Nonlinear Analysis Of Catalytic Reaction With Multiple Time Scales

The Belousov-Zhabotinsky(BZ)reaction is a kind of typically chemical oscillation reaction as well as Brusselator reaction,and the nonlinear dynamics of this kind of reaction has attracted most attentions from scholars at home and abroad.The nonlinear behaviors may become more complicated because of involving in different time scales for the presence of catalyst in reactions.In this dissertation,the fast-slow oscillation behaviors in BZ reaction and Brusselator reaction with different time scales have been investigated by using bifurcation theory of differential dynamics system,fast-slow dynamic analysis method,enveloping fast-slow analysis method,and numerical simulations.Various oscillation phenomena and their bifurcation mechanism have been found.The main respects of the reaction are following:The BZ reaction easily involved in different time scales when the periodic perturbation or fractional derivative is introduced.The novel bursting type is observed in the BZ reaction with low perturbation frequency and the corresponding generation mechanism is presented based on the fast-slow dynamical analysis and the transition portrait methods.In addition,the transition near the bifurcation point in the bursting phenomenon is analyzed,the details about typical hysteresis phenomenon of spiking state as well as their mechanisms are exhibited by theoretical analysis and numerical simulation.The external excitation tends to the fast process when the frequency of the periodic perturbation is much higher than system's inherent frequency,which case different fast-slow oscillation behaviors.Various periodic bursting oscillations are observed and their mechanism are presented through the enveloping fast-slow analysis method with one slow variable.The variation of the parameters will change the nonlinear behaviors of the system,and the route from simple movement to complex oscillation of period-doubling bifurcation is found.The integer-order system is generalized to fractional-order system with different time scales when introduced the fractional derivative in the BZ reaction.Thestabilities of the equilibrium points of integer-order and fractional-order systems are analyzed and compared based on the stability theories of differential equations.Furthermore,the fast-slow oscillation is firstly studied in fractional-order BZ reaction with two time scales coupled,and whose generation mechanism is explained by using the fast-slow dynamical analysis method.The effects of different fractional orders on the fast-slow oscillation behavior as well as the internal mechanism are both analyzed.An equivalent model of the classical Brusselator reaction is established when the coordinate transformation is introduced,which can be divided into the fast and slow subsystems.The stability condition and bifurcation phenomenon of the fast subsystem are analyzed,and the attraction domains of different equilibria are presented by theoretical analysis and numerical simulation respectively.Both the fast-slow oscillation and the mechanism in the new model are analyzed based on the fast-slow dynamical analysis method.