By Kazuo Nishimura, Alain Venditti, Nicholas C. Yannelis
This booklet provides the state of the art in non-linear dynamics and sunspots. those issues were the middle of a world convention on instability and public rules in a globalized global, prepared at Aix-Marseille college of Economics and GREQAM in honor of Jean-Michel Grandmont. He has made major contributions on normal equilibrium concept, financial idea, studying, aggregation, non-linear dynamics and sunspots. This publication assembles contributions by way of Jean-Michel Grandmont's colleagues, scholars and associates which were encouraged by way of his works and which are on the frontier of study during this area today.
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Additional info for Sunspots and Non-Linear Dynamics: Essays in Honor of Jean-Michel Grandmont
A geometric argument for the existence of real eigenvalues close to the singular critical set containing only slow-fast regular points is that the solution path will tend to evolve along the isocline for L, where f 2 (K , L , 0) = 0, which is a monotonic surface. Whilst we have only considered the cases in which slow-fast singular points exists for non-zero f 2,L (K , L , 0), there are also further results which hold for the zero case (for bifurcation results in slow-fast systems for the zero case, see Kuehn (2015, Chaps.
Node for → 0− , as trD F s ( K¯ , L, s ¯ ) > 0, then (i) the steady state is a saddle point for (b) If f 2,L ( K¯ , L, + ¯ 0+ ) < 0; (ii) set S p only contains slow-fast reg→ 0 , as det D F s ( K¯ , L, ular repeller points diverging from steady-state ( K¯ p , L¯ p ) for = 0, since s ( K¯ p , L¯ p , 0) > 0; and (iii) the regular steady-state is a stable node for f 2,L ¯ 0− ) < 0 and det D F s ( K¯ , L, ¯ 0− ) > 0. → 0− , as trD F s ( K¯ , L, 54 P. B. Brito et al. We can summarize the previous discussion in the following Proposition 3 which describes the types of DGE paths that can exist in the presence of a singular perturbation: Proposition 3 (DGE paths in the presence of a singular perturbation) Assume that ¯ if W s ( K¯ , L) ¯ is non-empty, there is a singular-perturbation and that K 0 ∈ W s ( K¯ , L), s ¯ ¯ ¯ or that K 0 = K , if W ( K , L) is empty.
However, for values of in a wider range around zero, the trace of the Jacobian tends to decrease which implies that the discriminant ⎡ ⎤ s ( K¯ , L, ¯ ) 2 4 f 1,L ( K¯ , L, ¯ ) f s ( K¯ , L, ¯ ) f 2,L 1 2,K ¯ ) = ⎣ f 1,K ( K¯ , L, ¯ )− ⎦ D F( K¯ , L, − 4 can become negative, and eigenvalues may become complex, leading to oscillatory dynamics. A geometric argument for the existence of real eigenvalues close to the singular critical set containing only slow-fast regular points is that the solution path will tend to evolve along the isocline for L, where f 2 (K , L , 0) = 0, which is a monotonic surface.