Hello

You can find one fine explanation at the below given link

https://youtu.be/wOQDGvCLOs8

see

Dynamical behavior of epidemiological models with nonlinear incidence rates

Wei-min Liu, Herbert W. Hethcote, Simon A. Levin

It would help to focus the discussion if you were more specific about what you mean by "nonlinear models." One of the earliest nonlinear model was Galileo's observation that a ball rolling down an inclined plane traveled a distance proportional to the square of the time. Newton's law of gravitation is a nonlinear model since it involves the product of the masses and the square of the distance between them.

I suspect you are thinking of nonlinear models of dynamical systems in discrete or continuous time governed by difference or differential equations since that is a topic of great current interest about which many books and articles have been written with titles that include terms like "nonlinear dynamics" and "chaos." Or perhaps your interest is in nonlinear time series analysis, which is another currently fashionable topic.

The exceptional dynamical behavior of certain nonliner models is suited to build enclosure-overcoming mathematics from the ground up: cf. Number form & Structure wave theory.

This allows to define number form metrics that enhance considerably the enclosure-confined paradigm and thus their possibilities of traditional science.

Let us consider nonlinear dynamical systems

x'=f(X,y)

y'=g(X,y)

Clearly we look at the qualitative behavior of the solution close to the equilibrium points and making use of Hartman. Theorem

Some of the features are:

1. May have multiple isolated equilibrium points.

2. Finite escape time: the state of a nonlinear model with unstable fixed point(s) can go to infinity in finite time.

3. However, for certain initial conditions, the states may be confined within some region for all time t>=0, (e.g. chaos).

4. Since multiple equilibrium points is possible, the concepts of local and global stability arise.

5. Stability of such a model near an equilibrium point does not necessarily guarantee stronger stability notions like asymptotic or exponential stability.

6. Hence, stability does not imply convergence and vice versa.

7. Some times, the Lyapunov first method cannot be used to prove stability of equilibrium point. Lyapunov second method or even the invariance principle may be required.

8. More interesting steady-state dynamics such as limit cycle, periodic oscillations, chaos can exist.

9. Homogeneity and additivity (i.e. superposition principle) are not satisfied by nonlinear models.

Marquez, H.J., 2003. Nonlinear control systems: analysis and design (Vol. 1). Hoboken: Wiley-Interscience.

Khalil, H.K. and Grizzle, J.W., 2002. Nonlinear systems (Vol. 3). Upper Saddle River, NJ: Prentice hall.

The really interesting question is how a nonlinear model gets qualified to ”quit“ the (unbounded) reference level where it is initiated: this goal is achieved with Number form theory.

See the following link,

https://www.quora.com/What-is-the-difference-between-linear-and-non-linear-features.