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Time series-based bifurcation diagram reconstruction. (English) Zbl 0988.37096
The main idea that the authors use is the ability of neural networks to learn general solutions to problems if given an appropriate set of specific examples. In the centre of interest is the problem of reconstructing bifurcation diagrams of dynamical systems without the knowledge of its functional form and its dependence of the parameter. By using time series at different parameter values a three-layer fully connected neural network is employed in the approximation of the map. The task of this network is to learn the dynamic of the system as function of the parameters from the time series.
For a special class of maps (linear in parameter, LIP) a scheme in locating a linear subspace in the network’s weight space with a qualitatively similar bifurcation structure as that of the map is discussed. Hereby the principal component analysis (PCA) plays an important role. With the predictor functions obtained by the neural network an algorithm in reconstructing the bifurcation diagrams for this class of maps is proposed by the authors. Other classes of predictors could also be used due to the flexibility of the algorithm.

MSC:
37M10 Time series analysis of dynamical systems
82C32 Neural nets applied to problems in time-dependent statistical mechanics
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[1] Prakash, M.; Murty, M.N., IEEE trans. neural networks, 8, 161, (1997)
[2] Levin, A.U.; Narendra, K.S., IEEE trans. neural networks, 7, 30, (1996)
[3] Parisini, T.; Zoppoli, R., IEEE trans. neural networks, 5, 436, (1994) · Zbl 0814.93064
[4] Saxen, H., Int. J. neural systems, 7, 195, (1996)
[5] Billings, S.A.; Hong, X., Neural networks, 11, 479, (1998)
[6] Connor, J.T.; Martin, R.D.; Atlas, L.E., IEEE trans. neural networks, 5, 240, (1994)
[7] Sompolinsky, H.; Tishby, N.; Seung, H.S., Phys. rev. lett., 65, 1683, (1990)
[8] Baldi, P.; Hornik, K., Neural networks, 2, 53, (1989)
[9] Hornik, K.; Stinchcombe, M.; White, H., Neural networks, 2, 359, (1989)
[10] Watkin, T.L.H.; Rao, A.; Biehl, M., Rev. mod. phys., 65, 499, (1993)
[11] Tokunaga, R.; Kajiwara, S.; Matsumoto, T., Physica D, 79, 348, (1994)
[12] Tokuda, I.; Kajiwara, S.; Tokunaga, R.; Matsumoto, T., Physica D, 95, 380, (1996)
[13] Bagarinao, E.; Nomura, T.; Pakdaman, K.; Sato, S., Physica D, 124, 258, (1998)
[14] W.H. Press, W.T. Vettering, S.A. Teukolsky, B.P. Plannery, Numerical Recipe, C: the Art of Scientific Computing, 2nd ed., Cambridge University Press, 1992.
[15] Lorenz, E., J. atmos. sci, 20, 130, (1963)
[16] C. Sparrow, The Lorenz E quations: Bifurcation, Chaos, and Strange Attractors, Springer, New York, 1982. · Zbl 0504.58001
[17] Abarbanel, H.D.I; Brown, R.; Sidorowich, J.J.; Tsimring, L., Rev. mod. phys., 65, 1331, (1995)
[18] Chon, K.H.; Kanters, J.K.; Holstein-Rathlou, N.H., Physica D, 99, 471, (1997)
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