| Abstract: |
| In this paper we study a simple model based on the cobweb demand-supply framework with costly innovators and free imitators. The evolutionary selection between technologies depends on a performance measure which is related to the degree of memory. The resulting dynamics is described by a two-dimensional map.\The map has a fixed point which may lose stability either via supercritical Neimark-Sacker bifurcation or flip bifurcation and several multistability situations exist. We describe some sequences of global bifurcations involving attracting and repelling closed invariant curves. These bifurcations, characterized by the creation of homoclinic connections or homoclinic tangles, are described through several numerical simulations. Particular bifurcation phenomena are also observed when the parameters are selected inside a periodicity region.
\The analysis gives us the opportunity to discover a peculiar behavior occurring within such region. In fact we find that, unlike what usually takes place inside an Arnold tongue in a neighborhood of the Neimark-Sacker bifurcation curve, (i.e. a periodic orbit exists and the closed invariant curve is made by a saddle-node connection), an eventuality of multistability with a cycle and a closed invariant curve may arise. This leads us to conjecture the existence of further global bifurcations recurring within the periodicity region emanating from the Neimark-Sacker bifurcation curve, and the Arnold tongue seems to be a subset of a larger periodicity region. |
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