ESTIMATING INVARIANT PROBABILITY DENSITIES FOR DYNAMICAL SYSTEMS

DEVIN KILMINSTER^{1}, DAVID ALLINGHAM^{1} AND ALISTAIR MEES^{1,2}

1Centre for Applied Dynamics and Optimization, The University of Western Australia,

35 Stirling Highway, Nedlands, WA 6009, Australia 2Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, NT, Hong Kong, China

(Received February 8, 2001; revised July 23, 2001)

Abstract. Knowing a probability density (ideally, an invariant density) for the trajectories of a dynamical system allows many significant estimates to be made, from the well-known dynamical invariants such as Lyapunov exponents and mutual information to conditional probabilities which are potentially more suitable for prediction than the single number produced by most predictors. Densities on typical attractors have properties, such as singularity with respect to Lebesgue measure, which make standard density estimators less useful than one would hope. In this paper we present a new method of estimating densities which can smooth in a way that tends to preserve fractal structure down to some level, and that also maintains invariance. We demonstrate with applications to real and artificial data.

Key words and phrases: Nonlinear dynamics, probability density, invariant measure, Radon transform.