Nilpotent Linear Part Research Articles

Global study of a family of cubic Liénard equations

We derive the global bifurcation diagram of a three-parameter family of cubic Liénard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.

Adjoint operator method and normal forms of higher order for nonlinear dynamical system

Normal form theory is a very effective method when we study degenerate bifurcations of nonlinear dynamical systems. In this paper by using adjoint operator method, normal forms of order 3 and 4 for nonlinear dynamical system with nilpotent linear part and Z2-asymmetry are computed. According to normal forms obtained, universal unfoldings for some degenerate bifurcation cases of codimension 3 and simple global characterizations, are studied.

Integrable analytic vector fields with a nilpotent linear part

We study the normalization of analytic vector fields with a nilpotent linear part. We prove that such an analytic vector field can be transformed into a certain form by convergent transformations when it has a non-singular formal integral. We then prove that there are smoothly linearizable parabolic analytic transformations which cannot be embedded into the flows of any analytic vector fields with a nilpotent linear part.

Splitting algorithm for nilpotent normal forms

An algorithm is given to normalize vector fields with nilpotent linear part using representation theory of . The theory is set up in such a way that the whole construction can also be used for Hamiltonian problems, or any other problem where one has a graded Lie algebra setting.

Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

AbstractA cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate toWe define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent toOur main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to