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Restrictions and unfolding of double Hopf bifurcation in functional differential equations |
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| Authors: | Pietro-Luciano Buono |
| Institution: | a Centre de recherches mathématiques, Université de Montréal, CP 6128, Succursale centre-ville, Montréal, Québec, Canada H3C 3J7 b Département de Mathématiques et de Statistique, Institut de Génie Biomédical, Université de Montréal, CP 6128, Succursale centre-ville, Montréal, Québec, Canada H3C 3J7 c Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montréal, Québec, Canada |
| Abstract: | The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhães (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and  -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations (n?2) with one delay feedback. |
| Keywords: | Delay-differential equations Double Hopf bifurcation Normal forms Centre manifold Unfolding |
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