2024-04-08 22:00:00
Systems used or designed in engineering fields are mostly inherently nonlinear. Some models can be approximated and simplified, which leads to linear modeling. However, in many cases, the approximations which lead to a linear system result in a loss of information compared to the initial system. It is therefore important to preserve the nonlinearities of a system as much as possible. Linear systems can be studied using the tools of linear algebra, but there are no such powerful generic tools to analyze all non-linear systems, at most we can classify these systems, for example from the type of dominant non-linearity, and propose an analysis of the behavior of the system. The introduction of digital tools in the mid-20th centurye century has enabled significant progress in this type of analysis. In general, a combination of theoretical studies and the use of digital tools will make it possible to better understand the impact of nonlinearities in a system. Numerical studies have made it possible to highlight complex behaviors that are impossible to observe in a linear system with real variables, such as behaviors described as chaotic.
Models of electronic, mechanical, automatic systems or those used in telecommunications transmissions often involve parameters that the user can modify so that the system acts adequately. For example, it may be useful for a system to remain in a fixed steady state; in other cases, it may be sufficient for the system to remain bounded (case of BIBO stability – Bounded Input Bounded Output, i.e. bounded input, bounded output); periodic type behaviors may also be of interest. In some cases, the user may not have complete control of the settings; these can vary in ranges where behaviors are drastically modified. For all systems, knowledge of possible behaviors when parameters evolve is of the greatest interest. A fundamental notion is that of bifurcation which corresponds to a qualitative modification of the behavior of a system when one or more parameters are modified. The system can go from a stable stationary state to periodic behavior, with a period which can change very quickly by variation of a parameter, or even to complete instability. Behaviors called “chaotic” that appear disordered can also occur.
This article is more particularly interested in the bifurcations which occur in discrete-time systems modeled by a point transformation or recurrence. It is possible to highlight not only specific bifurcations, but also global structures of bifurcations which are associated with certain types of nonlinearities. These global structures allow an overall understanding of a system and can particularly explain which mechanisms lead to chaos. We then obtain what we call a “road to chaos”.
In the first section, several types of nonlinearities will be studied using generic models: a quadratic or cubic nonlinearity, a modulo type nonlinearity or discontinuities of the function (or its derivative) which intervenes. in modeling. For each non-linearity, a specific structure of bifurcations will be highlighted: “nested boxes” structure for the first non-linearity, “boxes in rows” structure such as “addition of periods” or “increment of periods” for the second.
The second section will present examples of simple systems from several fields of engineering that present these different types of structures.
Thus, the study of a modulator in a transmission system, that of an oscillator and that of a hybrid electrical circuit will be proposed. In each of these examples, it is interesting to analyze the behavior of the system in order to either choose the parameters which will correspond to stable behavior, or on the contrary those which will lead to chaotic and limited behavior. Indeed, chaos corresponds to behavior that appears random, and it is possible to use chaotic signals to generate, for example, pseudo-random numbers.
The conclusion will give some perspectives for more complex studies.
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