Chaos Theory and Lyapunov Exponents

This criterion is based on a sophisticated mathematical approach, and therefore is not as easy to use as the preceding criteria. Nevertheless, this approach represents the most advanced technique in this field. It was developed by Strozzi and Zaldivar and co-workers [7-9, 16-19], 

The Lyapunov exponents describe the behavior of two neighboring points of a system in the phase space as a function of time. If the Lyapunov exponent is posi- 

The state variables define an ellipsoid in the state space, where Lf 0) and L,(t) are the lengths of the j-axis of the ellipsoid at t = 0 and t = t. This gives the evolution of the m-sphere in the state space. The volume is given by 

This criterion allows distinguishing two states in a batch reactor, no-runaway and runaway. For the semi-batch reactor, there are four different states, no-igni-tion, runaway, marginal ignition, and QFS. 

The advantage of this criterion is that it can be computed on-line, based only on temperature measurements, without the necessity of a model of the process [9]. The method uses a reconstruction of the phase space using only one state variable. This allows building a warning system for detecting a runaway situation this aspect is presented in Chapter 10. 

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