- Email: [email protected]
Comments on “an improved stability criterion of third-order non-linear differential. equations”
C’OMMENTS
ON “AN IMPROVED STABILITY NON-LINEAR DIFFERENTII
C’RITERION OF THIRD-ORDER EQLIATIONS”
That this work [I J is completely erroneous can bc seen from example. The authors claim that the origin of the equation
the following
is asymptotically stable. It is obvious, however, that the characteristic equation approximation of this equation has a positive real root and 50 the original system is unstable.
simple
of the first non-linear
RFFERENC‘F. I
R. P. AGAKW,\I. and D. P. Cir~~r.4 197X Journul of Sourd trml I ‘ihrrrriou 58. I 3. An Improved stability criterion of third-order non-linear differen& equations.
AUTHOR’S
REPLl
The author first wishes to express his sincere thanks to Professor Storey for his comments. The reply to these comments, which follows, includes (i) some notes on the stability of non-linear systems. aimed at removing confusion in certain matters among a number of research workers working in this field, and (ii) modifications of some stability regions in the original paper [l]. In this paper [l] and elsewhere the author has shown, by considering some examples, that it is not necessary that instability in the linear approximation implies that the nonlinear system is unstable, nor that stability in the linear approximation of a non-linear system around its singular point implies that the non-linear system is stable [2. 31. Stability and instability depend on the region under consideration. A system may be stable in a certain region and unstable outside of that region. Thus one cannot strictly establish the stability or instability of a system simply by mathematical calculations unless one gives the region. Moreover it is also possible that a system may be stable in practice but nevertheless appear to be unstable according to mathematical calculations. Thus the desired state of a system may be mathematically unstable and yet in practice the system may oscillate near this state in such a manner that its performance is acceptable. Many aircraft and missiles behave in this manner [4]. These points can be illustrated by some examples. The author [S] has shown that the system j - j3 + t’ = 0. 3 I0 0037~~~60X.7Y/100310+03
S&2.00/0
c
(1) lY7Y Academic
Press Inc. (London)
Limted
ON “AN IMPROVED STABILITY NON-LINEAR DIFFERENTII
C’RITERION OF THIRD-ORDER EQLIATIONS”
That this work [I J is completely erroneous can bc seen from example. The authors claim that the origin of the equation
the following
is asymptotically stable. It is obvious, however, that the characteristic equation approximation of this equation has a positive real root and 50 the original system is unstable.
simple
of the first non-linear
RFFERENC‘F. I
R. P. AGAKW,\I. and D. P. Cir~~r.4 197X Journul of Sourd trml I ‘ihrrrriou 58. I 3. An Improved stability criterion of third-order non-linear differen& equations.
AUTHOR’S
REPLl
The author first wishes to express his sincere thanks to Professor Storey for his comments. The reply to these comments, which follows, includes (i) some notes on the stability of non-linear systems. aimed at removing confusion in certain matters among a number of research workers working in this field, and (ii) modifications of some stability regions in the original paper [l]. In this paper [l] and elsewhere the author has shown, by considering some examples, that it is not necessary that instability in the linear approximation implies that the nonlinear system is unstable, nor that stability in the linear approximation of a non-linear system around its singular point implies that the non-linear system is stable [2. 31. Stability and instability depend on the region under consideration. A system may be stable in a certain region and unstable outside of that region. Thus one cannot strictly establish the stability or instability of a system simply by mathematical calculations unless one gives the region. Moreover it is also possible that a system may be stable in practice but nevertheless appear to be unstable according to mathematical calculations. Thus the desired state of a system may be mathematically unstable and yet in practice the system may oscillate near this state in such a manner that its performance is acceptable. Many aircraft and missiles behave in this manner [4]. These points can be illustrated by some examples. The author [S] has shown that the system j - j3 + t’ = 0. 3 I0 0037~~~60X.7Y/100310+03
S&2.00/0
c
(1) lY7Y Academic
Press Inc. (London)
Limted