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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Notes
- 1.
Indeed if \({h}_{1} > {s}_{1}\) we can write \(d = {h}_{1}n + {h}_{2}m - {s}_{1}n + {s}_{1}n = \left ({h}_{1} - {s}_{1}\right )n + \left ({h}_{2} + {s}_{2}\right )m\) and proceed in this way until the integer multiplying n is smaller than s 1 but still positive (while the second integer is not positive).
- 2.
Along this proof, the prime symbol denotes the derivative.
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Acknowledgements
The Authors address a special thank to Laura Gardini for her continuous encouragement and her valuable suggestions in developing new research in the Dynamic Systems field. Anna Agliari is also grateful for the overwhelming enthusiasm of Laura, a true inspiration for all her studies.
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Agliari, A., Fournier-Prunaret, D., Taha, A.K. (2013). Periodic Orbits and Their Bifurcations in 3D Maps with a Separate Third Iterate. In: Bischi, G., Chiarella, C., Sushko, I. (eds) Global Analysis of Dynamic Models in Economics and Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29503-4_15
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