V—
Relation to the Theory of Differential Equations

1. As we remarked in section III, the non-linear transformations discussed in this paper exhibit certain analogies with systems of differential equations. In the following we confine ourselves to discussing the plane case.

An important study in the theory of differential equations, particularly as applied to non-linear mechanics, is that of so-called autonomous systems¹¹ '¹² '¹³dx dy d=P(x,y), y = Q(X). (1)

The theory, initiated by Poincare, seeks to determine the properties of the solutions of (1) under very general conditions, and to deduce such properties for particular cases without actually solving the equations explicitly (i.e., obtaining the general integral). In particular, the trajectories, given parametrically as a function of t: x = x(t), = y(t) (2)

are investigated from a topological point of view. Fundamental is the classification of the singular points of the system (1), that is, the points x,y, where P(x,y) = Q(x,y) = 0. The behavior of trajectories in the neighborhood of singular points can be found by consideration of the linear approximation to (1); the real object of the theory, however, is to characterize and, where possible, predict behavior in the large. One of the most interesting phenomena connected with behavior in the large is the existence of closed trajectories, or limit cycles. The theorem of Poincare and Bendixson¹⁴ gives sufficient conditions for the existence of such. Unfortunately, the fulfillment of these conditions in particular cases is often hard to verify; to date no satisfactory theoretical method for dealing with an arbitrary given system has been found.*

2. If we write our general two-dimensional system of non-linear difference equations in the form: S— - S^(n-)S(n-l) + F(S(nl) a(n-l)) ~~~~~~~At ~~~~(3) a( a(-) a(n ) + G(S(n-), a(n-)) At

* See reference 15. The practical applications are largely confined to stability theory. Also reference 13 and the literature there cited.

the analogy with (1) is evident. The fixed points of (3) correspond to the singular points of (1), and the behavior of solutions in the neighborhood of a fixed point can be investigated via the linear approximation; this procedure, in fact, yields Ostrowski's criterion (see section III). If the fixed point is attractive, the asymptotic solution in its neighborhood can of course be obtained. In the case of repellent fixed points (or if the initial point is outside the region of attraction of all attractive fixed points), the sequence of iterates sometimes converges to a limit set which appears to resemble a Poincar6 limit cycle, i.e., a closed curve. In other cases, finite limit sets (periods) are obtained; on the other hand, one may observe limit sets of quite ambiguous geometrical, not to say topological, structure. These last two alternatives have no analogues in the case of differential equations.

In fact, the analogy between (3) and (1) is more apparent than real. The significant distinction lies, perhaps, in the fact that for our difference equations there is nothing corresponding to the trajectories of (1); successive iterates do not in general lie close to each other. This fact makes it difficult to use topological arguments to determine the character of the limit set. For sufficiently small At the sequence of iterates may resemble a trajectory to some extent, but the limit as At 0 is almost certain to be a single point.*