A convenient scheme for automatic computation is the successive evaluation of X_i(n+1), X_i((n+2)), and X_i,n+1 according to (9), (10) and (14). We have used this procedure in all the computations described in this study.
In phase space a numerical solution of (1) must be represented by a jumping particle rather than a continuously moving particle. Moreover, if a digital computer is instructed to represent each number in its memory by a preassigned fixed number of bits, only certain discrete points in phase space will ever be occupied. If the numerical solution is bounded, repetitions must eventually occur, so that, strictly speaking, every numerical solution is periodic. In practice this consideration may be disregarded, if the number of different possible states is far greater than the number of iterations ever likely to be performed. The necessity for repetition could be avoided altogether by the somewhat uneconomical procedure of letting the precision of computation increase as n increases.
Consider now numerica1 solutions of equations (4), obtained by the forward-difference procedure (11). For such solutions,

 Let S' be any surface of constant Q whose interior R' contains the ellipsoid E where dQ/ dt vanishes, and let S be any surface of constant Q whose interior R contains S'.
Since ∑F_i²  and dQ/ dt both possess upper bounds in R', we may choose ∆t so small that P_n+1 lies in R if P_n lies in R'. Likewise, since ∑F_i² possesses an upper bound and dQ/dt possesses a negative upper bound in R-R', we may choose ∆t so small that Q_n+1<Q_n if P_n lies in R- R'. Hence ∆t may be chosen so small that any jumping particle which has entered R remains trapped within R, and the numerical solution does not blow up. A blow-up may still occur, however, if initially the particle is exterior to R.
Consider now the double-approximation procedure (14). The previous arguments imply not only that P_(n+1) lies within R if P_n lies within R, but also that P_(n+2) lies within R if P_(n+1) lies within R. Since the region R is convex, it follows that P _n+1, as given by (14), lies within R if P_n lies within R. Hence if ∆t is chosen so small that the forward-difference procedure does not blow up, the double-approximation procedure also does not blow up.
We note in passing that if we apply the forward difference procedure to a conservative system where dQ/dt=0 everywhere,

In this case, for any fixed choice of t the numerical solution ultimately goes to infinity, unless it is asymptotically

approaching a steady state. A similar result holds when the double-approximation procedure (14) is applied to a conservative system. 

5. The convection equations of Saltzman

In this section we shall introduce a system of three ordinary differential equations whose solutions afford the simplest example of deterministic nonperiodic flow of which the writer is aware. The system is a simplification of one derived by Saltzman (1962) to study finite amplitude convection. Although our present interest is in the nonperiodic nature of its solutions, rather than in its contributions to the convection problem, we shall describe its physical background briefly.
Rayleigh (1916) studied the flow occurring in a layer of fluid of uniform depth H, when the temperature difference between the upper and lower surfaces is maintained at a constant value ∆T. Such a system possesses a steady-state solution in which there is no motion, and the temperature varies linearly with depth, if this solution is unstable, convection should develop.
In the case where all motions are parallel to the x–z-plane, and no variations in the direction of the y-axis occur, the governing equations may be written (see Saltzman, 1962)

Here ψ is a stream function for the two-dimensional motion, θ is the departure of temperature from that occurring in the state of no convection, and the constants g, α, v, and κ denote, respectively, the acceleration of gravity, the coefficient of thermal expansion, the kinematic viscosity, and the thermal conductivity. The problem is most tractable when both the upper and lower boundaries are taken to be free, in which case ψ and Ѳψ vanish at both boundaries.
Rayleigh found that fields of motion of the form

would develop if the quantity

now called the Rayleigh number, exceeded a critical value

The minimum value of R_c, namely 27p⁴/4, occurs when a²=1/2.

Saltzman (1962) derived a set of ordinary differential equations by expanding ψ; and θ in double Fourier series in x and z, with functions of t alone for coefficients, and