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A global analysis of Newton iterations for determining turning points

Vladimír Janovský, Viktor Seige (1993)

Applications of Mathematics

The global convergence of a direct method for determining turning (limit) points of a parameter-dependent mapping is analysed. It is assumed that the relevant extended system has a singular root for a special parameter value. The singular root is clasified as a b i f u r c a t i o n s i n g u l a r i t y (i.e., as a d e g e n e r a t e turning point). Then, the Theorz for Imperfect Bifurcation offers a particular scenario for the split of the singular root into a finite number of regular roots (turning points) due to a given parameter imperfection. The relationship...

A singular perturbation method for saddle connections and subharmonics of certain nonlinear differential equations with fixed saddle points.

Peter Smith (1990)

Revista Matemática de la Universidad Complutense de Madrid

Saddle connections and subharmonics are investigated for a class of forced second order differential equations which have a fixed saddle point. In these equations, which have linear damping and a nonlinear restoring term, the amplitude of the forcing term depends on displacement in the system. Saddle connections are significant in nonlinear systems since their appearance signals a homoclinic bifurcation. The approach uses a singular perturbation method which has a fairly broad application to saddle...

Abelian integrals related to Morse polynomials and perturbations of plane hamiltonian vector fields

Lubomir Gavrilov (1999)

Annales de l'institut Fourier

Let 𝒜 be the real vector space of Abelian integrals I ( h ) = { H h } R ( x , y ) d x d y , h [ 0 , h ˜ ] where H ( x , y ) = ( x 2 + y 2 ) / 2 + ... is a fixed real polynomial, R ( x , y ) is an arbitrary real polynomial and { H h } , h [ 0 , h ˜ ] , is the interior of the oval of H which surrounds the origin and tends to it as h 0 . We prove that if H ( x , y ) is a semiweighted homogeneous polynomial with only Morse critical points, then 𝒜 is a free finitely generated module over the ring of real polynomials [ h ] , and compute its rank. We find the generators of 𝒜 in the case when H is an arbitrary cubic polynomial. Finally we...

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