Liapunov characteristic exponent

The Belousov-Zhabotinskii reaction in an isothermal CSTR can undergo a series of transitions among periodic and chaotic states. One segment of this series of transitions is investigated in detail. Liapunov characteristic exponents are calculated for both the periodic and chaotic regions. In addition, the effect of external disturbances on the periodic behavior is investigated with the aid of a mathematical model. 

Figure 5. Liapunov characteristic exponent as a function of reciprocal residence...

It has been shown that the transition from the two peak periodic oscillation to the chaotic behavior occurs with a loss of stability of the periodic oscillation an unstable two peak oscillation is embedded In the chaotic region. During this transition the Liapunov characteristic exponent changes sign from negative to positive. Furthermore, calculations indicate that a small amplitude regular disturbance does not have a significant effect on the character of the oscillations. 

Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

The configuration entropy represents the size of the phase space, and its projection onto the configuration space may correspond to the extent of the configuration space. However, it does not give any information about the dynamics itself. The Liapunov exponents and the KS entropy that is the positive sum of the Liapunov exponents may give the characteristic of the dynamics. Thus, these properties were calculated in this work to compare the results... 

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