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00 Fig. 1; L1 / occurs (see Fig. 1; 1/, with L1 < 1, become respectively an attracting point and a saddle. The overlapping bifurcation diagrams in Fig. 2)) show the evolution of the attractors. Bifurcations and Chaotic Attractors in an Overlapping Generations Model 49 Fig. 6 Overlapping bifurcation diagrams varying the parameter Fig. 7 The fractal nature of the basins of attraction of period-6 and period-4 cycles The system shows a further interesting dynamic phenomenon non evidenced by the previous figures.
A a N ) would like to work in the capitalistic sector. We assume that E is taken as exogenously given by the Poor, that is, they do not internalize the impact of their production on natural resources; however, we will return to this issue later. In (4), the parameter a determines the difference between the wage in the capitalistic sector and the average output in the traditional sector that allows for the same level of utility. The population of the Rich is constituted by a continuum of identical individuals and the size of the population is represented by the positive parameter M .
FK gE hE fK : fE gK /; The following results can be easily proved. Lemma 1. J / > 0. Lemma 2. If > 0, then jJ j < 0 in A and jJ j > 0 in B. If < 0, then jJ j > 0 in the unique admissible fixed point. Lemma 3. If < 0, then w < 0. If > 0 and < 0, then w < 0. If > 0 and Â 1 E > 0, then E > 2 ˛N rı Ã is a sufficient condition for w < 0. It is now possible to discuss the stability properties of A and B, in the regime > 0, and of the unique admissible fixed point in the regime < 0. K ; E ; / is said “saddle-point stable” if J admits two eigenvalues with negative real parts.