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**Determining mixed linear-nonlinear coupled differential equations from multivariate discrete time series sequences.**
*(English)*
Zbl 1009.37501

Summary: A new method is described for extracting mixed linear-nonlinear coupled differential equations from multivariate discrete time series data. It is assumed in the present work that the solution of the coupled ordinary differential equations can be represented as a multivariate Volterra functional expansion. A tractable hierarchy of moment equations is generated by operating on a suitably truncated Volterra functional expansion. The hierarchy facilitates the calculation of the coefficients of the coupled differential equations.

In order to demonstrate the method’s ability to accurately estimate the coefficients of the governing differential equations, it is applied to data derived from the numerical solution of the Lorenz equations with additive noise.

The method is then used to construct a dynamic global mid- and high-magnetic latitude ionospheric model where nonlinear phenomena such as period doubling and quenching occur. It is shown that the estimated inhomogeneous coupled second-order differential equation model for the ionospheric peak plasma density can accurately forecast the future behaviour of a set of ionosonde stations which encompass the earth. Finally, the method is used to forecast the future behaviour of a portfolio of Japanese common stock prices. The hierarchy method can be used to characterise the observed behaviour of a wide class of coupled linear and mixed linear-nonlinear phenomena.

In order to demonstrate the method’s ability to accurately estimate the coefficients of the governing differential equations, it is applied to data derived from the numerical solution of the Lorenz equations with additive noise.

The method is then used to construct a dynamic global mid- and high-magnetic latitude ionospheric model where nonlinear phenomena such as period doubling and quenching occur. It is shown that the estimated inhomogeneous coupled second-order differential equation model for the ionospheric peak plasma density can accurately forecast the future behaviour of a set of ionosonde stations which encompass the earth. Finally, the method is used to forecast the future behaviour of a portfolio of Japanese common stock prices. The hierarchy method can be used to characterise the observed behaviour of a wide class of coupled linear and mixed linear-nonlinear phenomena.

### MSC:

37M10 | Time series analysis of dynamical systems |

86A10 | Meteorology and atmospheric physics |

86-08 | Computational methods for problems pertaining to geophysics |

37N20 | Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) |

91B28 | Finance etc. (MSC2000) |

37N40 | Dynamical systems in optimization and economics |

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\textit{A. D. Irving} and \textit{T. Dewson}, Physica D 102, No. 1--2, 15--36 (1997; Zbl 1009.37501)

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### References:

[1] | Volterra, V., Theory of Functionals and of Integral and Integro-differential Equations (1959), Dover: Dover New York |

[2] | Dewson, T.; Irving, A. D., Linear and non-linear response function estimation from multi-input systems, Physica D, 94, 19-35 (1996) · Zbl 0900.93134 |

[3] | Irving, A. D.; Dewson, T.; Hong, G.; Cunliffe, N. H., General nonlinear response of a single input system to stochastic excitations, Appl. Math. Modelling, 19, 45-56 (1995) · Zbl 0825.93993 |

[4] | Irving, A. D.; Clayton, B. R.; Dewson, T., Nonlinear thermoviscoelastic behaviour in complex materials, (IUTAM Symp. on Inhomogenity. IUTAM Symp. on Inhomogenity, Anisotropy and Nonlinearity in Solid Mechanics (1995), Elsevier: Elsevier Amsterdam), 125-132 · Zbl 0875.73059 |

[5] | Irving, A. D., Stochastic sensitivity analysis, Appl. Math. Modelling, 16, 1-12 (1992) · Zbl 0757.93074 |

[6] | A.D. Irving and T. Dewson, General mixed linear-nonlinear response of evolutionary causal physical processes, in preparation.; A.D. Irving and T. Dewson, General mixed linear-nonlinear response of evolutionary causal physical processes, in preparation. · Zbl 0900.93134 |

[7] | Drazin, P. G., Nonlinear systems, Cambridge Texts in Applied Mathematics (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0753.34001 |

[8] | Jackson, E. A., Perspectives of Nonlinear dynamics (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0701.70001 |

[9] | Takens, F., Detecting strange attractors in fluid turbulence, (Rand, D.; Young, L. S., Dynamical Systems and Turbulence (1981), Springer: Springer Berlin) · Zbl 0513.58032 |

[10] | Packard, N. H.; Crutchfield, J. D.; Farmer, R. S.; Shaw, R. S., Geometry from time series, Phys. Rev. Lett., 45, 712-716 (1980) |

[11] | Broomhead, D. S.; Jones, R.; King, G. P., J. Phys. A, 20, L563 (1987) |

[12] | Broomhead, D. S.; King, G. P., Extracting quantitative dynamics from experimental data, Physica D, 20, 217-236 (1986) · Zbl 0603.58040 |

[13] | Gibson, J. F.; Doyne Farmer, J.; Casdagli, M.; Eubank, S., An analytical approach to practical state space reconstruction, Physica D, 57, 1-30 (1992) · Zbl 0761.62118 |

[14] | Cremers, J.; Hubler, A., Construction of differential equations from experimental data, Z. Naturforsch. A, 42, 797-802 (1986) |

[15] | Gouesbet, G., Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series data, Phys. Rev. A, 43, 5321-5330 (1991) |

[16] | Rowlands, G.; Sprott, J. C., Extraction of dynamical equations from chaotic data, Physica D, 58, 251-259 (1992) · Zbl 1194.37141 |

[17] | Gouesbet, G.; Maquet, J., Construction of phenomenological models from numerical scalar time series, Physica D, 58, 202-215 (1992) · Zbl 1194.37132 |

[18] | Casdagli, M., Nonlinear prediction of chaotic time series, Physica D, 35, 335-356 (1989) · Zbl 0671.62099 |

[19] | Farmer, J. D.; Sidovowich, J. J., Predicting chaotic time series, Phys. Rev. Lett., 59, 845-848 (1987) |

[20] | Rivlin, R. S.; Ericksen, J. L., Stress-deformation relations for isotropic materials, J. Ration. Mech. Anal., 4, 323-425 (1955) · Zbl 0064.42004 |

[21] | Breeden, J. L.; Hubler, A., Reconstructing equations of motion from experimental data with unobserved variables, Phys. Rev. A, 10, 5817-5826 (1990) |

[22] | Gouesbet, G.; Letellier, C., Global vector-field reconstruction by using a multivariate polynomial L2 approximation on nets, Phys. Rev. E, 49, 6, 4955-4972 (1994) |

[23] | Letellier, C.; Le-Sceller, L.; Dutertre, P.; Gouesbet, G.; Fei, Z.; Hudson, J. L., Topological characterisation and global vector field reconstruction of an experimental electrochemical system, J. Phys. Chem., 99, 7016-7027 (1995) |

[24] | Wiener, N., Response of a nonlinear system in the presence of noise, MIT Radiation Laboratory Report No. 165 (1942) |

[25] | Barrett, J., Hermite functional expansions and the calculation of output autocorrelation and spectrum for any time-invariant nonlinear system, J. Electronics and control, 16, 107-113 (1964) |

[26] | Cameron, R.; Martin, W., The orthogonal development of nonlinear functionals in series of Hermite-Legendre Functionals, Ann. Math., 48, 385-392 (1947) · Zbl 0029.14302 |

[27] | Peng, J. H.; Ding, E. J.; Ding, M.; Yang, W., Synchronizing hyperchaos with a scalar transmitted signal, Phys. Rev. Lett., 76, 6, 904-907 (1996) |

[28] | J. Schweizer, A stochastic approach to spread spectrum communication using chaos, SPIE Vol. 2612, pp. 115-125.; J. Schweizer, A stochastic approach to spread spectrum communication using chaos, SPIE Vol. 2612, pp. 115-125. |

[29] | P. Celka, Experimental verification of chaotic self-synchronization in an integrated optical based system, SPIE Vol. 2612, pp. 15-24.; P. Celka, Experimental verification of chaotic self-synchronization in an integrated optical based system, SPIE Vol. 2612, pp. 15-24. |

[30] | Tufillaro, N. B.; Wyckoff, P.; Brown, R.; Schreiber, T.; Molteno, T., Topological time-series analysis of a string experiment and its synchronised model, Phys. Rev. E, 51, 1, 164-174 (1995) |

[31] | Brown, R.; Rulkov, N. F.; Tufillaro, N. B., Synchronization of chaotic systems: The effects of additive noise and drift in the dynamics of the driving, Phys. Rev. E, 50, 6, 4488-4508 (1994) |

[32] | Aguirre, L. A.; Billings, S. A., Dynamical effects of overparameterisation in nonlinear models, Physica D, 80, 26-40 (1995) · Zbl 0888.58060 |

[33] | Irving, A. D.; Dewson, T., The analysis of complex times series data, Appl. Math. Modelling, 20, 35-47 (1996) · Zbl 0844.62077 |

[34] | Priestly, M. B., Spectral Analysis and Time Series (1981), Academic Press: Academic Press London |

[35] | Bittencourt, J. A.; Chryssafidis, Comparison of IRI predictions with low ionospheric observations, Adv. Space Sci., 11, 10, 97-100 (1991) |

[36] | Rawer, K., Ionospheric mapping in the polar and equatorial zones, Adv. Space Sci., 16, 1, 9-12 (1995) |

[37] | Besprogvannaya, A. S., Empirical modelling of the F2 peak at 50-70 invariant latitude using magnetic conjugacy, Adv. Space Sci., 11, 10, 23-28 (1991) |

[38] | Lorenz, E. M., Deterministic nonperiodic flow, J. Atmospheric Sci., 20, 130-141 (1963) · Zbl 1417.37129 |

[39] | Merton, R. C., Optimum consumption and portfolio rules in a continuous-time model, J. Econom. Theory, 3, 373-413 (1971) · Zbl 1011.91502 |

[40] | Ito, K., On stochastic differential equations, Mem. Amer. Math. Soc., 4, 1-51 (1951) |

[41] | Ito, K.; McKean, H. P., Diffusion Processes and Their Sample Paths (1964), Academic Press: Academic Press New York · Zbl 0153.47302 |

[42] | Hull, J. C., Options, Futures and Other Derivative Securities (1993), Prentice-Hall: Prentice-Hall London |

[43] | Assaf, D.; Taksar, M.; Klass, M. J., A diffusion model of optimal portfolio selection in the presence of brokerage fees, Math. Oper. Res., 13, 277-294 (1988) · Zbl 0850.93886 |

[44] | Atkinson, C.; Wilmott, P., Portfolio management with transaction costs: An asymptotic analysis of the Morton and Pilska model, Math. Finance, 5, 4, 357-367 (1995) · Zbl 0866.90010 |

[45] | Davis, M.; Normann, A., Portfolio selection with transaction costs, Math. Oper. Res., 15, 676-713 (1990) · Zbl 0717.90007 |

[46] | Duffie, D.; Sun, T., Transaction costs and portfolio choice in a discrete-continuous time setting, J. Econom. Dynamics Control, 14, 577-597 (1990) |

[47] | Dumas, B.; Luciano, E., An exact solution to a dynamical portfolio choice problem under transaction costs, J. Finance, 46, 577-595 (1991) |

[48] | Eastham, J. F.; Hastings, K. J., Optimal impulse control of portfolios, Math. Oper. Res., 13, 588-605 (1988) · Zbl 0667.90009 |

[49] | Karatzas, I., Optimisation problems in the theory of continuous trading, SIAM J. Control Optm., 27, 1221-1259 (1989) · Zbl 0701.90008 |

[50] | Morton, A. J.; Pilska, R. S., Optimal portfolio management with fixed transaction costs, Math. Finance, 5, 4, 337-356 (1995) · Zbl 0866.90020 |

[51] | Neural Networks in the Capital Markets, (Refenes, A. N., Proc. 1st Int. Workshop (1993), London Business School) |

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