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\title{Detecting Chaos in Non-linear Dynamical Systems by Trajectory
Predictions and Innovation Tests}

\author{Jixiu Jiang and D Brynn Hibbert
\\ School of Chemistry, University of New South Wales,\\
Sydney, NSW 2052, Australia}

\date{}
\maketitle
A new method is described for detecting chaos in a time series which can be
described by a polynomial model, and for which control parameters can be
estimated using the standard Kalman filter. The procedure is based on
forecasting the trajectories of a non-linear dynamical system and testing
the innovation sequence of the Kalman filtering using the local overall
method tests (LOMT). The procedure has been applied to the population
logistic system. The results show that the technique can quickly diagnose
chaos $in\;situ$ in a low-dimensional non-linear dynamical system even when
the system is in filtering and controlling mode. It can discriminate chaotic
from periodic behaviour and from randomness and it is a good candidate for
detecting chaos in a non-linear tracking and/or controlling system.

\noindent
{\bf Keywords:}
{\small chaos, nonlinear dynamics, Gaussian map, innovation sequence, the extended Kalman filter.}

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