Mann iteration with power means

Abstract

We analyse the recurrence $x_{n+1}=f\left(z_n\right)$, where $z_n$ is a weighted power mean of $x_0,\dots ,x_n$, 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 $x_n.$ 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.

Keywords::

• forward-looking models
• learning
• Mann iterations
• non-autonomous difference equations

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

^1. Email: [email protected]

^2. Email: [email protected]

Additional information

Funding

This work was performed within the framework of COST Action IS1104 ‘The EU in the new economic complex geography: models, tools and policy evaluation’ and under the auspices of GNFM, Gruppo Nazionale di Fisica Matematica (Italy).