Business Cycle Dynamics: Models and Tools by Tönu Puu (auth.), Professor Dr. Tönu Puu, Dr. Iryna Sushko

By Tönu Puu (auth.), Professor Dr. Tönu Puu, Dr. Iryna Sushko (eds.)

Business cycle thought has been one of many quickest starting to be fields in glossy nonlinear financial dynamics. The e-book is established round types of multiplier-accelerator sort, rising from Samuelson's seminal paintings, later built into nonlinear codecs by way of Hicks and Goodwin. those types left open ends, because the instruments then to be had didn't enable extra systematic research. the current state of affairs is diversified, end result of the emergence of latest tools additionally focusing international research. the focal point on classical, causal or recursive versions implies a deviation from present major circulation enterprise cycle conception, according to "rational expectations", which in view of the potential of mathematical chaos turns into untenable. This ebook is a rejoinder to Puu and Sushko, Oligopoly Dynamics - types and Tools, (Springer 2002).

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It is for this reason that we propose this mechanism as a generic sequence of global bifurcations giving rise to two coexisting closed curves. More theoretical studies need to confirm such a conjecture. 8 Coexistence of Curves and Cycles and Their Interactions (Heteroclinic Loop) In this section we show a mechanism that causes the transition from an attracting closed invariant curve, say Fa, with a pair of cycles of period k outside it, a saddle S and an attracting one, C, inside a wider attracting closed invariant curve, say F^.

Any initial point (xo,2/o) € Q belongs to a quasiperiodic orbit dense in the corresponding invariant ellipse ofFi. 4, d = 10. (Indeed, because of numerical precision, we cannot show a true quasiperiodic case, but only its approximation by a periodic case of some high period). , the region Q defined above) and is invariant, when the map Fi is defined by a rotation matrix with a rational rotation number, but in such a case Q is not the largest invariant area. In fact, there are also other points outside the tangent ellipse £ which Iryna Sushko and Laura Gardini 54 are periodic with an orbit completely included in the region Ri.

Repelling). The case associated with the attracting fixed point P* is also interesting because it may explain the appearance of the repelling closed curve involved in the Neimark-Sacker bifurcation of subcritical type. Clearly the bifurcations described above may involve saddles and attracting or repelling cycles of period k{k > 1) instead offixedpoints: In such a case the mechanisms previously described occur for the map T^ and lead to k cyclical invariant closed curves, repelling or attracting, for the map T.

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