Nonlinear Analysis Of Chemical Oscillation Reaction With Different Time-scale Coupling

The chemical oscillation with projected engineering practices has attracted most attentions from scholars at home and abroad, because of involving in different time scales which have become one of the key problems in the fields of nonlinear dynamics and engineering science. In this dissertation, the nonlinear behaviors in the chemical oscillation reactions with different time-scale coupling have been investigated by using the theory of nonlinear dynamics, the slow-fast analysis and numerical simulation. Various oscillations as well as the induced bifurcation mechanism have been found. The main respects of the research are followings:Because the dimension of subsystems is different in the different systems, and the external excitation may alter the substructure of the original system, the slow-fast dynamical analysis should be used properly in every system. Based on the basic idea of slow-fast analysis and considering the factors about both of periodic excitation and the numbers of slow variable, three different slow-fast methods, such as the enveloping slow-fast analysis with one slow parameter, slow-fast analysis with two slow parameters and the enveloping slow-fast analysis with two slow parameters are proposed to reveal the bifurcation mechanism of bursting oscillation in the different systems, which can be used to analyze the induced mechanism of the bursters for different chemical oscillation systems in the chapter3and chapter4.CO oxidation on platinum group metals is the reaction with multiple time scales because the reaction rate on the surface is much faster than the subsurface. Upon using regression analysis to the measured data in the reaction, the mathematical model with multiple time scales is established in chapter3. The stability of equilibria is discussed in details, and different types of solutions of the differential equation may bifurcate from the equilibria with the variation of the parameters. With certain parameter condition, the system can exhibit periodic oscillations via the saddle-node homoclinic orbit bifurcation, which can evolve to periodic bursting owing to Hopf bifurcation of the fast subsystem. The bifurcation connecting the quiescent state and the repetitive spikes is presented to account for the occurrence of the Nk oscillations. Furthermore, the mechanism of sequence of the period-adding bifurcations is explored to reveal why the length of the sequences become longer with the variation of the parameter. By introducing periodic perturbation, forced bursting is found, and the related bifurcation mechanism is presented by using the enveloping slow-fast analysis with one slow parameter.The Belousov-Zhabotinsky (BZ) reaction is a kind of typically chemical oscillation reaction, if periodic perturbation given by light is introduced into the three-variable model, the nonlinear behaviors may become more complicated. The variation of periodic excitation will change the substructure of the system. The novel dynamical phenomena have been observed in the reaction under different external perturbation. For example, single-Hopf bursting may exist in the reaction with one slow process coming from the periodic excitation; forced bursting may take place in the reaction with fast periodic excitation; cusp bursting is presented in periodically perturbed reaction with two slow variables; if another excitation with fast frequency is introduced into the system, the quiescent states of cusp bursting may produce small amplitude oscillation; when two kinds of periodic excitations with slow frequency exist in the Belousov-Zhabotinsky reaction, the bursting oscillation with two kinds of spiking states and quiescent states is observed. Furthermore, the bifurcation mechanism for these bursting behaviors is given by means of the related slow-fast analysis mentioned in the chapter2.In the Belousov-Zhabotinsky reaction, two mathematical models are always involved, which is related to the presence and absence of the illumination respectively. Considering periodic switching signal associated with illumination to the reaction, a switched mathematical model is established. In suitable range of parameter values, different types of oscillations, such as2T-focus/cycle periodic switching,2T-focus/focus periodic switching and chaotic switching oscillation, are observed, the related mechanism of which is presented through the switching relationship. The distribution of eigenvalues and the invariant subspaces of the equilibrium points determined by two subsystems are discussed to interpret oscillation-increasing and oscillation-decreasing cascades of the periodic oscillations. The reaction with parameter perturbed is discussed, and the dynamical evolution with the variation of the parameters is analyzed. Furthermore, if a large gap associated with time scales between the switched time and the natural frequency of the subsystems, the slow-fast effect may appear in the switched system.