How the choice of the observable may influence the analysis of nonlinear dynamical systems. (English) Zbl 1099.37521

This paper discusses issues relating to the observability and controllability of nonlinear dynamical systems. Using the Rössler system, the authors show that a more general definition of the observability matrix is the Jacobian matrix of the coordinate transformation between the original phase space and the differential embedding induced by the observable. Some explicit examples are treated and the authors show that the more general definition of the observability matrix in some cases allows better indications concerning the ability to obtain a global model from a given observable.


37N99 Applications of dynamical systems
93B07 Observability
93B05 Controllability
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C10 Nonlinear systems in control theory
Full Text: DOI


[1] Takens, F., Detecting strange attractors in turbulence, Lect Notes Math, 898, 366-381 (1981) · Zbl 0513.58032
[2] Letellier, C.; Aguirre, L. A., Investigating nonlinear dynamics from time series: the influence of symmetries and the choice of observables, Chaos, 12, 549-558 (2002) · Zbl 1080.37600
[3] Aguirre, L. A., Controllability and observability of linear systems: some noninvariant aspects, IEEE Trans. Educ, 38, 33-39 (1995)
[4] Letellier, C.; Maquet, J.; Le Sceller, L.; Gouesbet, G.; Aguirre, L. A., On the non-equivalence of observables in phase space reconstructions from recorded time series, J Phys A, 31, 7913-7927 (1998) · Zbl 0936.81014
[5] Gouesbet, G.; Letellier, C., Global vector field reconstruction by using a multivariate polynomial \(L_2\)-approximation on nets, Phys Rev E, 49, 6, 4955-4972 (1994)
[6] Aguirre, L. A.; Billings, S. A., Retrieving dynamical invariants from chaotic data using NARMAX models, Int J Bifurc Chaos, 5, 2, 449-474 (1995) · Zbl 0886.58100
[7] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys Rev Lett, 64, 11, 1196-1199 (1990) · Zbl 0964.37501
[8] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys Lett A, 170, 421-428 (1990)
[9] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 8, 821-824 (1990) · Zbl 0938.37019
[10] Letellier, C.; Aguirre, L. A.; Maquet, J.; Aziz-Alaoui, M., Should all the species of a food chain be counted to investigate the global dynamics?, Chaos, Solitons & Fractals, 13, 1099-1113 (2002) · Zbl 1004.92039
[11] Maquet, J.; Letellier, C.; Aguirre, L. A., Scalar modeling and analysis of a 3D biochemical reaction model, J Theor Biol, 228, 3, 421-430 (2004) · Zbl 1439.92102
[12] Kailath, T., Linear systems (1980), Englewood Cliffs: Englewood Cliffs NJ, Prentice Hall · Zbl 0458.93025
[13] Chen, C. T., Linear systems theory and design (1999), Oxford University Press: Oxford University Press Oxford
[14] Klamka J. Controllability of dynamical systems. 7th conference on dynamical systems—theory and applications (Łódź), In: Proceedings (Décembre 2003), vol. I, p. 37-56, 8-11.; Klamka J. Controllability of dynamical systems. 7th conference on dynamical systems—theory and applications (Łódź), In: Proceedings (Décembre 2003), vol. I, p. 37-56, 8-11.
[15] Rössler, O. E., An equation for continuous chaos, Phys Lett A, 57, 5, 397-398 (1976) · Zbl 1371.37062
[16] Maquet, J.; Letellier, C.; Gouesbet, G., Global modelling and differential embedding, (Soofi, A.; Cao, L., Modeling and forecasting financial data: techniques of nonlinear dynamics (2002), Kluwer Academic Publisher), 351-374 · Zbl 0958.37065
[17] Lainscsek, C.; Letellier, C.; Gorodnitsky, I., Global modeling of the Rössler system from the \(z\)-variable, Phys Lett A, 314, 5-6, 409 (2003) · Zbl 1038.37063
[18] Corron, N. J.; Pethel, S. D., Control of long-period orbits and arbitrary trajectories in chaotic systems using dynamics limiting, Chaos, 12, 1, 1-7 (2002)
[19] Letellier, C.; Dutertre, P.; Maheu, B., Unstable periodic orbits and templates of the Rössler system: toward a systematic topological characterization, Chaos, 5, 1, 272-281 (1995)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.