Prognosis of critical transitions in a delay differential equations model of ENSO

Delay differential equations (DDEs) have been shown to be an efficient mathematical tool for modeling a wide range of delayed interactions and external forcings relevant to climate dynamics [1-3]. Research during the past 30 years has shown that the El-Niño/Southern-Oscillation (ENSO) dynamics is governed, by and large, by the interplay of several nonlinear mechanisms that can be studied in simple forced DDE models. These models provide a convenient paradigm for explaining interannual ENSO variability and shed new light on its dynamical properties. Initiated in the late 1980s, the DDE approach has been extended and advanced during the last decade [4-6]. This work focus on prognosis of critical transitions in the DDE models of ENSO. We use a consistent Bayesian approach to modeling stochastic (random) dynamical systems by time series[7]. In this approach, the key ("robust") dynamic properties of the system evolution can be described by a few variables, while other features may be considered as a stochastic disturbance. Stochastic models of this sort are of the form of random dynamical systems; they present a necessary and important step towards reconstructing the observed systems when their adequate first-principle mathematical models are either unknown or subjected to further verification. We construct stochastic model of evolution operator of unknown system by virtue of scalar time series generated by the system. The model operator includes deterministic as well as stochastic terms; both of them supposed to be inhomogeneous in the model state space and are parametrized by artificial neuron networks. We use as a data source one of the DDE model of ENSO [5], complimented by both dynamical noise reflecting influences of external forcing, and slow trend of control parameter making this system weakly non-autonomous. Applicability of reconstructed model for prognosis of qualitative changes (critical transitions) of system behavior is demonstrated for time interval greater than "observation" period. 1. Bhattacharya K., M. Ghil, and I. Vulis, Internal variability of an energy-balance model with delayed albedo effects,J.Atmos.Sci.,39,1747-1773,1982. 2. Ghil M. and S. Childress, Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory, and Climate Dynamics, Springer-Verlag,1987 3. Zebiak S. and Cane M. A., A model El Nino Southern Oscillation,Mon.Wea.Rev.,115,2262-2278,1987. 4. Munnich M., M. Cane and S. E. Zebiak, A study of self-excited oscillations of the tropical ocean-atmosphere system. Part II: Nonlinear cases,J.Atmos.Sci.,48(10),1238-1248,1991. 5. Tziperman E., L. Stone, M. Cane, and H. Jarosh, El Niño chaos: Overlapping of resonances between the seasonal cycle and the Pacific ocean-atmosphere oscillator, Science,264,72-74,1994. 6. Ghil M., I. Zaliapin, and S. Thompson, A delay differential model of ENSO variability: parametric instability and the distribution of extremes.,Nonlin.Proc.Geophys.,15,417-433,2008. 7. Molkov Ya.I., Loskutov E.M., Mukhin D.N., and Feigin A.M.,Random dynamical models from time series, Phys.Rev.E, sub judice,2010.