Multistability and role of noninvertibility in a discrete-time business cycle model

We study a discrete-time business cycle model in income and capital, a Kaldor-type model, in order to discuss the problem of coexistence of attractors and the related problem of basins of attraction. The model considered is particularly useful for pedagogical purposes because economically meaningful ranges of parameters exist such that an attractor characterized by oscillatory motion (which may be periodic, quasi-periodic or chaotic) coexists with two stable steady states, and consequently the choice of the initial conditions is crucial to decide if economic fluctuations are obtained or not in the long run. Moreover, the map whose iteration gives the time evolution of the system may be invertible or noninvertible according to the parameter constellations considered. These features allow us to compare the different behaviors of the model in these two regimes to stress the role of noninvertibility in the global dynamical properties, due to the geometric action of folding the phase space. In particular, we describe the creation of non-connected basins, and we show that the two regions of the phase space separated by a closed invariant curve are not invariant. Such properties have no analogue neither in continuous time models nor in discrete time models described by invertible maps.