Does A Chaotic System Dynamic Really Exist In Nature Or Is It A Misconception Dynamics?: A Hypothesis

Abstract

The theory of nonlinear dynamieal systems deals with deterministie systems that exhibit a eomplieated and random-Iooking behavioL Life seienees have been one of the most applieable areas for the ideas of ehaos beeause of the eomplexity of biologieal systems. it is widely appreeiated that ehaotie behavior dominates physiologieal systems. However, as an extension of this trend, a new hypothesis is proposed that the existenee of embedded nonlinear systems suggest a new rationale fundamentally whieh is different from the classic approaeh. A biologieal system can be eonsidered as a simple explanation of transitions breaking up generic orbits onto higher dimensions with eovering maps by preventing ehaos. We seek to diseuss and understand how a biologieal system can deerease its vulnerability to sensitivity at system transitions what we define those transitions as injeetive immersions of differentiable smooth manifolds with eaeh eorresponding to a transition to different state Iike synehronization, anti-synehronization and oseillator-death when network strueture varies abruptly and asynehronously. We can then eonsider a biologieal system if an existenee of such a unique immersed smooth submanifoJd into higher dimensional space can be shown that there is no ehaotie dynamies assoeiated with a map from one manifold to another one when the system is perturbed. We then will introduee an open problem whether Melnikov funetion is a eontinuously deereasing funetion for smail perturbations which this distanee funetion serves as a diseriminate funetion for implieations of the ehaos transitions.