## Using invariants to determine change detection in dynamical system with chaos.(English)Zbl 1142.93435

Summary: The problem of change detection in dynamical systems originated from ordinary differential equations and real world phenomena is covered. Until now suitable methods for detecting changes for linear systems and nonlinear systems have been elaborated but there are no such method for chaotic systems. In this paper we propose the method of change detection based on the fractal dimension, which is the one of characteristics dynamical system invariants. The application of the method is illustrated with simulations.

### MSC:

 93000 Stochastic systems in control theory (general) 9.3e+12 Filtering in stochastic control theory

### Keywords:

change detection; dynamical systems

### Software:

TISEAN; Fracdim; Massdal; qcc
Full Text:

### References:

 [1] Alon N, Matias Y, Szegedy M (1996) The space complexity of approximating the frequency moments. In: STOC ’96: proceedings of the twenty-eighth annual ACM symposium on theory of computing, ACM Press, New York, pp 20–29 · Zbl 0922.68057 [2] Cheng B, Tong H (1994) Orthogonal projection, embedding dimension and the sample size in chaotic time series from statistical perspective. Phil Trans R Soc Lond A 348:325–339 · Zbl 0859.62077 [3] Graham Cormode. Massdal public code bank. Source code avalible at [ http://www.cs.rutgers.edu/%7Emuthu/massdal-code-index.html ] [4] Falconer K (1990) Fractal geometry. Wiley, New York · Zbl 0689.28003 [5] Fraser AM, Swinney HL (1986) Independent coordinates for strange attractors from mutual information. Phys Rev A 33 · Zbl 1184.37027 [6] González FA, Dasgupta D (2003) Anomaly detection using real-valued negative selection. Genet Program Evolv Mach 4(4):383–403 · Zbl 02003806 [7] Grassberger P (1990) An optimized box-assisted algorithm for fractal dimensions. Phys Lett A 148:63–68 [8] Haykin S, Principe J (1998) Making sense of complex world. IEEE Signal Process Mag 15:66–81 [9] Hegger R, Kantz H, Schreiber T (1999) Practical implementation of nonlinear time series methods: the tisean package. Chaos 9:413–435 · Zbl 0990.37522 [10] Hively LM, Protopopescu VA (2000) Timely detection of dynamical change in scalp eeg signals. Chaos 10:864–875 · Zbl 1016.92019 [11] Ilin A, Valpola H (2001) Detecting process state changes by nonlinear blind source separation. In: Proceedings of international conference on independent component analysis and signal separation [12] Ilin A, Valpola H, Oja E (2004) Nonlinear dynamical factor analysis for state change detection. IEEE Trans Neural Netw 15(3):559–575 [13] Kennel MB, Brown R (1992) Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys Rev A 45:3403–3411 [14] Letellier C, Maquet J, Le Sceller L, Gouesbet G, Aguirre LA (1998) On the non-equivalence of observables in phase space reconstructions from recorded time series. J Phys A 31:7913–7927 · Zbl 0936.81014 [15] Lucas JM, Saccucci MS (1990) Exponentially weighted moving average control schemes: properties and enhancements. Technometrics 32(1):1–29 [16] Miyata T, Watanabe T (2003) Bi-lipschitz maps and the category of aproximate resolutions. Glasnik Maematicki 58(38):129–155 · Zbl 1050.54013 [17] Molteno TC (1993) Fast O(N) box-counting algorithm for estimating dimension. Phys Rev E 48(5):3263–3264 [18] Montgomery DC (2001) Introduction to statistical quality control, 4th edn. Wiley, New York [19] Sauer T, Yorke JA, Casdagli M (1991) Embedology. J Stat Phys 65(3–4):579–616 · Zbl 0943.37506 [20] Schreiber T, Kantz H (1995) Dimension estimates and physiological data. Chaos 5:143–154 [21] Schuster HG (1988) Deterministic chaos. VGH Verlagsgesellschaft, Weinheim [22] Luca Scrucca. Quality control package for R-project. Avalible at http://www.r-project.org/ · Zbl 1145.62353 [23] Takens F (1981) Detecting strange attractors in turbulence. Lecture Notes in Mathematics vol 898, pp. 366–381 · Zbl 0513.58032 [24] Tricot C (1995) Curves and fractal dimension. Springer, Heidelberg · Zbl 0847.28004 [25] Wong A, Wu L (2003) Fast estimation of fractal dimension and correlation integral on stream data. Inf Process Lett 93 [26] Leejay Wu, Christos Faloutsos. Fracdim, jan 2001. Perl package available at http://www.andrew.cmu.edu/$$\sim$$lw2j/downloads.html
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.