REVIEW OF THE VAN DER POL EQUATION AND ITS MODIFICATIONS

Abstract

The studies carried out on the Boole transformation B(x) = x− 1
x and its parameterizations, for the most part, have been
done from the perspective of the infinite ergodic theory and recently the ergodicity for the absolutely continuous (respect to
Lebesgue measure) invariant probability measure for Bα(x) = α(x− 1
x ) with α ∈ (0, 1). These studies only consider the
cases where the family of functions do not have fixed points and even in that case the dynamic behavior is not completely
described. Starting from this, consider a generalization of the Boole transformation of the form fabc (x) = ax− b
x +c with
a > 0, b > 0 and c ∈ R and a study of their dynamics is carried out for a wide set of parameter spaces. Specifically, a region
of the set of parameters is obtained where fabc is transitive and presents a relation, by means of topological conjugation,
with the symbolic dynamics (associated with the Shift) for a subset of the space of two symbols, which justifies its chaotic
behavior in this case. It is shown that there is an open set of the parameter space where fabc is uniformly robustly transitive.

Also, it is proved that for a > 1, b > 0 and c ∈ R, f has two hyperbolic fixed points and between these two points there
exists a uniformly robustly transitive invariant Cantor set, whose dynamic is equivalent to unilateral shift of two symbols.