Quadratic neural unit for adaptive prediction of transitions among local attractors of Lorenz system

The goal of the paper is to demonstrate an adaptive prediction of trajectory transitions between local attractors of deterministic chaotic systems using low-dimensional dynamic quadratic neural unit with periodically forced neural inputs. The forthcoming transitions of higher dimensional chaotic systems are predicted by a low dimensional discrete dynamic neural unit implemented as a special adaptive forced oscillator. The real-time sample-by-sample evaluation of complex system behavior is based on monitoring of parameters of an adaptive model during its adaptation. The behavior of chaotic systems in the state-space is adaptively transformed to system behavior in an approximated parameter-space using special higher order nonlinear neural unit forced with periodical inputs. Added forcing inputs naturally allow a low dimensional dynamic neural unit to better approximate higher dimensional behavior; the forcing neural inputs are initially configured upon analysis of frequency spectra of the evaluated time series. It is demonstrated that monitoring of system parameters during adaptation of a dynamic neural architecture can reveal important attributes of complex behavior in real time, and should be considered as a novel methodology for early detection of transients between local attractors of higher dimensional chaotic systems (transient chaos) by lower dimensional adaptive evaluating systems, i.e. a neural unit with forcing inputs. The stable learning technique combining consequent adaptation of the static and dynamic neural unit is discussed. Results are shown on the application of monitoring and detection of significant transitions among local basins of attraction in deterministic chaotic time series generated first by Lorenz system in a chaotic mode and second by a high dimensional complex nonlinear dynamic system.