Modeling And Solving Of Two Nonlinear Phenomena

People are more familiar to describe the dynamics law with the linear method on understanding and interpretation of natural phenomena.Linear equations are easily solved and have relatively simple properties(such as superposition principle).So far,the physical phenomena and other natural phenomena or social phenomena are actually complicated,their dynamic properties can't be solved by linear method,and need to be established a nonlinear model.In addition to the rare number of these nonlinear equations,almost of all are no analytic solutions and they have some special properties different from one of linear.Therefore,research of nonlinear dynamics is coming.Two nonlinear phenomena are modeling and studying in this paper.The first is to establish nonlinear dynamic equations for the core oscillator of cyanobacteria biological clock.The circadian rhythm is the near periodic for adapting environment,therefore,various of physiological functions in organisms are set up.The biological clock which widely existed in the living body is a kind of invisible "clock" in order to adapt the environment changes.According to rhythm of physiological activities of cyanobacteria and the interactions among the core oscillator KaiA,KaiB and KaiC protein of biological clock,we established relationships of regulating and feedback cycle of key proteins in the core oscillator of biological clock of cyanobacteria,and then established nonlinear dynamical equations of the protein secretion,stimulation and controlled.Numerical solutions of the protein secretion with the time variation was getting including periodic oscillatory,double periodic and chaotic solutions.The circadian rhythm nonlinear characteristics of core oscillator of biological clock of cyanobacteria were obtained using theoretical simulation.The second is to study the nonlinear characteristics of forced oscillation by using Pohl resonator.Continuous actions of the periodic external force cause the forced oscillation.Using the experimental data from Pohl resonator,the nonlinear equations of forced vibrations were established after a theoretical correction of the bearing friction.The periodic oscillation and chaos maps of numerical solutions are obtained.The results of obvious nonlinear characteristic of max amplitude of the forced oscillation amplitude were obtained.