A note on Liapunov's method

The purpose of this communication is to exhibit, in an important special case, a reformulation of Liapunov’ s method which makes the calculations much less onerous and from which one can draw interesting conclusions which do not obviously follow from the classical presentation. Let _ x ̧ ˆ f ̧...x1 ; . . . ;xn† be a system of n ® rst-order di€ erential equations, let V ...x1 ; . . . ;xn† be a well-behaved function and write W ˆ ...1=2† dV =dt. We call V a Liapunov function for the value c if W 4 0 at all points of V ˆ c, and a strict Liapunov function for the value c if W < 0 there. If V is a strict Liapunov function for c then the trajectory through a point of V ˆ c crosses from V > c to V < c there ; in other words, the region V > c is one which trajectories can only leave, and the region V < c is one which they can only enter. The same is often true if V is merely a Liapunov function for c, but in this case a more detailed analysis is needed. If the hypersurface V ˆ c is not connected, then similar results hold for each of its connected components. In practice one normally expresses this di€ erently. Let C be the set of values which V takes on the hypersurface W ˆ 0, let c be any real number not in C, and assume for simplicity that the hypersurface V ˆ c is connected. Then W has ® xed sign on V ˆ c, so that if W < 0 on V ˆ c then V is a strict Liapunov function for c, and if W > 0 on V ˆ c then ¡V is a strict Liapunov function for ¡c. If W ˆ 0 is bounded, then C can be found simply by ® nding the extremal points of V on W ˆ 0; but if W ˆ 0 is unbounded more complicated arguments are needed. The calculation of C becomes rapidly more onerous as the degrees of V and W increase ; and for this reason many workers have con® ned themselves to the case when V and W are inhomogeneous quadratic. It seems not to have been noticed that in this case there is a simpler way of deciding which V are Liapunov functions, and that this has implications which are not obvious from the classical Liapunov approach. Here I shall present this new approach in general terms. In a subsequent paper it will be applied to the Lorenz equations.