Nonlinear effects in chemical dynamics and chemical kinetics: Chaos in physical chemistry.

Abstract

This work shows that nonlinear dynamic systems are quite different from linear dynamic systems. The usual phase plots of linear and nonlinear dynamical systems are also distinctly different, even before damping or forcing terms are added. It is also shown that the usual phase plot allows for the visualization of unobservable information that is present in the times series. The higher-order phase plots give yet additional information that is not present in the existing methods of plotting the data. Higher-order phase plots were originated and applied for the first time to a dynamical system (Morse oscillator) for the purpose of earlier detection of nonlinear effects. The dynamics of a weakly forced and weakly damped Morse oscillator is examined. The novel tool of higher-order phase plots is used to visualize the importance of the higher harmonics in the phase which are essential for the dynamics to be complicated and dissociative. Expansions of the higher-order phase plots in regions about x$\sp{(n-1)}$ = 0, x$\sp{(n)}$ = 0 are considered and it is shown that there is a topological change that occurs sequentially for each higher-order phase plot. After the topological change, which occurs at a critical value of initial total energy E(0) for a particular value of forcing F, the higher-order phase space structure has a circular loop. As F or E(0) is further increased the phase space trajectory loops an increasing number of times in the higher-order phase plot. It is shown that for F = $1.0\times10\sp{-3}$ the topological change occurs around E(0) = 0.96 for the fifth-order phase plot and around E(0) = 0.94 for the eleventh-order phase plot. This is also illustrated with a series of higher-order phase plots (2$\sp{nd}$-10$\sp{th}$) for F = $1\times10\sp{-3}$ and E(0) = 0.97. These plots indicate that although the 5$\sp{th}$ order phase plot forms loops the 4$\sp{th}$ forms only half-loops. Thus the higher-order phase plots are increasingly sensitive probes of the phase space dynamics as the order increases. Qualitatively this is because, as the order increases, part of each higher-order phase space structure is increasingly close to the point (x$\sp{(n)}$,x$\sp{(n-1)}$) = (0,0). For larger values of F the topological change occurs at a smaller value of E(0) for each higher-order phase plot, as the radius of the loop centered on x$\sp{(n-1)}$ = 0, x$\sp{(n)}$ = 0 is larger. While the phase space trajectory loops the energy is approximately constant with small oscillations. The circular loop in the higher-order phase plot is the higher-order space structure that is expected for the weakly forced and weakly damped free particle. The significance of the circular loop is that the lower the order of the higher-order phase plot in which the phase space trajectory loops, the closer the Morse oscillator is to dissociation. From this viewpoint, the Morse oscillator dissociates when the value of F or E(0) becomes sufficiently large that the topological change occurs in the usual phase plot. That is, the Morse oscillator dissociates when the phase space structure becomes open for the usual phase plot. (Abstract shortened by UMI.)