On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

In this paper, autonomous differential equations of the second order are considered, the right-hand sides of which are polynomials of degree n with respect to the first derivative with periodic continuously differentiable coefficients, and the corresponding vector fields on the cylindrical phase space. The free term and the leading coefficient of the polynomial is assumed not to vanish, which is equivalent to the absence of singular points of the vector field. Rough equations are considered for which the topological structure of the phase portrait does not change under small perturbations in the class of equations under consideration. It is proved that the equation is rough if and only if all its closed trajectories are hyperbolic. Rough equations form an open and everywhere dense set in the space of the equations under consideration. It is shown that for n > 4 an equation of degree n can have arbitrarily many limit cycles. For n = 4, the possible number of limit cycles is determined in the case when the free term and the leading coefficient of the equation have opposite signs.

Download Full-text

Stable Index Pairs for Discrete Dynamical Systems

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-053-2 ◽

1997 ◽

Vol 40 (4) ◽

pp. 448-455 ◽

Cited By ~ 3

Author(s):

Tomasz Kaczynski ◽

Marian Mrozek

Keyword(s):

Dynamical Systems ◽

Vector Fields ◽

Discrete Dynamical Systems ◽

Discrete Case ◽

Smooth Vector ◽

Small Perturbations ◽

Index Pairs ◽

Open Question ◽

Stable Index

AbstractA new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

Download Full-text

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500904 ◽

2014 ◽

Vol 24 (07) ◽

pp. 1450090 ◽

Cited By ~ 8

Author(s):

Tiago de Carvalho ◽

Durval José Tonon

Keyword(s):

Vector Field ◽

Normal Form ◽

Vector Fields ◽

Normal Forms ◽

Piecewise Smooth ◽

Topological Equivalence ◽

Smooth Vector ◽

Codimension One ◽

Piecewise Smooth Vector Fields ◽

Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

Download Full-text