# Non-linear dynamics vs Chaos

I am confusing between non linear dynamics and chaos. Chaos is also a non-linear dynamics right? then what is the difference between chaos and non-linear dynamics?

What I understood about chaos is that, it is some errors or small deviations which repeat itself in space-time and with time it amplifies. Also, this depends on the initial condition as well. This is also a non-linearity right?

-
–  Wouter Apr 21 at 17:59
There are some rare interesting examples of chaotic motion, that even if you know the initial conditions accurately; you won't be able to predict their future with any accuracy. For instance, a particle resting at the top of to the curve $y=-x^{\frac 4 3}$, can leave the top at any moment, without any thermal perturbations. –  Ali Apr 21 at 20:52

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1:

Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

Like explained in the text:

• aperiodic long-termed behavior = the system never settles down into a stable configuration.

• deterministic = you rule out the possibility that the irregular motion is due to noise or random input, i.e. you want it to be due to the nonlinearity of the system.

• sensitive dependence on initial conditions = even if you start with two initial conditions that are very close to each other, the result of each will be tremendously different. I.e. you can't say what will happen to other (close) points if you know how one point behaves.

A nice example is the double rod, see Wiki:

Double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.

A linear system on the other hand is a system that, when you augment the input, e.g., twice, the output will also be twice as big. In nonlinear system however the change in the input can be totally different from the change in output.

I must however add a caveat: according to wikipedia chaos is possible in linear systems if the system is infinite dimensional.

For clarification: chaos is not another word for instability. Take for example the system $$\dot{x}=\frac{dx}{dt}=x.$$ The solution for this system is an exponential $x=x(0)e^{t}$, which implies that nearby trajectories separate exponentially from each other. But this is not chaos! Thi sis because we know what will eventually happen: i.e. the trajectories are repelled to infinity and will never return. Here the fixed point of the system is infinity!

Chaos however is inherently unpredictable and aperiodic, it excludes fixed points and periodic solutions.

References:

1. Nonlinear Dynamics And Chaos by Steven H. Strogatz
-
With respect to the "sensitive dependence on initial conditions" qualification: dx/dt = x certainly has 'sensitive dependence', since any difference will be magnified exponentially, but it's linear. So what distinguishes that from, say, the Lorenz equations? –  Venge Apr 21 at 19:46
–  metacompactness Apr 21 at 19:59
@metacompactness : what is the book from which you took this definition? –  Anne O'Nyme Apr 22 at 16:42
@AnneO'Nyme chapter 8 of this book. –  metacompactness Apr 22 at 16:48

It's true that not all nonlinear systems are chaotic, but that all chaotic systems are nonlinear (or infinite-dimensional linear). The sensitivity to initial conditions is an important point and the commenter raises a good question.

Consider the Lorenz system in a non-chaotic parameter regime (or for that matter, any stable oscillatory system): as time goes to extremely big numbers, we can be sure that we'll end up lying somewhere on a fixed trajectory, regardless of the initial conditions (assuming we started somewhere in the basin of attraction for the attractor). So that's a big limitation on where we'll end up.

Now consider the Lorenz system in a chaotic parameter regime, the strange attractor. What defines the situation as such is the fact that we cannot limit where we will end up in the same way; there is no eventually-fixed trajectory in a chaotic system.

The addition here, and more mathematically direct way of saying what I just said, is that a chaotic system has at least one positive Lyapunov exponent. For any dynamical system, one can calculate one Lyapunov exponent for each dimension of the system; the Lorenz system has three, for instance, for a given parameter regime. The Wikipedia article defines it well, so I won't belabor the definition: a Lyapunov exponent defines the rate of divergence (or convergence) of a system's trajectories in its phase space. You can imagine, then, for a stable attractor, these trajectories would get infinitely closer together as time goes to infinity. This would then be a negative Lyapunov exponent.

Lyapunov exponents can only be calculated numerically, but the more time you have, the surer you can be of their values. Since a chaotic system never converges to a stable trajectory in phase space, at least one Lyapunov exponent will be positive. Finally, all of them cannot be positive because this would be a system which goes to infinity as time goes to infinity, and therefore does not have any kind of attractor.

https://en.wikipedia.org/wiki/Lyapunov_exponent

-

As you have commented, there is a minor but very important omission. The sensitivity to initial conditions is infinitely big. Or better, say, two vectors from the state space, modeling the initial conditions, no matter how close they are would eventually diverge from each others corresponding trajectories.

-