space of skew products over the horseshoe with the fiber M having the following properties. Each C2-skew product from [Formula: see text] possesses a statistical attractor with an ε-invisible part, for an extraordinary value of ε (ε = (m + 1)-n), whose size of invisibility is comparable to that of the whole attractor, and the Lipschitz constants of the map and its inverse are no longer than L. The set [Formula: see text] is a ball of radius O(n-3) in the space of skew products over the horseshoe with the C1-metric. In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Moreover, for skew products which have an m-sphere as a fiber, it consists of structurally stable skew products. Our construction develops the example of [Ilyashenko & Negut, 2010] to skew products which have an m-dimensional compact manifold as a fiber, m ≥ 2.

A Note on Patterns of Wear Within the Tread Area

1979 ◽  
Vol 7 (1) ◽  
pp. 3-13
Author(s):  
F. C. Brenner ◽  
A. Kondo
Keyword(s):  
Content Analysis ◽  
Groove Depth ◽  
Climatic Data ◽  
Periodic Behavior ◽  
Harmonic Content ◽  
Straight Line ◽  
Data Points

Abstract Tread wear data are frequently fitted by a straight line having average groove depth as the ordinate and mileage as the abscissa. The authors have observed that the data points are not randomly scattered about the line but exist in runs of six or seven points above the line followed by the same number below the line. Attempts to correlate these cyclic deviations with climatic data failed. Harmonic content analysis of the data for each individual groove showed strong periodic behavior. Groove 1, a shoulder groove, had two important frequencies at 40 960 and 20 480 km (25 600 and 12 800 miles); Grooves 2 and 3, the inside grooves, had important frequencies at 10 240, 13 760, and 20 480 km (6400, 8600, and 12 800 miles), with Groove 4 being similar. A hypothesis is offered as a possible explanation for the phenomenon.

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