RUNGE--KUTTA integration schemes should be applied to nonlinear systems with knowledge of the caveats involved. The absolute-stability boundaries may be very different from the linear case, so a linear stability analysis may well be misleading. A problem may occur if a reduction in step length happens to take one outside the absolute-stability region due to the shape of the boundary. In this case, the usual step-control schemes would have disastrous results on the problem, as step-length reduction in an attempt to increase accuracy would have the opposite effect.
Even inside the absolute-stability boundary, all may not be well due to the existence of stable ghost fixed points in many problems. Since basin boundaries are finite, starting too far from the real solution may land one in the basin of attraction of a ghost fixed point. Contrary to expectation, this incorrect behaviour is not prevented by insisting that the method be convergent.
Stiffness needs a new and better definition for nonlinear systems. We have provided a verbal description, but a mathematical definition is still lacking. There is a lot of scope to investigate further the interaction between stiffness and chaos. Explicit Runge--Kutta schemes should not be used for stiff problems, due to their inefficiency: Backward Differentiation Formulae methods, or possibly implicit Runge--Kutta methods, should be used instead.
Dynamics is not only interested in problems with fixed point solutions, but also in periodic and chaotic behaviour. This is something that has not in the past been fully appreciated by some workers in numerical analysis who have tended to concentrate on obtaining results, such as those of nonlinear stability theory, that require properties like contractivity which are too restrictive for most dynamical systems.
There are results that tie the limit set of the Runge--Kutta map to that of the ordinary differential equation from which it came, but they are not as powerful as those which relate the dynamics of Poincaré maps to their differential equations. Structurally-stable behaviour in the ordinary differential equation is correctly portrayed in the Runge--Kutta map, but additional limit-set behaviour may be found in the map that is not present in the differential equation. Nonhyperbolic behaviour will probably not be correctly represented by the Runge--Kutta method.
Shadowing theory offers hope that it will be possible to produce numerical methods with built-in proofs of correctness of the orbits they produce, at least for hyperbolic orbits. However, it does not seem to be possible to do this with Runge--Kutta methods, and the Taylor series method, for which it is possible, has some severe disadvantages. More research needs to be done in this area.
Hamiltonian systems should be integrated with symplectic Runge--Kutta methods so that dissipative perturbations are not introduced. Even using symplectic integration, Hamiltonian systems still need to be handled with care. As in dissipative systems, nongeneric behaviour like integrability will not be reproduced in the numerical method. A more general problem is that approximate symplectic integrators cannot conserve energy. Round-off error is more of a problem for symplectic integration than in other cases, because it introduces dissipative perturbations to the system that one is trying to avoid.
A lot more work is needed on predicting the stability and accuracy of methods for integrating nonlinear and chaotic systems. At present, we must make do with Runge--Kutta and other methods, but be wary of the results they are giving us---caveat emptor!