Applications Of Bifurcation Theory In Chemical Reaction Models

Three nonlinear dynamical systems with theoretical significance and extremely extensive applied value are taken as research objects in this paper. These systems are Belousov-Zhabotinsky(BZ) reaction, power system with three machines and four buses, cancer growth model. Based on bifurcation theory, stability and qualitative theory of dynamical system equation, chaos theory, their nonlinear dynamics are investigated mainly in the following hints: 1) In theory, the existence, stability and type of equilibrium points are discussed by applying stability and qualitative theory of dynamical system equation; The existence of static bifurcations is verified by applying the center manifold relevant theory and the local bifurcation theory in time domain; The existence of Hopf bifurcations is analyzed by applying bifurcation theory in frequency domain; The approach of the fourth-order harmonic balance and sixth-order harmonic balance is applied to provide the high-precision approximate analytical expressions of periodic solutions generated from Hopf bifurcations and their frequencies and amplitudes; Moreover, the stability and location of these periodic solutions is judged by means of harmonic balance approach; 2) In numerical simulation, the local bifurcation diagrams of equilibrium points and periodic solutions, projections and time series are derived by applying the software of matlab and Auto2007; The validity of theoretical analysis is further verified by computing the eigenvalues of equilibrium points, Lyapunov exponents, and the Floquet multipliers of periodic orbits. Meanwhile, other complex dynamics are also found.This paper consists five chapters as follows.The research background of three nonlinear dynamical systems and some basic numerical and theoretical methods about dynamical systems are briefly introduced in chapter one.The bifurcation phenomena in a BZ reaction model are investigated in chapter two. The existence of two supercritical Hopf bifurcations is strictly proved by applying Hopf bifurcation theory in frequency domain. The fourth-order harmonic balance approximate analytical expressions, frequencies and amplitudes of periodic solutions generated from these Hopf bifurcations are given and the stability and location of these periodic solutions is also judged. Numerical simulations present the verification for theoretical research, and show some complex oscillations, including quasi-periodic oscillation, period-doubling cascade and chaos resulted from the period-doubling cascade.Chapter three takes a power system with three machines and four buses as research object. The existence of saddle-node bifurcations is analyzed in time domain, and Hopf bifurcation is in frequency domain. The fourth-order harmonic balance approximations of periodic solutions generated from Hopf bifurcations are provided and the approximations about frequencies and expressions of periodic solutions are verified by utilizing the software Auto2007 and numerical solutions. The further numerical analysis shows that the periodic solutions undergo various bifurcations, such as period-doubling bifurcation, cyclic fold bifurcation, torus bifurcation, and system presents some complex dynamics, such as chaos, quasi-periodic oscillations.The bifurcations in a three-dimensional cancer growth model are considered in chapter four. Especially, it is firstly found that equilibrium points of this model undergo a saddle-node bifurcation. The examinations of transcritical bifurcations and saddle-node bifurcation are strictly analyzed in time domain. The existence of Hopf bifurcation is verified in frequency domain. Higher accurate predictions than Chapter two and three on frequency, amplitude and analytical expression for periodic solution generated from Hopf bifurcation are provided by applying the sixth-order harmonic balance method, on which the stability and location of the periodic orbit is also detected. In addition, numerical simulations show that equilibrium points and periodic orbits undergo various local bifurcations, including period-doubling bifurcation, cyclic fold bifurcation, period-doubling cascade and chaos.In chapter five, the entire study work of this article is minutely summarized, and the pivotal research contents in future are determined.