Chaotic Dynamics

Multistability and cyclic attractors in duopoly games

“…If the parameters' values are such that the cyclic chaotic intervals of the map p are chaotic``in strict sense'' 1 [14] then it is also proved that for the two-dimensional map the cyclic rectangles are chaotic in the strict sense, which means, also in the two-dimensional case, that almost all the trajectories in such intervals are aperiodic, the periodic points are dense in the rectangles, no attracting cycles exists, and an absolutely continuous invariant (two-dimensional) measure exists.…”

“…[14,15]), the sides of the chaotic rectangles of the two-dimensional map are formed by segments of lines through these critical points and parallel to the coordinate axes. Hence the boundaries of these cyclic chaotic rectangles are completely known on the basis of the knowledge of the critical points p k of the one-dimensional map p and those of qY given by q k 1 r 2 p k .…”

“…8b, obtained with l 2 3X85 and l 1 P 2X77Y 3X04 , some noticeable sequences of local and global bifurcations can be identi®ed. Sequences of fold and¯ip bifurcations, similar to those occurring in unimodal maps, can be classi®ed in terms of a``box-within-a-box '' structure, as described in Appendix B (see also [6,7,14]). As long as p has two distinct attractors we can consider two parallel sequences of boxes, one related to the asymptotic behavior of the critical point a and the other related to that of b .…”

“…[2,5,14]). The asymptotic dynamics take place in two absorbing intervals, t a and t b given in (21), giving rise to two disjoint attracting sets for F as long as l`l à 1 (this bifurcation value shall be commented on below).…”

“…A box of the second kind is opened by a¯ip bifurcation of a k-cycle that creates an attracting 2k-cycle. A more detailed description can be found in [9,14,15].…”