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I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in deep. This is the equation used in the book: $$ \frac{d^2\theta}{dt^2} = -\frac{g}{l}\sin\theta - q\frac{d\theta}{dt}+F_d\sin(\Omega_dt) $$

Obviously the authors know when (whit which conditions) the pendulum switch to a chaotic regime.

My question is the following: looking at the ODE, we can see (predict) the existence of chaotic behavior? If yes, this is valid for all the ODEs or only for a strict family of ODEs?

[1] Computational physics, N. J. Giordano and H. Nakanishi, Second Edition, Pearson Prentice Hall, 2006

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See here for some examples applying the usual techniques of nonlinear dynamics to this system. – poorsod Aug 22 '12 at 15:17
@poorsod Thank you for the reference: it seems (same graphs) the book I used. However my question was much more general: can we say that a system hide chaotic behavior only at looking ODEs? – R. M. Aug 22 '12 at 15:47
@RM I have written a reply in a full answer. – poorsod Aug 22 '12 at 16:10

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In general, no. It is possible to recognise a (system of) differential equation(s) as being nonlinear purely by inspection, but there are plenty of non-chaotic nonlinear systems. Chaos is a stronger (and, unfortunately, not well-defined) condition.

Also, many (most?) systems, including the famed Lorenz attractor, only exhibit chaos under certain conditions. The Lorenz attractor, for example, undergoes a bifurcation at $r=1$; for $r<1$ the origin is a stable attractor. There's no way you can tell the bifurcation is there, or in what region the system is chaotic, just by looking at the form of the equations, because the same equations can describe both a non-chaotic and chaotic solution.

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What physical condition does the $r<1$ condition correspond to? Is that the boundary between orbiting one body and orbiting both of them? – AlanSE Aug 22 '12 at 16:56
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These days, the Lorenz system is often studied 'in the abstract' - that is, without referring the parameters $r$, $b$ and $\sigma$ to any physical system. However Lorenz developed the equations as a toy model of fluid convection in a box. In this interpretation, $r$ is the Rayleigh number of the fluid (as a ratio of the critical Rayleigh number: $r = \text{Ra}/\text{Ra}_{crit}$). So $r < 1$ corresponds to $\text{Ra} \lt \text{Ra}_{crit}$ - ie no convection. – poorsod Aug 22 '12 at 18:27
Note that in this interpretation of the Lorenz model, $x$, $y$ and $z$ do not refer to spatial co-ordinates. They are parameters related to dynamical properties of the convection, like temperature profiles and velocities. – poorsod Aug 22 '12 at 18:31
Even if I understand your point, Chaos is well defined and the first sentence can be misleading... – J. C. Leitão 8 hours ago

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