Estimates of planar regions of asymptotic stability

Weight functions for a class of Liapunov functions in the plane

“…These papers show that in many cases there are many admissible values of 9. In [1] it is shown that, in the special case of the van der Pol equation, one can choose 6 so as substantially to improve the estimate of the region of asymptotic stability of an isolated critical point over a well-known estimate.…”

“…In this paper we generalize the work done in [1]. In particular, we consider weight functions of the form 6(x, y) = j" (j)(s) ds + X2 f Ks> 0) (1-2)…”

“…As was pointed out in [1], an unsolved problem is the general problem of finding 6 so as to maximize the estimate of the region of asymptotic stability among admissible values of 9. This problem remains unsolved.…”

“…Further, the global behavior of (3.2) and the estimates by (2.3) depend heavily upon the behavior of g and F. We consider the case where (3.2) has an asymptotically stable equilibrium point and a single neighboring unstable equilibrium point. A separate analysis may be given under different conditions; for example, the case where the origin is a unique equilibrium point surrounded by a unique limit cycle (see [1]). To consider the regions of asymptotic stability we state the following theorem which has been proved elsewhere [4].…”

Weighted Liapunov functions for a class of third-order autonomous differential equations

“…Further, this latter paper gave a class of weighted Liapunov functions for certain second-order systems. Further work utilizing weighted Liapunov functions for second-order equations was published by the current author in [1] and by S. Duchich and the current author in [2], A separate paper by A. Skidmore [8] extended Leighton's 1963 work on third-order equations to fourth-order equations, but no "weighting" was considered in either paper.…”