Download Business Cycles Dynamics: Models and Tools by Iryna Sushko PDF

By Iryna Sushko

Company cycle idea has been one of many quickest starting to be fields in glossy nonlinear monetary dynamics. The booklet is founded round versions of multiplier-accelerator variety, rising from Samuelson's seminal paintings, later built into nonlinear codecs via Hicks and Goodwin. those types left open ends, because the instruments then on hand didn't enable extra systematic research. the current state of affairs is assorted, as a result of the emergence of recent tools additionally focusing worldwide research. the point of interest on classical, causal or recursive versions implies a deviation from present major circulation enterprise cycle conception, in keeping with ''rational expectations'', which in view of the potential for mathematical chaos turns into untenable.

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Example text

The dynamical behavior of a subcritical Neimark-Sacker bifurcation is very importand in the economic literature (as well as in other applied models). In fact, the existence of a repelling closed curve which bounds the basin of attraction of the stable fixed point implies that small shocks of the system have no effects on its dynamical behavior, while large enough shocks may lead to another attractor. This requires the coexistence of the fixed point with a different attracting set, and may cause hysteresis phenomena.

As we shall see, in such a case we have a particular kind of noninvertibility in which a whole half-plane R2 is mapped into one straight line, so that the map is of so-called {ZQ — Z^ — Z\) type. While for 6 7^ 0 the map F can be either invertible (for h > 0), or noninvertible (for 6 < 0) of {Zo — Z2) type, so that we can compare the results of the center bifurcation in these cases. The map F is given by two linear maps Fi and F2 defined, respectively, below and above the straight line LC-i = {{x,y) :y = x-\- d/a] .

13d. l3e: A repelling closed curve Tu appears, replacing the saddle-focus connection (and replacing it in the role of separatrix between 1 Some Methods for the Global Analysis 37 the basins of attraction of A and Tg). Once more, the occurrence of this global bifurcation can be checked observing the behavior of the branches ai and oji involved in it. Summarizing, we have seen that the coexistence of two closed invariant curves, one attracting and one repelling, in discrete maps can be achieved by a double mechanism: Starting from a repelling cycle and a saddle cycle, a first saddle connection (or tangle) causes the appearance of the attracting one associated with an (unstable) heteroclinic connection saddle - repelling cycle that plays the role of separatrix of basins, which is then replaced by the second closed curve, repelling, whose appearance is associated with a second saddle connection (or tangle).

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