Delay Model, SSA and Brownian Motion

This chapter presents three other tools to approach complex, nonlinear and chaotic dynamics. We will consider the Delay-model, the Singular Spectrum analysis and the Brownian motions (fractional or non-fractional). Firstly, we present the delay-model which is applied to the logistic equation. According to the Medio's work, a discrete-delay is integrated into the construction of an economic model by means of a convolution. The lengths of lags are distributed in a random way in the population. The delay is in fact modelled by means of a random variable which is characterized by its probability distribution. We will notice that in this way, the system built rocks more tardily to the chaos. We will observe a shift of bifurcation points, but also an unhooking in the trajectory.

Delay-model applied to the logistic equation. We will use the equation with the first-order differences used by Robert May. The central element of the model is the concept of “delay”. For a macroeconomic consumption model for example, if we postulate that there is an (unspecified) great number of agents, and that all these agents answer to a certain stimulation with given discrete-lags, the lengths of lags are different for various agents and are distributed in a random way in the population. In a global model, in the whole population, the reaction times are aggregate. In the described case, we can model the reaction-time by means of a random variable that will represent the global length of the lag.