On Mitra's Sufficient Condition for Topological Chaos: Seventeen Years Later
Liuchun Deng, M. Ali Khan
Economics Letters,
March
2018
Abstract
This letter reports an easy extension of Mitra’s “easily verifiable” sufficient condition for topological chaos in unimodal maps, and offers its application to reduced-form representations of two economic models that have figured prominently in the recent literature in economic dynamics: the check- and the M-map pertaining to the 2-sector Robinson–Solow–Srinivasan (RSS) and Matsuyama models respectively. A consideration of the iterates of these maps establishes the complementarity of the useful 2001 condition with the 1982 (LMPY) theorem of Li–Misiurewicz–Pianigiani–Yorke when supplemented by a geometric construction elaborated in Khan–Piazza (2011).
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On Growing through Cycles: Matsuyama's M-map and Li-Yorke Chaos
Liuchun Deng, M. Ali Khan
Journal of Mathematical Economics,
January
2018
Abstract
Recent work of Gardini et al. (2008), building on earlier work of Mitra (2001) and Mukherji (2005), considers the so-called M-map that generates a dynamical system underlying Matsuyama’s (1999) endogenous growth model. We offer proofs of the fact that there do not exist 3- or 5-period cycles in the M-map, and an example (a numerical proof) of the existence of a 7-period cycle. We use the latter, and a construction in Khan and Piazza (2011), to identify a range of parameter values of the M-map that guarantee the existence of cycles of all periods, except 3 and 5. Our argumentation relies on, and reports, the first four iterations of the M-map that may have independent interest.
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