We consider a class of three-dimensional maps T having the property that their third iterate has separate components. We show that the cycles of T can be obtained by those of a one-dimensional map (one of the components of T 3) and we give a complete classification of such cycles. The local bifurcations of the cycles of T are studied as well, showing that they are of co-dimension 3, since at the bifurcation value three eigenvalues simultaneously cross the unit circle. To illustrate the obtained results we consider as an example a delayed logistic map.
Agliari, A., Fournier Prunaret, D., Kaddous Taha, A., Periodic orbits and their bifurcations in 3-D maps with separate third iterate, in Bischi, G., Chiarella, C., Sushko, I. (ed.), Global Analysis of Dynamic Models in Economics and Finance. Essays in Honour of Laura Gardini, Springer Verlag, BERLIN HEIDELBERG 2012: 397- 427. 10.1007/978-3-642-29503-4_15 [http://hdl.handle.net/10807/32883]
Periodic orbits and their bifurcations in 3-D maps with separate third iterate
Agliari, Anna;
2012
Abstract
We consider a class of three-dimensional maps T having the property that their third iterate has separate components. We show that the cycles of T can be obtained by those of a one-dimensional map (one of the components of T 3) and we give a complete classification of such cycles. The local bifurcations of the cycles of T are studied as well, showing that they are of co-dimension 3, since at the bifurcation value three eigenvalues simultaneously cross the unit circle. To illustrate the obtained results we consider as an example a delayed logistic map.
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