TY - JOUR
T1 - Mann iteration with power means
AU - Bischi, G. I.
AU - Cavalli, Fausto
AU - Naimzada, A.
PY - 2015
Y1 - 2015
N2 - We analyse the recurrence (Formula presented.) , where (Formula presented.) is a weighted power mean of (Formula presented.) , which has been proposed to model a class of non-linear forward-looking economic models with bounded rationality. Under suitable hypotheses on weights, we prove the convergence of the sequence (Formula presented.) Then, to simulate a fading memory, we consider exponentially decreasing weights. Since, in this case, the resulting recurrence does not fulfil the hypotheses of the previous convergence theorem, it is studied by reducing it to an equivalent two-dimensional autonomous map, which shares the asymptotic behaviours with a particular one-dimensional map. This allows us to prove that a long memory with sufficiently large weights has a stabilizing effect. Finally, we numerically investigate what happens when the memory ratio is not sufficiently large to provide stability, showing that, depending on the power mean and the memory ratio, either a delayed or early cascade of flip bifurcations occurs.
AB - We analyse the recurrence (Formula presented.) , where (Formula presented.) is a weighted power mean of (Formula presented.) , which has been proposed to model a class of non-linear forward-looking economic models with bounded rationality. Under suitable hypotheses on weights, we prove the convergence of the sequence (Formula presented.) Then, to simulate a fading memory, we consider exponentially decreasing weights. Since, in this case, the resulting recurrence does not fulfil the hypotheses of the previous convergence theorem, it is studied by reducing it to an equivalent two-dimensional autonomous map, which shares the asymptotic behaviours with a particular one-dimensional map. This allows us to prove that a long memory with sufficiently large weights has a stabilizing effect. Finally, we numerically investigate what happens when the memory ratio is not sufficiently large to provide stability, showing that, depending on the power mean and the memory ratio, either a delayed or early cascade of flip bifurcations occurs.
KW - Forward-looking models
KW - Learning
KW - Mann iterations
KW - Non-autonomous difference equations
KW - Forward-looking models
KW - Learning
KW - Mann iterations
KW - Non-autonomous difference equations
UR - http://hdl.handle.net/10807/85361
U2 - 10.1080/10236198.2015.1080252
DO - 10.1080/10236198.2015.1080252
M3 - Article
SN - 1023-6198
VL - 21
SP - 1212
EP - 1233
JO - Journal of Difference Equations and Applications
JF - Journal of Difference Equations and Applications
ER -