zbMATH — the first resource for mathematics

Micro-chaos in digital control. (English) Zbl 0863.93050
The authors study the ‘micro chaos’ map \(F:x \to ax - b\text{Int} (x)\), describing the local dynamics of a digitally controlled unstable system, being subject to a linear effect of sampling and a nonlinear effect of round-off. These effects frequently cause chaotic oscillations on a microscopic scale near the equilibrium. The authors prove the existence of a hyperbolic strange attractor for a large set of parameter values of the map \(F\) and study its domain of attraction – being of full measure but possibly having a fractal boundary. Finally, they describe the dynamics on the fractal attractor using symbolic dynamics.

93C62 Digital control/observation systems
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI
[1] Abraham, R., Marsden, J. E., and Ratiu, T.,Manifolds, Tensor Analysis and Applications, Springer-Verlag, New York (1988). · Zbl 0875.58002
[2] Bowen, R., Markov partition for Axiom A diffeomorphisms,Am. J. Math. 92 (1970) 725–747. · Zbl 0208.25901 · doi:10.2307/2373370
[3] Bowen, R., Invariant measures for Markov maps of the interval.Commun. Math. Phys. 69 (1979) 1–17. · Zbl 0421.28016 · doi:10.1007/BF01941319
[4] Boyarsky, A., and Scarowsky, M., On a class of transformations which have unique absolutely continuous invariant measures.Trans. AMS 255 (1979) 243–262. · Zbl 0418.28010 · doi:10.1090/S0002-9947-1979-0542879-2
[5] Cullen, C.,Matrices and Linear Transformations, 2nd ed., Dover, New York (1972). · Zbl 0287.15001
[6] Delchamps, F. D., Stabilizing a linear system with quantized state feedback.IEEE Trans. Autom. Contr. 35 (1990) 916–924. · Zbl 0719.93067 · doi:10.1109/9.58500
[7] Domokos, G., Digital modelling of chaotic motion,Stud. Sci. Math. Hung. 25 (1990) 323–341. · Zbl 0644.58011
[8] Enikov, E., and Stépán, G., Micro-chaotic behavior of digitally controlled machines, inProc. 15th ASME Biennial Conference on Mechanical Vibration and Noise (Boston, 1995). · Zbl 0831.70015
[9] Guckenheimer, J., and Holmes, P. J.,Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectorfields, Springer-Verlag, New York (1983). · Zbl 0515.34001
[10] Ledrappier, F., Some relations between dimensions and Ljapunov exponents,Commun. Math. Phys. 81 (1981) 229–238. · Zbl 0486.58021 · doi:10.1007/BF01208896
[11] Li, T., and Yorke, J. A., Ergodic transformations from an interval into itself,Trans. AMS 235 (1978) 183–192. · Zbl 0371.28017 · doi:10.1090/S0002-9947-1978-0457679-0
[12] Mané, R.,Ergodic Theory and Differentiable Dynamics, Springer-Verlag, New York (1987).
[13] Stépán, G., \(\mu\)-chaos in digitally controlled mechanical systems,Nonlinearity and Chaos in Engineering Dynamics, J. M. T. Thompson and S. R. Bishop, Eds., John Wiley & Sons, Chichester (1994) 143. · Zbl 0869.70020
[14] Tél, T., inDirections in Chaos, ed. Hao Bai-Liu, World Scientific, Singapore, vol. 3 (1990).
[15] Tobias, S. A.,Machine Tool Vibration, Blackie, Londom (1965).
[16] Ueda, K., Amano, A., Ugawa, K., Takamatsu, H., and Sakuta, S., Machining high-precision mirrors using newly developed CNC machine,CIRP Ann. 40, (1991) 554–562.
[17] Ushio, T., and Hsu, S., Chaotic rounding error in digital control systems,IEEE Trans. Circuits Syst. 34, (1987) 133–139. · Zbl 0619.93058 · doi:10.1109/TCS.1987.1086113
[18] Wiggins, S.,Global Bifurcations and Chaos-Analytical Methods, Springer-Verlag, New York (1988). · Zbl 0661.58001
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.