Abstract
The formal analysis of bifurcations from homoclinic orbits in low-dimensional ordinary differential equations is here extended to deal with ordinary differential equations in n dimensions, and to certain partial differential equations in one space variable on the infinite real axis. For ordinary differential equations, results are equivalent to various cases treated by Shil’nikov: depending on the eigenvalues at the fixed point, an infinite number of periodic orbits can bifurcate at the critical parameter value. By contrast, homoclinic bifurcations for partial differential equations can produce an infinite number of quasi-periodic (modulated travelling wave) solutions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arneodo, A., P. Coullet and C. Tresser 1982 Oscillations with chaotic behaviour: an illustration of a theorem by Shil’nikov. J. Stat. Phys. 27, 171–182.
Gaspard, P., R, Kapral and G. Nicolis 1984 Bifurcation phenomena near homoclinic systems: a two-parameter analysis. J. Stat. Phys. 35, 697–727.
Glendinning, P. 1988 Global bifurcations in flows. In: New directions in dynamical systems, ed. T. Bedford and J. Swift. LMS Lect. Note Ser. 127, pp. 120–149.
Glendinning, P. and C. Sparrow 1984 Local and global behaviour near homoclinic orbits. J. Stat. Phys. 35, 645–696.
Lorenz, E.N. 1963 Deterministic non-periodic flow. J. Atmos. Sci. 20, 131–141.
Shil’nikov, L.P. 1965 A case of the existence of a countable number of periodic motions. Sov. Math. Dokl. 6, 163–166.
Shil’nikov, L.P. 1967 The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighbourhood of a saddle-focus. Sov. Math. Dokl. 8, 54–58.
Shil’nikov, L.P. 1970 A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type. Mat. Sb. 10, 91–102.
Sparrow, C.T. 1982 The Lorenz equations: bifurcations, chaos, and strange attractors. Springer-Verlag, Berlin.
Stuart, J.T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. J. Fluid Mech. 9, 353–370.
Tresser, C. 1984 About some theorems by L.P. Shil’nikov. Ann. de l’Inst. Henri Poinc. 40, 441–461.
Widnall, S.E. 1984 Growth of turbulent spots in plane Poiseuille flow. In: Turbulence and Chaotic Phenomena in Fluids, ed.T. Tatsumi, pp.93–98. North-Holland, Amsterdam.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Plenum Press, New York
About this chapter
Cite this chapter
Fowler, A.C. (1990). Homoclinic Bifurcations in Ordinary and Partial Differential Equations. In: Coullet, P., Huerre, P. (eds) New Trends in Nonlinear Dynamics and Pattern-Forming Phenomena. NATO ASI Series, vol 237. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-7479-4_40
Download citation
DOI: https://doi.org/10.1007/978-1-4684-7479-4_40
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4684-7481-7
Online ISBN: 978-1-4684-7479-4
eBook Packages: Springer Book Archive