Abstract
In the case of autonomous dynamical systems, it is better to base symmetry considerations on trajectories than on full solutions. In this setting topological arguments can be used; a special role is played in this context by time-independent Lie-point symmetries. As an application of this approach, we obtain results on the existence of stationary and/or periodic solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V. I. (1983).Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, Berlin.
Arnold, V. I. (1990).Equations Differentielles Ordinaires, 2nd ed., Mir, Moscow.
Bluman, G. W., and Kumei, S. (1989).Symmetries and Differential Equations, Springer, Berlin.
Chow, S. N., and Hale, J. (1982).Methods of Bifurcation Theory, Springer, Berlin.
Cicogna, G. (1984).Bollettino Unione Matematica Italiana,3, A-131.
Cicogna, G. (1990).Journal of Physics A,23, L1339.
Cicogna, G., and Degiovanni, M. (1984).Nuovo Cimento,82B, 54.
Cicogna, G., and Gaeta, G. (1989).Nonlinear Analysis,13, 475.
Crawford, J. D. (1991).Reviews of Modern Physics,63, 991.
Crawford, J. D., and Knobloch, K. (1991).Annual Review of Fluid Mechanics,23, 341.
Crawford, J. D., and Omohundro, S. (1984).Physica D,13, 161.
Gaeta, G. (1990).Physics Reports,189, 1.
Gaeta, G. (1991a). The Jones polynomial and periodic orbits of nonlinear oscillators, Preprint, CPTh Ecole Polytechnique; to appear inInternational Journal of Theoretical Physics.
Gaeta, G. (1991b). Reduction and equivariant branching lemma: Dynamical systems, evolution PDEs, and gauge theories, Preprint, CPTh Ecole Polytechnique; to appear in Acta Appl. Math.
Gaeta, G. (1992).Symmetries of Nonlinear Equations and Physics, in preparation.
Golubitsky, M., and Stewart, I. (1985).Archive for Rational Mechanics and Analysis,87, 107.
Golubitsky, M., Stewart, I., and Schaeffer, D. G. (1988).Singularity and Groups in Bifurcation Theory, Vol. II, Springer, New York.
Guillemin, V., and Pollack, A. (1974).Differential Topology, Prentice-Hall, Englewood Cliffs, New Jersey.
Hirsch, M., and Smale, S. (1978).Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York.
Milnor, J. (1965).Topology from the Differential Viewpoint, University of Virginia Press.
Olver, P. J. (1986).Applications of Lie Groups to Differential Equations, Springer, Berlin.
Ovsjannikov, L. V. (1962).Group Properties of Differential Equations, Novosibirsk, Russia.
Ruelle, D. (1989).Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, London.
Sattinger, D. H. (1979).Group Theoretic Methods in Bifurcation Theory, Springer, Berlin.
Stephani, H. (1989).Differential Equations. Their Solution Using Symmetries, Cambridge University Press, Cambridge.
Wulfman, C. E. (1974).Journal of Physics A,12, L73.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gaeta, G. Autonomous systems, dynamical systems, LPTI symmetries, topology of trajectories, and periodic solutions. Int J Theor Phys 32, 191–199 (1993). https://doi.org/10.1007/BF00674404
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00674404