Lyapunov exponents

Chaotic dynamical systems are predictable for finite time scales, but not for infinite ones. Let consider two initial states of the system with almost identical initial conditions. Their trajectories in the phase space will move apart, given a metric, at an exponential rate while moving on an attractor. This rate is described by the largest Lyapunov exponent in the system, ^max. If any Lyapunov exponent is positive, the system will be chaotic by definition. Yet, should one of them be zero, the system can be described by a set of ordinary differential equations [10].

Figure 6 shows the d Lyapunov exponents for the differenced time series as a function of the number of samples following the suggested algorithm by Sano and Sawada [11]. The main interest in this test is to find Lyapunov exponents greater or equal than zero. The two Lyapunov exponents exhibited by the time series are negative and thus, this time series cannot be considered coming from a chaotic dynamical system neither be described by a set of differential equations.

image127

Fig. 6. Lyapunov exponents against the number of samples.

4. Conclusions

The global solar radiation data were measured at the radiometric station of the University of Almeria (Spain) during eight years. Results have shown the non-existence of any attractor in the phase space for the global irradiance time series. Negative Lyapunov exponents exclude a chaotic behaviour that might allow a better short term prediction than autorregresive models, and the idea of the existence of a nonlinear differential equation system. These results match with those

obtained from applying local linear models for prediction, of which estimations suggest that the data are best described by a linear stochastic process.

Acknowledgements

This work was supported by the project ENE2007-67849-C02-02 of the Ministerio de Ciencia y Tecnologia of Spain.

References

[1] R. Festa, S. Jain, C. F. Ratto, Renewable Energy 2 (1992) 23.

[2] J. Boland, Solar Energy 55 (1995) 377.

[3] H. Morf, Solar Energy 62 (1998) 101.

[4] E. N. Lorenz, J. Atmos. Sci. 20 (1963) 130.

[5] L. Alados-Arboledas, F. J. Batlles, F. J. Olmo, Solar Energy 54 (1995) 183.

[6] F. Takens, (1981). Dynamical Systems and Turbulence, Lecture Notes in Math, Springer-Verlag, Berlin.

[7] A. M. Fraser and H. L. Swinney, Phys. Rev. A, 33 (1986) 1134.

[8] R. Hegger, H. Kantz, and T. Schreiber, Chaos, 9 (1999) 413.

[9] M. B. Kennel, R. Brown, and H. D.I. Abarbanel, Phys. Rev. A, 45 (1992) 3403.

[10] M. Casdagli, J. Roy. Stat. Soc., 54 (1991) 303.

[11] M. Sano and Y. Sawada, Phys. Rev. Lett., 55 (1987) 1082.

[12] J. D. Farmer and J. J. Sidorowich, Phys. Rev. Lett., 59 (1987) 845.

Добавить комментарий

Ваш e-mail не будет опубликован. Обязательные поля помечены *