UNFOLDING A DEGENERATE NONLINEAR OSCILLATOR: A CODIMENSION TWO BIFURCATION*
1980; Wiley; Volume: 357; Issue: 1 Linguagem: Inglês
10.1111/j.1749-6632.1980.tb29711.x
ISSN1749-6632
Autores Tópico(s)Chaos control and synchronization
ResumoAnnals of the New York Academy of SciencesVolume 357, Issue 1 p. 473-488 UNFOLDING A DEGENERATE NONLINEAR OSCILLATOR: A CODIMENSION TWO BIFURCATION* Philip Holmes, Philip Holmes Department of Theoretical and Applied Mechanics Cornell University Ithaca, New York 14853Search for more papers by this author Philip Holmes, Philip Holmes Department of Theoretical and Applied Mechanics Cornell University Ithaca, New York 14853Search for more papers by this author First published: December 1980 https://doi.org/10.1111/j.1749-6632.1980.tb29711.xCitations: 30 † This research was partially supported by a National Science Foundation grant, no. ENG 78-02891. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL References 1 Marsden, J. E. & M. McCracken. 1976. The Hopf Bifurcation and Its Applications. Springer-Verlag. Berlin . 2 Carr, J. & R. Muncaster. 1979. Centre Manifolds and Amplitude Expansions, I: Ordinary Differential Equations. Preprint. Heriot-Watt University. Edinburgh . 3 Hassard, B. D., N. D. Kazarinoff & Y. H. Wan. 1980. Theory and Applications of the Hopf Bifurcation. Cambridge University Press. Cambridge . In press. 4 Hassard, B. D. & Y. H. Wan. 1978. Bifurcation formulae derived from center manifold theory. J. Math. Anal. Appl. 63(1): 297– 312. 5 Holmes, P. J. & J. E. Marsden. 1978. Bifurcations to divergence and flutter in flow-induced oscillations: an infinite dimensional analysis. Automatica 14(4): 367– 84. 6 Takens, F. 1974. Singularities of vector fields. Publications of the Institut des Hautes Études Scientifiques, Bures-sur-Yvette, Paris 43: 47– 100. 7 Arnol'd, V. I. 1972. Lectures on bifurcations in versal families. Russ. Math. Surv. 27: 54– 123. 8 Holmes, P. J. 1981. Center manifolds, normal forms and bifurcations of vector fields with applications to coupling between periodic and steady motions. Physica D: Nonlinear Phenomena. In press. 9 Cuckenheimer, J. 1979. On a Codimension Two Singularity. Preprint. University of California at Santa Cruz. Santa Cruz , California . 10 Month, L. A. 1979. On approximate first integrals of Hamiltonian systems with an application to nonlinear normal modes in a two degree of freedom nonlinear oscillator. Ph. D. Thesis, Cornell University. Ithaca, N.Y. 11 Langford, W. F. 1979. Periodic and steady mode interactions lead to tori. SIAM J. Appl. Math. 37(1): 22– 48. 12 Chillingworth, D. R. J. 1976. Differential Topology with a View to Applications. Pitman. London . 13 Smale, S. 1967. Differentiable dynamical systems. Bull. Am. Math. Soc. 73: 747– 817. 14 Moser, J. 1973. Stable and Random Motions in Dynamical Systems. Princeton University Press. Princeton . N.J . 15 Newhouse, S. E. 1980. Lectures on dynamical systems. In Proc. CIME Summer School, Bressanone. Italy, 1978. Birkhauser. Boston . 16 Nitecki, Z. 1971. Differentiable Dynamics. MIT Pres Cambridge . Mass . 17 Takfns, F. 1973. Introduction to global analysis. Commun. 2 Math. Inst. Rijksuniversiteit Utrecht. 18 Li, L. C. 1979. Unpublished notes. Cornell University. Ithaca , N.Y . 19 Holmes, P. J. 1980. On a codimension two singularity with Z1/Z2 symmetry. In preparation. 20 Hartman, P. 1964. Ordinary Differential Equations. Wiley. New York . 21 Andronov, A. A., E. A. Leontovich, I. I. Gordon & A. G. Maier. 1971. Theory of Bifurcations of Dynamic Systems on a Plane. Program for Scientific Translations. Jerusalem, Israel. 22 Melnikov, V. K. 1963. Stability of the center 10 time periodic pertubation. Trans. Moscow Math. Soc. 12(1): 3– 57. 23 Holmes, P. J. 1980. Averaging and chaos in forced oscillations. SIAM J. Appl. Math. 38(1): 65– 80. 24 Hirsch, M. W., C. C. Pugh & M. Shub. 1977. Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag. New York . 25 Arnol'd, V. I. & A. Avez. 1968. Ergodic Problems of Classical Mechanics. W. A. Benjamin. New York . 26 Newhouse, S. E. 1977. The abundance of wild hyperbolic and non-smooth stable sets for diffeomorphisms. Reprint. Institut des Hautes Etudes Scientifique. Bures-sur-Yvette, Paris . Citing Literature Volume357, Issue1Nonlinear DynamicsDecember 1980Pages 473-488 ReferencesRelatedInformation
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