Imbedding of a one-dimensional endomorphism into a two-dimensional diffeomorphism. Implications

Abstract

The above-mentioned properties for f = 1 - a x², remain the same qualitatively, when f(x, a) is a continuous, and continuously differentiable function, having only one extremum, and satisfying other conditions (cf. p. 121 of ⁷). For f = ax ± x³, from the known bifurcation structure of T_o ⁷, it is possible to obtain the properties of T_b as for the quadratic case. Consider now the ordinary differential equations of one of the following types : either three-dimensional, autonomous, or two-dimensional with periodical coefficients of the independent variable. Each of these equations has a parameter μ, such that μ = o gives a one unit decrease of the dimension. The method of sections of Poincaré gives a generalization of T_b, x_n+1 = f(x_n, a) + y_n h(x_n, y_n), y_n+1 b g(x_n, y_n) ⁷, b = 0(μ^α), a > o, f, g, h being functions such that this mapping T is a difformorphism ⁷. Then Tb can be considered as a first approach to the study of T.