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

  • Christian Mira
Seminars and Communications
Part of the Lecture Notes in Physics book series (LNP, volume 179)


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 To7, it is possible to obtain the properties of Tb 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 Tb, xn+1 = f(xn, a) + yn h(xn, yn), yn+1 b g(xn, yn) 7, b = 0(μα), 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.


Accumulation Point Continuity Property Bifurcation Curve Invariant Curve Basic Cycle 
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Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Christian Mira
    • 1
  1. 1.Equipe “Systèmes Non Linéaires” I.N.S.A.Toulouse CedexFrance

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