On Shilnikov attractors of three-dimensional flows and maps

Scenarios for the creation of hyperchaotic attractors in 3D maps

“…Very close to the point ff 4 , there is one more important codimension-two bifurcation, which significantly contributes to the organisation of the bifurcation diagram. On the curve PD 4 , we observe a point of the degenerate period-doubling bifurcation which gives rise to a saddlenode bifurcation curve SN 8 . A pair of stable and saddle period-8 orbits P 8 and S 8 is born when crossing the upper part of this curve.…”

“…Respectively, the perioddoubling bifurcation transforms the stable orbit P 4 to the saddle of type (2,1) and the saddle orbit S 4 to the saddle of type (1,2). Further, we explain bifurcations associated only with the 8 , and SN 16 are colored in blue and the period-doubling bifurcation curves PD 4 , PD 8 , PD 16 , pd 4 , pd 8 , and pd 16 -in green (solid curves are used for bifurcations of stable orbits, while dashed-of saddle ones); the points ff 4 , ff 8 , and ff 16 correspond to the fold-flip bifurcations; ff ∞ is a limit point for the fold-flip bifurcations. left fold-flip point ff 4 , keeping in mind that the bifurcations at the right fold-flip point are the same.…”

“…The bottom boundary of this region consists (as for the period-4 orbit P 4 ) of two pieces PD 8 corresponding to the supercritical period-doubling bifurcation. The same as for P 4 , the period-doubling bifurcation curve PD 8 for P 8 is tangent to the curve SN 8 at the fold-flip bifurcation point ff 8 . The bottom branch of this curve (pd 8 ) corresponds to the period-doubling bifurcation of the saddle orbit S 8 .…”

“…The same as for P 4 , the period-doubling bifurcation curve PD 8 for P 8 is tangent to the curve SN 8 at the fold-flip bifurcation point ff 8 . The bottom branch of this curve (pd 8 ) corresponds to the period-doubling bifurcation of the saddle orbit S 8 . Close to the point ff 8 , on the curve PD 8 , we again observe the degenerate period-doubling bifurcation which gives rise to a saddle-node bifurcation curve SN 16 and so on.…”

“…The bottom branch of this curve (pd 8 ) corresponds to the period-doubling bifurcation of the saddle orbit S 8 . Close to the point ff 8 , on the curve PD 8 , we again observe the degenerate period-doubling bifurcation which gives rise to a saddle-node bifurcation curve SN 16 and so on.…”