Diagnosing chaos in non-linear dynamical systems by trajectory predictions and innovation tests of the Kalman filter

We report the use of the Kalman filter to detect the onset of chaos in a non-linear system for which the model is exactly known. The procedure is based on predicting the trajectories of a non-linear dynamical system and testing the innovation sequence for the predicted trajectory using the local overall method tests (LOMT). This is applied to the Rossler–Wegmann model of the Zhabotinskii reaction system. The Rossler–Wegmann model is a three dimensional set of non-linear ordinary differential equations, the solution to which gives the time-evolution of the concentrations of the major species in the Zhabotinskii reaction. With suitable parameters and starting values of concentrations, stable, oscillatory and chaotic solutions may be found. Five thousand points were generated from the model, for each of the three species in the model, by fourth order Runge–Kutta integration. For a complex oscillatory case the LOMT showed no chaos, and when parameters that lead to chaos were used the LOMT quickly revealed the mismatch between predicted and actual trajectories. It is concluded that the Kalman filter with LOMT is a quick and accurate method of diagnosing chaos, which could be used in a monitoring and on-line controlling system for a chemical process.