Topological conjugations of normal forms including higher order terms
by
J. C. Valverde
Departamento de Matemáticas, Universidad de Castilla-La Mancha, Avda. de España, s/n, 02071-ALBACETE, SPAIN
Coauthors: F. Balibrea (Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-MURCIA, SPAIN)

Let us consider two parametric families of maps

f:R×R --> R,     f(x, \mu)=f\mu(x)

and

g:R×R --> R,     g(y, \nu)=g\nu(x)

where \mu, \nu in R will be the parameters and x, y in R the variables.

f and g are called locally topologically equivalent near the origin, if there exists a map

(x, \mu)\rightsquigarrow (h\mu(x), r(\mu)),

defined in a small neighborhood of (x, \mu)=(0, 0) in the direct product R×R and such that

• r:R --> R is a homeomorphism defined in a small neighborhood of \mu = 0, with r(0)=0;
• h_\mu:R --> R is a parameter-dependent homeomorphism defined in a small neighborhood U_\mu of x=0, with h₀(0)=0 and mapping orbits of the first system in U_\mu onto orbits of the second one in h_\mu(U_\mu), preserving the direction of time.

The appearance of a topologically non-conjugate system under variation of the parameter is called a bifurcation.

Sometimes, for bifurcations of a family f, near a fixed point of a map belonging to the family, it is possible to construct a simple polynomial family

p(y, \nu),     y in R, \nu in R

which present at the map corresponding to the parameter value \nu = 0 the fixed point y=0, satisfying the same bifurcation conditions. With same bifurcation conditions we refer to present the same eigenvalue (of unit modulus) at the fixed point and some other conditions that will have the form of inequalities (equalities)

Di(p) =/= 0     i=1, ..., r        (Di(p) = 0     i=1, ..., s)

where D_i are some algebraic functions of partial derivatives of p, evaluated on (y, \nu)=(0, 0). The inequalities (equalities) D_i which only involve partial derivatives with respect to the state variable y are called nondegeneracy conditions (degeneracy conditions), while those involving parameters are known as transversality conditions.

A family of the form p is said to be a topological normal form for a bifurcation if any family f with the same bifurcation conditions is locally topologically conjugate to it, near the corresponding fixed points.

Here, we study how to construct an appropriate map like (h_\mu(x), r(\mu)) in order to obtain normal forms as simple as possible.

Date received: February 27, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-36.