MARCH 1963 EDWARD N. LORENZ 133
< ε. Periodic trajectories are special cases of quasi-periodic trajectories.
   A trajectory which is not quasi-periodic will be called nonperiodic. If P(t) is nonperiodic, P(t1+τ) may be arbitrarily close to P(t1) for some time t1 and some arbitrarily large time interval τ, but, if this is so, P(t+τ) cannot remain arbitrarily close to P(t) as t → ∞. Non­periodic trajectories are of course representations of deterministic nonperiodic flow, and form the principal subject of this paper.
   Periodic trajectories are obviously central. Quasi-periodic central trajectories include multiple periodic trajectories with incommensurable periods, while quasi-periodic noncentral trajectories include those which approach periodic trajectories asymptotically. Non­periodic trajectories may be central or noncentral.
We can now establish the theorem that a trajectory with a stable limiting trajectory is quasi-periodic. For if P0(t) is a limiting trajectory of P(t), two distinct points P(t1) and P(t1+τ), with τ arbitrarily large, may be found arbitrary dose to any point P0(t0). Since P0(t) is stable, P(t) and P(t+τ) must remain arbitrarily close to P0(t+t0-t1), and hence to each other, as t → ∞, and P(t) is quasi-periodic.
   It follows immediately that a stable central trajectory is quasi-periodic, or, equivalently, that a nonperiodic central trajectory is unstable.
   The result has far-reaching consequences when the system being considered is an observable nonperiodic system whose future state we may desire to predict. It implies that two states differing by imperceptible amounts mar eventually evolve into two considerably different states. If, then, there is any error whatever in observing the present state-and in any real system such errors seem inevitable-an acceptable prediction of an instantaneous state in the distant future may well be impossible.
   As for noncentral trajectories, it follows that a uniformly stable noncentral trajectory is quasi-periodic, or, equivalently, a nonperiodic noncentral trajectory is not uniformly stable. The possibility of a nonperiodic noncentral trajectory which is stable but not uniformly stable still exists. To the writer, at least, such trajectories, although possible on paper, do not seem charac­teristic of real hydrodynamical phenomena. Any claim that atmospheric flow, for example, is represented by a trajectory of this sort would lead to the improbable conclusion that we ought to master long-range forecasting as soon as possible, because, the longer we wait, the more difficult our task will become.
   In summary, we have shown that, subject to the conditions of uniqueness, continuity, and boundedness prescribed at the beginning of this section, a central trajectory, which in a certain sense is free of transient properties, is unstable if it is nonperiodic. A noncentral trajectory, which is characterized by transient properties, is not uniformly stable it it is nonperiodic, and,
if it is stable at all, its very stability is one of its transient properties, which tends to die out as time pro­gresses. In view of the impossibility of measuring initial conditions precisely, and thereby distinguishing between a central trajectory and a nearby noncentral trajectory, all nonperiodic trajectories are effectively unstable from the point of view of practical prediction.

4. Numerical integration of nonconservative system

   The theorems of the last section can be of importance only if nonperiodic solutions of equations of the type considered actually exist. Since statistically stationary nonperiodic functions of time are not easily described analytically, particular nonperiodic solutions can prob­ably be found most readily by numerical procedures. In this section we shall examine a numerical-integration procedure which is especially applicable to systems of equations of the form (4). In a later section we shall use this procedure to determine a nonperiodic solution of a simple set of equations.
   To solve (1) numerically we may choose an initial time t0 and a time increment Δt, and let

We then introduce the auxiliary approximations



where Pn and P(n+1) are the points whose coordinates are (X1,n ,..., XM,n) and (X1(n+1),...,XM(n+1)).
The simplest numerical procedure for obtaining approximate solutions of (1) is the forward-difference procedure,

In many instances better approximations to the solutions of (1) may be obtained by a centered-difference procedure

   This procedure is unsuitable, however, when the deter­ministic nature of (1) is a matter of concern, since the values of X1,n ,…, XM,n do not uniquely determine the values of X1,n+1, ..., XM.n+1.
A procedure which largely overcomes the disadvant­ages of both the forward-difference and centered-difference procedures is the double-approximation procedure, defined by the relation

Here the coefficient of Δt, is an approximation to the time derivative of Xi at time t0+(n+1/2)Δt. From (9) and (10), it follows that (13) may be rewritten

 
     
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