Abstract
As in many physical and non physical systems chaos can have harmful consequences, the possibility is discussed of reducing or suppressing it without radically modifying the system.
An heuristic reasoning is proposed, then it is shown on a Duffing-Holmes oscillator, that a resonant effect can kill chaos when parametric perturbations are introduced with suitable frequencies.
Preview
Unable to display preview. Download preview PDF.
References
F.C. Moon, Chaotic Vibrations. An Introduction for Applied Scientists and Engineers, John Wiley & Sons, N.Y. 1987.
M. Pettini et al., Phys. Rev. A38, 344 (1988), and references quoted therein.
M.N. Rosenbluth, R.Z. Sagdeev, J.B. Taylor, and G.M. Zaslavsky, Nucl. Fusion 6, 297 (1966).
M. Rasetti, Modern Methods in Equilibrium Statistical Mechanics, World Scientific, Singapore 1987, p.73.
J.D. Hanson, and J.R. Cary, Phys. Fluids 27, 767 (1984).
F.M. Izrailev, and B.V. Chirikov, Numerical experiments on stabilization of stochastic instability with the use of computer in interactive regime (in russian), I.Ya.F. preprint 74-13, Novosibirsk 1974.
K. Matsumoto, and I. Tsuda, J. Stat. Phys. 31, 87 (1983).
I. Purica, Mathematical models for the correlation of equipment quality with their behavior in operation, Doctoral Thesis, National Committee for Nuclear Energy — IBNE, Bucarest 1987.
V.I. Arnold, Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordinaires, Editions MIR, Moscow 1980 (French translation).
V.I. Arnold, and A. Avez, Ergodic Problem of Classical Mechanics, W.A.Benjamin Inc., N.Y. 1968.
J. Guckenheimer, and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, N.Y. 1983.
E.C. Zeeman, Nonlinearity 1, 115 (1988).
V.I. Arnold, Les Méthodes Mathématiques de la Méchanique Classique, Editions MIR, Moscow 1976.
R. Abraham, and J.E. Marsden, Foundations of Mechanics, W.A.Benjamin Inc., N.Y. 1967.
B. Doubrovine, S. Novikov, and A. Fomenko, Géometrie Contemporaine, Vol. I, Editions MIR, Moscow 1979 (French translation), p. 354.
R. Lima, and M. Pettini, Suppression of chaos by resonant parametric perturbations, preprint, CPT-CNRS Marseille 1989.
R.A. Mahaffey, Phys. Fluids 19, 1387 (1976).
B.V. Chirikov, Phys. Rep. 52, 263 (1979).
G. Benettin, L. Galgani, and J.M. Strelcyn, Phys. Rev. A14, 2338 (1979).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1990 Springer-Verlag
About this paper
Cite this paper
Pettini, M. (1990). Controlling Chaos through parametric excitations. In: Lima, R., Streit, L., Vilela Mendes, R. (eds) Dynamics and Stochastic Processes Theory and Applications. Lecture Notes in Physics, vol 355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-52347-2_34
Download citation
DOI: https://doi.org/10.1007/3-540-52347-2_34
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52347-5
Online ISBN: 978-3-540-46969-8
eBook Packages: Springer Book Archive