If you need teaching examples, it is better to choose models that work perfectly or nearly perfectly in the real life (i.e. the "models" that are actually the laws of nature): e.g., lots of models in mechanics (celestial mechanics/gravity/Kepler laws, nonlinear pendulum, etc.) or in chemistry (bilinear reaction kinetics law and its upgrades).
In the "more real" life nonlinear models are usually only rough approximation of the real processes, and so some complications arise. The good examples of this are the basic models of spread of an infectious disease (so-called SIR-model and its variations SI, SIS, SEIR, etc...). Although the equations are effectively the same as in chem.kinetics, the way they are applied to the real data is quite different.
If you need teaching examples, it is better to choose models that work perfectly or nearly perfectly in the real life (i.e. the "models" that are actually the laws of nature): e.g., lots of models in mechanics (celestial mechanics/gravity/Kepler laws, nonlinear pendulum, etc.) or in chemistry (bilinear reaction kinetics law and its upgrades).
In the "more real" life nonlinear models are usually only rough approximation of the real processes, and so some complications arise. The good examples of this are the basic models of spread of an infectious disease (so-called SIR-model and its variations SI, SIS, SEIR, etc...). Although the equations are effectively the same as in chem.kinetics, the way they are applied to the real data is quite different.
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