- Email: [email protected]
On a proposed screening test for limit cycles
Letters
to the Editors
4 Uppal and Luus state “It was shown by Luus and Lapldus [2] that tlus system exhttnts two hmtt cycles, an unstable hmlt cycle mstde a larger stable lmut ” The above was shown by numerlcal analysis methods There 1s an element of uncertamty m the result merent approaches may lead to dtierent results in numerical methods as explamed m the ongmal acle of mme The use of such results Introduces the element of uncertmnty The Proof of Daoud’s screemng test IS simple and depends Oust as Bendlxson theorem and Dulac’s Cnterlon do) on Green’s Theorem
1543
29 October
1976, accepted
-dy =Yk dt
Y)
This cnterion IS based on a theorem to the effect that “If two closed trajectories, one completely enclosmg the other, are to exist m a simple-connected domam m the phase-plane of the system (la,b), then &X/ax + aY/ay (called SUM) must change sign or be zero everywhere wltlnn the area enclosed between the two closed trajectones provided no smgular pomt exists In that region ” In proposmg this extension of BendIxson’s theorem, Daoud pomts out that It provides a necessary but not sufficient condition for the existence of more than one hnut cycle, m the same way as Bendlxson’s cntenon for a smgle hmlt cycle Unfortunately he appears to be unaware that this cntenon, and, mdeed, a more powerful version of It, was previously enunciated by Dulac m 1937[2] and has since been reported m at least one textbook[3] Like Bendlxson’s cntenon, the proposed test IS essentially a negative one, that IS, It allows a declslon as to whether several limit cycles may exist, not whether they do exist Whether ths critenon 1s sufficiently powerful to be useful as a screenmg test m any particular system IS another matter Certamly, a re-exammanon of tbe example consldered by Daoud would not encourage ttis view His example 1s drawn from the study of Ans and Amundson [4] and concerns the behaviour of a first-order ureverslble exothermlc reactlon m a contmuous stured-tank reactor It IS described by eqns (2) and (3)
DAOUD
REFJWENCES
30 November
Dear Sus, In a recent article, Daoud[l] proposes a negative cntenon for the existence of two or more hmlt cycles m the autonomous system
T
Wales
[ll Uppal A and Luus R, Chem Engng Scl 1977 32 1541 [2] Luus R and Lapldus L , A ICh E J 1972 18 1%0
On a proposed screening test for hit (Recerved
ALBERT
School of Mathematdcs Umverslty of New South Kensmgton NSW Australia
1976, received
cycles for publtcatcon
2 May
1977)
however do not affect the remamder of his treatment Equations (2) and (4) above are the corrected forms ) Consldenng a lme of constant 1 m the 9. 5 field, he finds that SUM along such a hne must be of the form either SUM=-L+A45
or
SUM=+P+Me
(L, M, P all positive
constants)
dependmg on the particular values of 7 and k chosen Since for a Bven 9. “SUM can change srgn at most only once as 6 mcreases from zero,” he concludes that “at most only one hmlt cycle may exist in the remon of Interest ” This IS a surpnsmg conclusion In the first place, there 1s no apparent connection between the number of changes of sign of SUM along a path parallel to one of the co-ordmate axes and the conchtrons of the theorem In the second place, the results of such an approach are bkely to be different depending on which co-ordmate duectlon 1s chosen
,-
$=-2(~-175)~k(q-2)($-175) +eexp(25-50/q) dS G=
(2)
l-[-[exp(25-SO/q)
(3) I-
for which SUM = - 3 - 2vk + 3 75k + (SOy$-
1) exp (25 -50/q)
(Note that eqns 4 and 9 in Daoud [ 11 both contam
misprints
(4) which
It3
IS
20
Temperature
Fig
1 Contours
of SUM(v,
21
22
7
t), for k = 15 0
Letters
1544
-
to the Editors
-7
-
K)
-
Of
I18
19
requuement that the Integral of SUM over the region enclosed between two successive lmut cycles should be zero It IS poten tmlly satisfied by any dlstibution of SUM whrch contams both posttlve and negative sub-reLpons wlthm that regon, the number or &stnbutlon of such sub-regons being unmaterlal Thus all the tralectories shown m l%g 2, supposmg them to exist, would be allowed under the condlfions of the theorem, Smce all adjacent paus of tralectones enclose sub-regions of both posltlve and negative values of SUM Far from the test having “defirutely Indicated the absence of multiple hmlt cycles m the phase plane” for the system under conslderatlon, it mcbcates the possibility of any number of limit cycles, from zero upwards The usefulness of this as a screerung procedure for the example consldered seems open to questIon School of Chenucal Engmeenng Umverslty of New South Wales Kensmgton, N S W, Australia
\
t-
NOTATION
k L. M. P
.
a proportlonahty constant DoSltlve constants
ix
-: 20
Temperature
21
P SOUTER
22
r)
Fig 2 Contours of SUM(q, t), for k = 9 01, showing possible location of closed trajectories allowed by Dulac’s cnterton As is clear from Fig 1, winch shows the contours of SUM m the 7, g plane for k = 15 0, had he chosen to evaluate SUM at constant values of 5, for varymg 7, he would have observed two changes of sign rather than one and could well have concluded that two hnut cycles may exist m the region of mterest, for smtable values of k Actually, both these conclusions would be mappropnate Dulac’s cntenon for an annular region 1s eqmvalent to the
aY
SUM ax+ar t time, dlmenslonless x, Y X, Y
vanables fun&Ions
time
of x, y
Greek symbols r) dImensionless q dlmenslonless
temperature concentration REFERENCES
111Daoud A T , Chem Engng SC! 1976 31 510 PI Dulac H , CR Acad SCI (Pans) 1937 204 1703
A A, Vltt A A and Kharkrn S E, Theory of Oscdlators Pergamon Press, Oxford 1966 [41 AIW R and Amundson N R , Chem Engng Scr 1958 7 121, 132
[31 Andronov
The vahdity of a screening test for hmit cycles (Recelued for publuxtlon 2 May 1977) Dear sus, The three Issues that are rarsed m Souters letter are (1) Dulac’s cnterlon and my screenmg test (2) Mlspnnts in eqns (4) and (9) m my article (3) The vahdlty of my conclusion I now answer each of the above separately 1 DULAC’S CRITERION AND MY SCREXNING TEST
I have looked up the two references mentloned by Mr Souter regardmg Dulac’s crltenon Theorem of sum (which 1s my screening test) considers the region enclosed by any two closed tralectones provided no singular pcnnts exist m that region Dulac cnterlon considers a different function m an annular region m the phase plane It 1s true that the proofs of both cnteria (Dulac’s and screemng test) and the Bendixson theorem depend on Green’s theorem 2MlSPRlNTS
Souter 1s right regardmg the mlsprmts m eqns (4) and (9) m my article However they do not interfere with the subsequent treatment at all
There are two mlsprmts Mmpnnt No 1 Equation (4) appears dq t = 9 = -2(v The coefficient
which
are
as
- 1 75) - k(v - 2)(q - 175)+ of eso(“z-“n)must
esW”*-““’
(4)
be 5 The correct form is
dq - = q = -2(r, - 1 75) - k(q - 2)(q - 1 75) + 6 e5ff”2-*‘4) dr (4) Equation (8) LSobtamed by dtierentlatmg (partially with respect to q) of eqn (4) The result m (8) 1s result of chfferentlatmg the correct form of eqn (4) This shows that this mlsprmt 1s due to typmg Mtspnnt No 2 Equation (9) appears
as
to the Editors
4 Uppal and Luus state “It was shown by Luus and Lapldus [2] that tlus system exhttnts two hmtt cycles, an unstable hmlt cycle mstde a larger stable lmut ” The above was shown by numerlcal analysis methods There 1s an element of uncertamty m the result merent approaches may lead to dtierent results in numerical methods as explamed m the ongmal acle of mme The use of such results Introduces the element of uncertmnty The Proof of Daoud’s screemng test IS simple and depends Oust as Bendlxson theorem and Dulac’s Cnterlon do) on Green’s Theorem
1543
29 October
1976, accepted
-dy =Yk dt
Y)
This cnterion IS based on a theorem to the effect that “If two closed trajectories, one completely enclosmg the other, are to exist m a simple-connected domam m the phase-plane of the system (la,b), then &X/ax + aY/ay (called SUM) must change sign or be zero everywhere wltlnn the area enclosed between the two closed trajectones provided no smgular pomt exists In that region ” In proposmg this extension of BendIxson’s theorem, Daoud pomts out that It provides a necessary but not sufficient condition for the existence of more than one hnut cycle, m the same way as Bendlxson’s cntenon for a smgle hmlt cycle Unfortunately he appears to be unaware that this cntenon, and, mdeed, a more powerful version of It, was previously enunciated by Dulac m 1937[2] and has since been reported m at least one textbook[3] Like Bendlxson’s cntenon, the proposed test IS essentially a negative one, that IS, It allows a declslon as to whether several limit cycles may exist, not whether they do exist Whether ths critenon 1s sufficiently powerful to be useful as a screenmg test m any particular system IS another matter Certamly, a re-exammanon of tbe example consldered by Daoud would not encourage ttis view His example 1s drawn from the study of Ans and Amundson [4] and concerns the behaviour of a first-order ureverslble exothermlc reactlon m a contmuous stured-tank reactor It IS described by eqns (2) and (3)
DAOUD
REFJWENCES
30 November
Dear Sus, In a recent article, Daoud[l] proposes a negative cntenon for the existence of two or more hmlt cycles m the autonomous system
T
Wales
[ll Uppal A and Luus R, Chem Engng Scl 1977 32 1541 [2] Luus R and Lapldus L , A ICh E J 1972 18 1%0
On a proposed screening test for hit (Recerved
ALBERT
School of Mathematdcs Umverslty of New South Kensmgton NSW Australia
1976, received
cycles for publtcatcon
2 May
1977)
however do not affect the remamder of his treatment Equations (2) and (4) above are the corrected forms ) Consldenng a lme of constant 1 m the 9. 5 field, he finds that SUM along such a hne must be of the form either SUM=-L+A45
or
SUM=+P+Me
(L, M, P all positive
constants)
dependmg on the particular values of 7 and k chosen Since for a Bven 9. “SUM can change srgn at most only once as 6 mcreases from zero,” he concludes that “at most only one hmlt cycle may exist in the remon of Interest ” This IS a surpnsmg conclusion In the first place, there 1s no apparent connection between the number of changes of sign of SUM along a path parallel to one of the co-ordmate axes and the conchtrons of the theorem In the second place, the results of such an approach are bkely to be different depending on which co-ordmate duectlon 1s chosen
,-
$=-2(~-175)~k(q-2)($-175) +eexp(25-50/q) dS G=
(2)
l-[-[exp(25-SO/q)
(3) I-
for which SUM = - 3 - 2vk + 3 75k + (SOy$-
1) exp (25 -50/q)
(Note that eqns 4 and 9 in Daoud [ 11 both contam
misprints
(4) which
It3
IS
20
Temperature
Fig
1 Contours
of SUM(v,
21
22
7
t), for k = 15 0
Letters
1544
-
to the Editors
-7
-
K)
-
Of
I18
19
requuement that the Integral of SUM over the region enclosed between two successive lmut cycles should be zero It IS poten tmlly satisfied by any dlstibution of SUM whrch contams both posttlve and negative sub-reLpons wlthm that regon, the number or &stnbutlon of such sub-regons being unmaterlal Thus all the tralectories shown m l%g 2, supposmg them to exist, would be allowed under the condlfions of the theorem, Smce all adjacent paus of tralectones enclose sub-regions of both posltlve and negative values of SUM Far from the test having “defirutely Indicated the absence of multiple hmlt cycles m the phase plane” for the system under conslderatlon, it mcbcates the possibility of any number of limit cycles, from zero upwards The usefulness of this as a screerung procedure for the example consldered seems open to questIon School of Chenucal Engmeenng Umverslty of New South Wales Kensmgton, N S W, Australia
\
t-
NOTATION
k L. M. P
.
a proportlonahty constant DoSltlve constants
ix
-: 20
Temperature
21
P SOUTER
22
r)
Fig 2 Contours of SUM(q, t), for k = 9 01, showing possible location of closed trajectories allowed by Dulac’s cnterton As is clear from Fig 1, winch shows the contours of SUM m the 7, g plane for k = 15 0, had he chosen to evaluate SUM at constant values of 5, for varymg 7, he would have observed two changes of sign rather than one and could well have concluded that two hnut cycles may exist m the region of mterest, for smtable values of k Actually, both these conclusions would be mappropnate Dulac’s cntenon for an annular region 1s eqmvalent to the
aY
SUM ax+ar t time, dlmenslonless x, Y X, Y
vanables fun&Ions
time
of x, y
Greek symbols r) dImensionless q dlmenslonless
temperature concentration REFERENCES
111Daoud A T , Chem Engng SC! 1976 31 510 PI Dulac H , CR Acad SCI (Pans) 1937 204 1703
A A, Vltt A A and Kharkrn S E, Theory of Oscdlators Pergamon Press, Oxford 1966 [41 AIW R and Amundson N R , Chem Engng Scr 1958 7 121, 132
[31 Andronov
The vahdity of a screening test for hmit cycles (Recelued for publuxtlon 2 May 1977) Dear sus, The three Issues that are rarsed m Souters letter are (1) Dulac’s cnterlon and my screenmg test (2) Mlspnnts in eqns (4) and (9) m my article (3) The vahdlty of my conclusion I now answer each of the above separately 1 DULAC’S CRITERION AND MY SCREXNING TEST
I have looked up the two references mentloned by Mr Souter regardmg Dulac’s crltenon Theorem of sum (which 1s my screening test) considers the region enclosed by any two closed tralectones provided no singular pcnnts exist m that region Dulac cnterlon considers a different function m an annular region m the phase plane It 1s true that the proofs of both cnteria (Dulac’s and screemng test) and the Bendixson theorem depend on Green’s theorem 2MlSPRlNTS
Souter 1s right regardmg the mlsprmts m eqns (4) and (9) m my article However they do not interfere with the subsequent treatment at all
There are two mlsprmts Mmpnnt No 1 Equation (4) appears dq t = 9 = -2(v The coefficient
which
are
as
- 1 75) - k(v - 2)(q - 175)+ of eso(“z-“n)must
esW”*-““’
(4)
be 5 The correct form is
dq - = q = -2(r, - 1 75) - k(q - 2)(q - 1 75) + 6 e5ff”2-*‘4) dr (4) Equation (8) LSobtamed by dtierentlatmg (partially with respect to q) of eqn (4) The result m (8) 1s result of chfferentlatmg the correct form of eqn (4) This shows that this mlsprmt 1s due to typmg Mtspnnt No 2 Equation (9) appears
as