Keywords: anticipating synchronization, delayed systems, self-sustained oscillations, nonlinear dynamics, coupling design, periodically forced systems.

When exploring different dynamic systems, the need often arises to predict the behavior of the system. This work presents a method of forecasting of forced weakly nonlinear self-sustained oscillations. It relies on the phenomenon of anticipating synchronization [1]. Primary it was applied for the systems with chaotic dynamics [2].

The phenomenon of synchronization refers to the collective timing of coupled systems and manifests itself in physical, chemical as well as biological systems [3]. Recently the notion of synchronization was generalized and applied to different chaotic systems [4,5,6]. Anticipated synchronization refers to a particular regime, which appears in unidirectionally coupled systems in a drive-response ("master"-"slave") configuration. In this regime two dynamic systems synchronize in such a way that the "slave" system anticipates the trajectory of the "master" one. Anticipated synchronization has been studied theoretically and experimentally in many systems [7]. It is remarkable that neither nonlinearity nor chaotic dynamics of the master and drive systems are the necessary condition for the anticipating synchronization [8].

In this work the phenomenon of the anticipating synchronization is applied to predict the dynamics of the periodically driven self-sustained oscillator. A diagonal coupling matrix is introduced in order to supply the long-term prediction of the behavior of the drive system. The efficiency of such a scheme is studied analytically using a simple model of coupled circles. This investigation has enabled us to estimate the optimal values of the coupling strength as well as the ranges of the anticipating synchronization domain. The analytical results obtained have provided the forecasting of the dynamics of a periodically forced self-sustained oscillator. The numerical simulations of the dynamics of unidirectionally coupled nonlinear systems, van der Pol oscillators, have justified the the analytical results and have demonstrated the realization of the regime of anticipating synchronization. The prediction of the quasi-periodical and periodical oscillations was demonstrated. The prediction time attained is equal to the period of the external periodical oscillations. The information processed in an auxiliary slave system can then be used to decide whether to activate the control of the master system in a manner like [9].

1

H.U. Voss, "Anticipating chaotic synchronization", Phys. Rev. E, 61, 5115-5119, 2000. doi:10.1103/PhysRevE.61.5115

2

E.N. Lorenz, "Deterministic Nonperiodic Flow", J. Atmos. Sci., 20, 130-141, 1963. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

3

A. Pikovsky, M. Rosemblum, J. Kurths, "Synchronization: A universal concept in nonlinear sciences", Cambridge University Press, Cambridge, England, 2001.

4

V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, "Stochastically synchronized oscillations in dissipative systems", Radiophysics and Quantum Electronics, 29, 1050-1060, 1986. doi:10.1007/BF01034476

5

L.M. Pecora, T.L. Carroll, "Synchronization in chaotic systems", Phys. Rev. Lett., 64, 821-824, 1990. doi:10.1103/PhysRevLett.64.821

6

S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, "The synchronization of chaotic systems", Phys. Rep., 366, 1-101, 2002. doi:10.1016/S0370-1573(02)00137-0

7

M. Ciszak, O. Calvo, C. Masoller, C.R. Mirasso, R. Toral, "Anticipating the Response of Excitable Systems Driven by Random Forcing", Phys. Rev. Lett., 90, 204102-1-204102-4, 2003. doi:10.1103/PhysRevLett.90.204102

8

O. Calvo, D.R. Chialvo, V.M. Eguíluz, C. Mirasso, R. Toral, "Anticipated synchronization: A metaphorical linear view", Chaos, 14, 7-13, 2004. doi:10.1063/1.1620991

9

M. Ciszak, C. Mirasso, R. Toral, O. Calvo, "Predict-prevent control method for perturbed excitable systems", Phys. Rev. E, 79, 046203, 2009. doi:10.1103/PhysRevE.79.046203

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