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Subsidiary bifurcations near bifocal homoclinic orbits

Published online by Cambridge University Press:  04 October 2011

Paul Glendinning
Paul Glendinning
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW
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Abstract

Suppose that an ordinary differential equation in ℝ4 has an orbit Γ bi-asymptotic to a stationary point O of the flow. If the characteristic equation of the linear flow near O has roots (λ0 ± ιω0, λ1±ιω1) with —λ0 > λ1 > 0 and ωi > 0 for i = 0,1, we show that there are sequences of more complicated orbits bi-asymptotic to O in generic one-parameter perturbations of the equations. When ω0 ≠ 2nω1 for all n ∈ ℕ, there are sequences of such orbits on both sides of the bifurcation value of the parameter, in contrast to a similar case for flows in ℝ3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 597 - 605
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Fowler, A. C. and Sparrow, C.. Bifocal homoclinic orbits in four dimensions. (Preprint, 1984.)Google Scholar
[2] Gaspard, P.. Generation of a countable set of homoclinic flows through bifurcation. Phys. Lett. A 97 (1984), 1–4.Google Scholar
[3] Gaspard, P.. Generation of a countable set of homoclinic flows through bifurcation in multidimensional systems. Bull. Class. Sci. Acad. Roy. Belg. Serie 5 70 (1984), 61–83.Google Scholar
[4] Gaspard, P., Kapral, R. and Nicolis, G.. Bifurcation phenomena near homoclinic systems: a two-parameter analysis. J. Statist. Phys. 35 (1984), 697–727.CrossRefGoogle Scholar
[5] Glendinning, P. and Sparrow, C.. Local and global behaviour near homoclinic orbits. J. Statist. Phys. 35 (1984), 645–696.CrossRefGoogle Scholar
[6] Shil'nikov, L. P.. A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type. Math. USSR-Sb. 10 (1970), 91–102.CrossRefGoogle Scholar
[7] Tresser, C.. About some theorems by L. P. Shil'nikov. Ann. Inst. M. Poincaré Probab. Statist. 40 (1984), 441–461.Google Scholar