Summary
A single-degree-of-freedom nonlinear parametrically excited oscillator is considered. Such oscillators provide models for mechanical systems such as shells and plates under periodical load. Chaotic motions and a strange attractor are found to exist applying the theory of Lyapunov exponents. Some difficulties in practical application of the computational procedure for Lyapunov exponents are discussed. Three particular zones for different values of the excitation coefficient are shown to exist, with different type of long-term behavior.
Übersicht
Ein parametererregter nichtlinearer Schwinger mit einem Freiheitsgrad wird untersucht. Solche Schwinger dienen als Modelle für verschiedene mechanische Systeme, z. B. für Platten und Schalen unter periodischer Belastung. Mit Hilfe der Theorie der Lyapunovschen Exponenten ist die Existenz chaotischer Bewegungen und eines seltsamen Attraktors festgestellt worden. Einige Schwierigkeiten bei der Anwendung des numerischen Verfahrens für die Lyapunovschen Exponenten werden diskutiert. Für verschiedene Werte des Erregungskoeffizienten wurden drei wichtige Zonen gefunden, in denen verschiedenes Langzeitverhalten des Systems stattfindet.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lorenz, E.: Deterministic non-periodic flow. J. Atmos. Sci. 20 (1963) 130–141
Holmes, P.; Moon, F.: Strange attractors and chaos in nonlinear mechanics. Trans. ASME J. Appl. Mech. 50 (1983) 1021–1032
Dowell, E.: Chaotic oscillations in mechanical systems. Comp. Mech. 3 (1988) 199–216
Thompson, J.; Stewart, H.: Nonlinear dynamics and chaos. London: Wiley 1986
Guckenheiner, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Berlin, Heidelberg, Tokyo, New York: Springer 1983
Lichtenberg, A.; Lieberman, M.: Regular and stochastic motion. Berlin, Heidelberg, Tokyo, New York: Springer 1983
Grundmann, H.: Zur dynamischen Stabilität des schwach gekrümmten zylindrischen Schalenfeldes. Ing. Arch. 39 (1970) 261–272
Kisliakov, S.: Parametererregte Schwingungen eines schwach gekrümmten zylindrischen Schalenfeldes. Theor. and Appl. Mech. 1 (1977) 26–34
Kisliakov, S.: Some problems in the theory of dynamic stability for deformable systems. Adv. Mech. 5 (1982) 3–29
Kisliakov, S.: Strange attractors and parametrically excited chaotic motions of thin shells. In: Proc. Int. Conf. Nonlinear Mechanics, pp. 1038–1043. Beijing: Science Press 1985
Izrailev, F.: Rabinovich, M.; Ugodnokov, A.: Approximate description of three-dimensional dissipative systems with stochastic behavior, Phys. Lett./A 86 (1981) 321–325
Wolf, A.; Swift, J.; Swinney, J.; Vastano, J.: Determining Lyapunov exponents from time series Physica/D 16 (1985) 285–317
Froeschlé, C.: The Lyapunov characteristic exponents and, applications. J. Méchanique Théor. Appl., Numéro Spécial (1984) 101–132
Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems. A method for computing all of them. Meccanica 15 (1980) 21–30
Horozov, E.: Private communication. Sofia (1987)
Farmer, J.; Ott, E.; Yorke, J.: The dimension of chaotic attractors. Physica/D 7 (1983) 153–180
Haken, H.: At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point. Phys. Lett./A 94 (1983) 71–72
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kisliakov, S.D., Popov, A.A. Lyapunov exponents and a strange attractor for the damped nonlinear mathieu equation. Arch. Appl. Mech. 61, 49–56 (1991). https://doi.org/10.1007/BF00788137
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00788137