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Bilinear systems and chaos. (English) Zbl 0823.93026
Motivated by the chaotic behavior of Lorenz equations, the authors presents a conjecture on chaos in a bilinear system in $\bbfR\sp 3$ of the form $\dot x= Ax+ Bxu$, $x\in \bbfR\sp 3$, $u\in \bbfR$. The main results of this article are two theorems on the existence of a pair of symmetric homoclinic orbits and on the chaotic behavior of a generalized Lorenz equation (GLE). The computer simulation shows that the GLE has a chaotic behavior similar to the Lorenz equation although there is some difference between the theoretical analysis and the numerical simulation.

MSC:
93C15Control systems governed by ODE
37D45Strange attractors, chaotic dynamics
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Full Text: Link EuDML
References:
[1] F. Alberting, E. D. Sontag: Some connections between chaotic dynamical systems and control systems. Proceedings of First European Control Conference, Grenoble, 1991, pp. 159-163.
[2] S. Čelikovský, A. Vaněček: Bilinear systems as the strongly nonlinear systems. Systems Structure and Control (V. Strejc, Pergamon Press, Oxford 1992, pp. 264-267.
[3] S. Čelikovský: On the stabilization of homogeneous bilinear systems. Systems and Control Lett. 21 (1993), 6. · Zbl 0794.93089 · doi:10.1016/0167-6911(93)90055-B
[4] W. J. Freeman: Strange attractors that govern mammalian brain dynamics shown by trajectories of EEG potential. IEEE Trans. Circuits and Systems 35 (1988), 791-783.
[5] R. Genesio, A. Tesi: Chaos prediction in nonlinear feedback systems. IEE Proc. D 138 (1991), 313-320. · Zbl 0754.93024 · doi:10.1049/ip-d.1991.0042
[6] R. Genesio, A. Tesi: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28 (1992), 531-548. · Zbl 0765.93030 · doi:10.1016/0005-1098(92)90177-H
[7] R. Genesio, A. Tesi: A harmonic balance approach for chaos prediction: the Chua’s circuit. Internat. J. of Bifurcation and Chaos 2 (1992), 61-79. · Zbl 0874.94042 · doi:10.1142/S0218127492000070
[8] A. L. Goldberger: Nonlinear dynamics, fractals, cardiac physiology and sudden death. Temporal Disorder in Human Oscillatory Systems, Springer-Verlag, Berlin 1987.
[9] J. Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York 1986.
[10] A. V. Holden (ed.): Chaos. Manchester Univ., Manchester 1986. · Zbl 0743.58005
[11] H. Hyotyniemi: Postponing chaos using a robust stabilizer. Preprints of First IFAC Symp. Design Methods of Control Systems, Pergamon Press, Oxford 1991, pp. 568-572.
[12] L. O. Chua (ed.): Special Issue on Chaotic Systems. IEEE Proc. 6 (1987), 8, 75.
[13] E. N. Lorenz: Deterministic non-periodic flow. J. Atmospheric Sci. 20 (1965), 130-141.
[14] G. B. Di Massi, A. Gombani: On observability of chaotic systems: an example. Realization and Modelling in Systems Theory. Proc. International Symposium MTNS-89, Vol. II, Birkhäuser, Boston -- Basel -- Berlin 1990, pp. 489-496. · Zbl 0733.93009
[15] R. R. Mohler: Bilinear Control Processes. Academic Press, New York 1973. · Zbl 0343.93001
[16] J. M. Ottino: The mixing of fluids. Scientific Amer. 1989, 56-67.
[17] C. T. Sparrow: The Lorenz Equations: Bifurcation, Chaos and Strange Attractors. Springer-Verlag, New York 1982. · Zbl 0504.58001
[18] A. Vaněček: Strongly nonlinear and other control systems. Problems Control Inform. Theory 20 (1991), 3-12. · Zbl 0747.93033
[19] A. Vaněček, S. Čelikovský: Chaos synthesis via root locus. IEEE Trans. Circuits and Systems 41 (1994), 1, 54-60. · Zbl 0843.93024 · doi:10.1109/81.260222
[20] A. Vaněček, S. Čelikovský: Synthesis of chaotic systems. Kybernetika, accepted.
[21] S. Wiggins: Global Bifurcations and Chaos. Analytical Methods. Springer-Verlag, New York 1988. · Zbl 0661.58001