Download Sunspots and Non-Linear Dynamics: Essays in Honor of by Kazuo Nishimura, Alain Venditti, Nicholas C. Yannelis PDF

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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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.

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