Bifurcation structures in nonlinear discrete-time systems

The systems used or designed in engineering fields are for the most part intrinsically non-linear. Some models can be approximated and simplified, leading to linear modeling. However, in many cases, the approximations that lead to a linear system result in a loss of information with respect to the original system. It is therefore important to preserve the non-linearities of a system as far as possible. Linear systems can be studied using the tools of linear algebra, but there are no equally powerful generic tools for analyzing all non-linear systems. At most, we can classify these systems, for example on the basis of the type of non-linearity that dominates, and propose an analysis of the system's behavior. The introduction of numerical tools in the mid-twentieth century enabled a significant advance in this type of analysis. In general, a combination of theoretical studies and the use of numerical tools will provide a better understanding of the impact of non-linearities in a system. Numerical studies have highlighted complex behaviors that are impossible to observe in a linear system with real variables, such as those described as chaotic.

Models of electronic, mechanical or automatic systems, or those used in telecommunications transmissions, often include parameters that the user can modify to ensure that the system acts appropriately. For example, it may be useful for a system to remain in a fixed stationary state; in other cases, it may be sufficient for the system to remain bounded (case of BIBO stability – Bounded Input Bounded Output); periodic-type behavior may also be of interest. In some cases, the user may not have complete control over the parameters, which may vary in ranges where the behavior is drastically altered. For all systems, knowledge of the possible behaviors when parameters change is of the utmost interest. A fundamental notion is that of bifurcation, which corresponds to a qualitative change in a system's behavior when one or more parameters are modified. The system can go from a stable stationary state to periodic behavior, with a period that can change very rapidly when a parameter is varied, or even to complete instability. So-called "chaotic" behaviors that appear disordered can also occur.

This article focuses on bifurcations occurring in discrete-time systems modeled by a point or recursion transformation. It is possible to highlight not only specific bifurcations, but also global bifurcation structures that are associated with certain types of nonlinearities. These global structures provide an overall understanding of a system, and in particular can explain which mechanisms lead to chaos. The result is a so-called "road to chaos".

In the first section, several types of nonlinearities will be studied using...