TITLE: How to know whether an Ordinary Differential Equation is Chaotic?
QUESTION [19 upvotes]: Assuming we have an ordinary differential equation (ODE) such as Lorenz system:
$$
\dot x=\sigma(y-x)\\
\dot y=\gamma x-y-xz\\
\dot z=xy-bz
$$
where 
$$
\sigma = 10\\
\gamma = 28\\
b = \frac{8}{3}\\
x(0)=10\\
y(0)=1\\
z(0)=1
$$
This system is known to be chaotic because of its behavior [1], [2].
However, we normally judge about a system by the output results plot. But how can I judge about a system whether it is chaotic or not just by looking at its formulation in state space representation without plotting it?
Or if there is no way for 100% judging, at least is there any way to guess it?

REPLY [9 votes]: An introduction to the Lorenz system can be found in [1,2].
If there is no general tool to prove that a continuous dynamical system is chaotic, there are at least several tools to prove that a system is not chaotic (see e.g. [3]). Here is a short non-exhaustive list of features which allow a first-order autonomous ODE system
$$
\dot{X} = F(X) \, \qquad\text{where}\qquad X\in\mathbb{R}^n \;\text{and}\; F \in C^1
$$
to be chaotic:

$F$ is nonlinear
the phase space dimension $n$ is equal to $3$ or larger (consequence of the Poincaré-Bendixson theorem)
there is at least one eigenvalue of the Jacobian matrix $\partial F/\partial X$ evaluated at the equilibria of the system that has a non-negative real part (consequence of the Hartman-Grobman theorem)

There are several case-dependent methods for the analysis of chaos. In the case of periodically forced Hamiltonian systems, a dedicated tool is Melnikov's method.