doi:10.1016/j.chaos.2003.12.012 
Copyright © 2003 Elsevier Ltd. All rights reserved.
Tongues of periodicity in a family of two-dimensional discontinuous maps of real Möbius type
a Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska st., Kiev 01601, Ukraine
b Department of Economics, University of Urbino, 61029, Urbino (PU), Italy
c Centre for Regional Science, Umeå University, SE-901 87, Umea, Sweden
Accepted 4 December 2003.
Available online 25 January 2004.
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Abstract
In this paper we consider a two-dimensional piecewise-smooth discontinuous map representing the so-called “relative dynamics” of an Hicksian business cycle model. The main features of the dynamics occur in the parameter region in which no fixed points at finite distance exist, but we may have attracting cycles of any periods. The bifurcations associated with the periodicity tongues of the map are studied making use of the first-return map on a suitable segment of the phase plane. The bifurcation curves bounding the periodicity tongues in the parameter plane are related with saddle-node and border-collision bifurcations of the first-return map. Moreover, the particular “sausages structure” of the bifurcation tongues is also explained.
Fig. 1. Bifurcation diagram of the map F at r=0.01.
Fig. 2. A trajectory of the map F for r=0.01, a=1 and c=0.65. The trajectory is tangent to the boundary of R1.
Fig. 3. In (a) it is shown the segment [AB] suitable for the first-return map. In (b) it is shown the one-dimensional map g(x), restriction of F2 to LC.
Fig. 4. Saddle-node bifurcation of the first-return map 
(x) at r=0.01, a=1.5, c=0.3174.
Fig. 5. First-return map (x) at r=0.01, a=1.5, c=0.3. Two fixed points are clearly visible, one stable and one unstable.
Fig. 6. First-return map (x) at r=0.01, a=1.5, c=0.2868. Border-collision bifurcation.
Fig. 7. First-return map (x) at r=0.01, a=1.26, c=0.149.
Fig. 8. Three-return map 3(x) at r=0.01, a=1.26, c=0.149. Three stable and three unstable fixed points are clearly visible.
Fig. 9. Attracting 8-cycle of F in the phase plane at r=0.01 and (a,c)=(1.25,0.365) belonging to the first area of the sausage structure of the period-8 tongue. Two periodic points belong to the region R2 and six belong to R1.
Fig. 10. Attracting 8-cycle of F in the phase plane at r=0.01 and (a,c)=(1.5,0.3) belonging to the second area of the sausage structure of the period-8 tongue. Three periodic points belong to the region R2 and five belong to R1.
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