An Elementary Study of Chaotic Behaviors in 1-D Maps

Abstract

In this article, we have discussed basic concepts of one-dimensional maps like Cubic map, Sine map and analyzed their chaotic behaviors in several senses in the unit interval. We have mainly focused on Orbit Analysis, Time Series Analysis, Lyapunov Exponent Analysis, Sensitivity to Initial Conditions, Bifurcation Diagram, Cobweb Diagram, Histogram, Mathematical Analysis by Newton's Iteration, Trajectories and Sensitivity to Numerical Inaccuracies of the said maps. We have tried to make decision about these mentioned maps whether chaotic or not on a unique interval of parameter value. We have performed numerical calculations and graphical representations for all parameter values on that interval and have tried to find if there is any single value of parameter for which those maps are chaotic. In our calculations we have found there are many values for which those maps are chaotic. We have showed numerical calculations and graphical representations for single value of the parameter only in this paper which gives a clear visualization of chaotic dynamics. We performed all graphical activities by using Mathematica and MATLAB.

IFT20 and WWTR1 govern bone homeostasis via synchronously regulating the expression and stability of TβRII in osteoblast lineage cells

“…[39][40][41] Dysfunction of osteoblasts could delay bone formation and cause a spectrum of diseases, such as osteoporosis and dwarfism. 17,42 Here, by combined deletions of IFT20 and WWTR1 in osteoblasts using OSX-Cre mice, we found for the first time that loss of IFT20 and WWTR1 in osteoblasts significantly increased MAT accumulation and reduced bone mass, showing a severe osteoporosislike phenotype in 12-week-old dKO mice. Mechanistically, we revealed that IFT20 and WWTR1 play essential roles in osteoblasts for controlling MAT and bone homeostasis via regulating the stability and expression of TβRII.…”

“…15,16 Dysfunction of osteoblasts could result in various bone diseases, including osteoporosis. 17,18 Moreover, disruption of the balance between adipogenesis and osteogenesis has been reported to contribute to osteopenia accompanied by progressive marrow adiposity. Nevertheless, what signaling molecules or critical genes control the commitment and function of osteoprogenitor cells are still undiscovered.…”