scholarly journals A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields

2007 ◽  
Vol 340 (2) ◽  
pp. 437-463 ◽  
Author(s):  
Robert L. Bryant ◽  
Gianni Manno ◽  
Vladimir S. Matveev
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Two Dimensional ◽  
Projective Vector

scholarly journals On the Computation of Degenerate Hopf Bifurcations forn-Dimensional Multiparameter Vector Fields

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Michail P. Markakis ◽  
Panagiotis S. Douris
Keyword(s):  
Vector Fields ◽  
Center Manifold ◽  
Normal Forms ◽  
Computer Assisted ◽  
Hopf Bifurcations ◽  
Symbolic Form ◽  
Parametric System ◽  
Analytical Expressions

The restriction of ann-dimensional nonlinear parametric system on the center manifold is treated via a new proper symbolic form and analytical expressions of the involved quantities are obtained as functions of the parameters by lengthy algebraic manipulations combined with computer assisted calculations. Normal forms regarding degenerate Hopf bifurcations up to codimension 3, as well as the corresponding Lyapunov coefficients and bifurcation portraits, can be easily computed for any system under consideration.

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