Our physical environment can be seen as a very complex system where various components interact. Naturally, in order to comprehend some details of the dynamics of our environment, scientists aim at isolating a smaller subsystem. By experimenting and measuring they investigate the behavior of the subsystem and by identifying laws and formulating mathematical models, ie, a set of evolution equations, they try to explain the observations. And science does not stop after obtaining a model; we study the past and present time to make predictions for the future. A mathematical analysis of a model can help to make such predictions.

Before that, a model must be built, which is an art in itself; identification and description of the type of time evolution, state variables, parameters and nonlinearities can be a difficult task. We will not address this issue here, but rather use models as examples for our analysis. Good deterministic models (with nonlinear interactions), may be found in some physical settings, eg, electronic devices, chemical reaction systems, neuron networks or laboratory populations. Even when the dynamics of a system is deterministic, providing good predictions can be nearly impossible. Indeed, one of the main problems of analysis in the twentieth century is that it is notoriously difficult to analyze and solve nonlinear equations. Therefore, our predictions usually have only a limited range. However, even with limitations predictions can be useful. For instance, an electrical engineer wants to know whether the voltage output in an electrical circuit is reliable. If a mathematical analysis of the circuit can provide operating