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Discrete Dynamics in Nature and Society

Volume 2016, Article ID 6891302, 4 pages

http://dx.doi.org/10.1155/2016/6891302

## A Note on Discrete Multitime Recurrences of Samuelson-Hicks Type

^{1}Department of Law and Economics, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy^{2}Decisions Lab, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy^{3}Department of Economics and Statistics, University of Siena, Siena, Italy

Received 11 August 2016; Accepted 5 October 2016

Academic Editor: Gafurjan Ibragimov

Copyright © 2016 Bruno Antonio Pansera and Francesco Strati. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By this work, we aim at fostering further research on the applications of multitime recurrences. In particular, we shall apply this method by generalizing the Samuelson-Hicks model so as to make the new concept of time that this method proposes clear. In particular, the multitime approach decomposes a point of time into a vector, taking into account how different coordinates of time referring to the same date can affect the dynamics of a model.

#### 1. Introduction

This work belongs to a series of papers (see [1–4]) which explore the notion of* multitime* in different branches of mathematical analysis, such as economics and physics. Since 1932, the adjective* multitime*, introduced by Paul Dirac, appears so as to cope with an environment in which there might be more than one dimension of time.

We make some considerations concerning the physical idea of the multitime notion: we recall that a* coordinate space* is an index numbering degrees of freedom, and the coordinate of time is the usual physical time in which a system evolves. This model is satisfactory, unless we turn our attention to relativistic problems. Moreover, some physical phenomena (and social sciences too) are observed in a two-time environment , in which is the* intrinsic time* and is the* observer time*. In some real phenomena, there is no reason to prefer one coordinate to another. Following these assumptions, we refer to multitime as* vector parameters of evolution* in* multitime geometric evolution* and* multitime optimal control problems*.

The second relevant aspect to analyze is the multitime wave functions considered by Paul Dirac in 1932 via -time evolution equations . The Dirac PDEs system is consistent (completely integrable) if and only if for . The consistency condition is easy to achieve for noninteracting particles, while it turns out to be harder in the presence of interactions. All of these laws cannot be applied to relevant problems, if one aims at a plain covariant formulation of relativistic quantum mechanics.

Eventually, following the literature, we can divide the laws of evolution of physical theories into two branches: the* single-time evolution laws* (ODEs) and the* multitime evolution laws* (PDEs). It is clear that, in order to turn a single-time evolution into a multitime evolution, changing the ODEs into PDEs is enough by accepting that time is function of certain parameters; let us say that .

Many authors studied the different applications of these notions: we recall some significative papers as [4–10] in which the PDEs constraints usually show significant challenges for optimization principles. The multivariable maximum principle was studied in the presence of PDE constraints (see [1]).

The multivariate recurrences are based on multiple sequences; they come from areas like analysis of algorithms, computational biology, information theory, queueing theory, filters theory, statistical physics, and so forth. We consider a lattice of points with positive integer coordinates in . A multivariate recurrence is a set of rules which transfer a point to another, together with initial conditions, which is capable of covering the hole lattice.

A multivariate recurrence relation is an equation that recursively defines a multivariate sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms. Some defined recurrence relations can have very complex (chaotic) behaviors, and they are part of a field of mathematics known as nonlinear analysis.

We shall extend the* Floquet theory* which had been firstly utilized for periodic linear ODEs [11] and then extended to difference equations [12–15]. In [3, 16], the authors have extended this theory to the multitemporal first-order PDEs. In Floquet theory, it is necessary to find the associated monodromy matrix and its eigenvalues (called Floquet multipliers) in order to pass from a constant coefficient state to a periodic one. It can be crucial in a recurrence formulation (as noted in [14]) in which time-independent coefficients are unrealistic. An application of the Floquet theory can be found in [14]; here, we shall discuss a further possible extension. In this case, there is a generalization of the periodicity to a* sheet of time* rather than a point of time approach. In few words, it can be stated that, by a multitime approach, a point of time opens up to a vector of multiple times at that date (see Figure 1).