What is the difference between nonlinear mechanics and the chaos theory? originally appeared on Quora, the place to gain and share knowledge, empowering people to learn from others and better understand the world. You can follow Quora on Twitter, Facebook, and Google Plus.

Nonlinear systems come in all shapes and sizes. Linear dynamics is a useful abstraction, but if you look long enough at any real system you consider linear, you will discover nonlinearities. It could be saturation, hysteresis, jump discontinuities, different sorts of nonlinear behavior coming from the system.

And that is perfectly fine. There are only a few disciplines of science and engineering so thoroughly investigated as linear systems: we know everything about them and as long as a real life system fits a linear model reasonably, the use of linear systems and linear interpretation of dynamics is fine. Set your linear differential equations up and solve them.

However, that water tank over there just seems to defy your linear model fitting: it has a square root relationship of level and flow somewhere. You may try to linearise it around a point, but such a model would be valid in just a narrow interval. And that’s where you make a nonlinear model and observe nonlinear dynamics. It’s fun, it’s exciting and probably hard. You often can’t get an exact solution, but a numerical one, but lo and behold all the rich dynamics you can get from it! The limit cycles, for example.

If you have a nonlinear system of a sufficient order (in continuous systems, that would mean greater or equal than three), your system might reach even more interesting dynamics for certain values of its parameters. It could exhibit sensitive dependence on initial conditions (two very close trajectories tend to diverge from each other exponentially fast), its trajectories in phase space could be confined to an area of non-integer dimension (strange attractor), etc. This is what we call chaos, and chaos theory addresses the phenomena in chaos and the ways to transition to it from the state of order. Chaotic systems are nonlinear (although technically you could have an infinite-order linear system which would be chaotic), but most nonlinear systems you observe can never be chaotic. Their order is either too low or their parameter space does not have any combination of parameters suitable for demonstration of chaos.

A traditional problem of nonlinear dynamics is one of a non-linear oscillator: it’s not chaotic, it doesn’t try to be chaotic, it’s just more complex than a linear oscillator (and more realistic).

Chaos is an exception in the world of nonlinear dynamics. An interesting and valuable one, but also one confined to narrow windows of parameter space and only certain dynamical systems.

Nonlinear dynamics, on the other hand, rules the world.

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