# Chaos is predictable?

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

The authors know under which conditions the pendulum switch to a chaotic regime, probably because they used previous literature or because they experimented numerically with this system.

Looking at the ODE, we can predict the existence of chaotic behavior? If yes, it is possible to know, qualitatively, how parameters have to be tuned in order to find this chaotic behavioutr?

[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. – Benjamin Hodgson 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. – Benjamin Hodgson Aug 22 '12 at 16:10

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? – Alan Rominger Aug 22 '12 at 16:56
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. – Benjamin Hodgson 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. – Benjamin Hodgson 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 May 21 '13 at 12:32
– Benjamin Hodgson May 22 '13 at 10:34

The answer is a clear no. First off, non linear ODEs may have no solution for some initial conditions or on the contrary have several solutions. For unicity and existence it is needed that the derivative be continuous what may not be obvious at a simple glance for systems of several non linear ODEs.

Once the unicity and existence warranted, in chaotic systems there are always control parameters. A given non linear ODE system will have simple or complex but non chaotic solutions for a range of control parameters and it may have chaotic solutions for another range of control parameters.

The logistic equation even if it is not an ODE (X(n+1) = µ.Xn.(1-Xn)) will show chaotic behaviour only for some values of µ. There is no known general and simple rule allowing to know whether for a given non linear ODE system, there exists a set of values of the control parameters for which the solutions are chaotic.

However it is slightly easier from a physical point of view. Chaotic orbits in physics have 2 properties - exponential divergence (called sensibility to initial conditions) and dissipation which makes sure that the orbits don't explode to infinity. This is topologically seen like "stretching" and "folding" in the phase space. So if you take a forced system which will be dissipative AND its equations of motion will be non linear, then you will have a good chance to find chaotic regimes.

The case of Hamiltonian chaos (gravitational N body problem) is different as it doesn't involve stretching and folding but orbit instabilities on a torus.

Beyond these 2 cases where chaotic solutions may be looked for, there is of course the domain of PDEs which gives rise to spatio temporal chaotic solutions (as opposed to the mere temporal chaos for the case of ODEs).

Even if the question didn't concern the PDE, the answer is obviously no for them too even if chaotic behaviour is more frequent in spatio temporal than in the temporal domain.

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