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Stability Regions of Certain Linear Second Order Periodic Differential Equations
Trends in the Theory and Practice of Non-Linear Analysis V. Lakshmikantham (Editor) 0 Elsevier Science Publishers B.V.(NorthHolland), 1985
67
STABILITY REGIONS OF CERTAIN LINEAR SECOND OFtDER PERIODIC DIFFERENTIAL EQUATICNS
Bernfeld and M. Pandian
S.R.
Department of L%thematics University of Texas a t Arlington Arlhg-ton, Texas 76019 USA
I m m I o N The difficulty in fb%llg the Flcquetmultipliers of H i l l ' s equation x
(1)
+ p(t)ic + q ( t ) x =
0
is w e l l known (see [l] or [21), where p ( t ) and q ( t ) are 2n-periodic. Consequently the s t a b i l i t y properties of (1)are very d i f f i c u l t to ascertain i n genzral unless p ( t ) and q ( t ) are constant. Recall that (1) is stable i f ard only i f a l l solutions are bowrded and is unstable i f there e x i s t s an sol tion. I f the two m l t i p l i e r s pl, p2 of (1) are within the u n i t c i r q + 0 2< 1, then (pl( > 1 o r I p 2 ( > 1 then (1) is unstable. If both m l t i p l i e r s have sinple elementary divisors then (1)is
(1) is stable whereas if either p12
+ p22
= 1
stable: i f one of the multipliers does not have a simple elewntary divisor than (1) is unstable.
equations have heen analyzed i n various contexts. Fbr example i f OIE wants to study the s t a b i l i t y properties of periodic solutions of autonmxls second order mnlinear equations then a useful technique is to analyze the s t a b i l i t y properties of the corresponding variational equation w h i c h is of the form (1). k@reuverin problems of mathemtical physics, such as i n the theory of elastic vibrations, p a r t i a l a r cases of (1) such as the Mathieu equation ( p ( t ) z 0, q ( t ) = a + b cos t) a d other forms of H i l l ' s equation are used [l]. Second order periodic
In scIty3 problerns in ecology and mnlkar &mica1 kinetics, for example, one i s often led to (1) when certain parameters are m t identically constant but rather fluctuate periodically around a c o n s d t value.
As irdicated above p ( t ) a d q ( t ) may deped on parameters. Consequently Iuxmledge of the d e ~ + & ~ =of the multipliers on these p a r m t e r s yields information on the dependence of the s t a b i l i t y regions i n the p a r a t e r space. Such regions have been depicted for the Mathieu q u a t i o n [2] ard other forms of H i l l ' s equation [2]. bre0ve.r i n problms i n bifurcation theory it is also important to have infonmtion on changes of the s t a b i l i t y behavior of (1)as paramters vary, again leading to an analysis of the depdeme of the m l t i p l i e r s on parmters. I n this preliminary rpte we begin with a study of the aepenaenCe of m l t i p l i e r s on parameters for a particular case of (1).
S.R. Bernfeld and M. Pandian
68
(2)
x + (A + B sin(t+a))jc
+ (c + D cos(t+B))x
= 0,
where A, B, C, D, a,B, are real constants. (the case A = B = 0, 8 = 0 is the Mathieu equation) We will look at the critical case A = C = Q B = D = 1 a d later study a neighborhood in parameter space of the critical case. Thus consider
x
(3)
+
sin(t+a)l
+ cos(t+B)x
= 0;
we will attempt to determine the behavior of the +SKI multipliers p l ( a , B ) , p2(a,B) on a ard 6. In fact we may restrict a,B to S = [0,2n] x [0,2n] by observing (x,v) = (y,w) if ard only if x z y n-cd 2n and v z w n-cd 2n, is an equivalence relationship. Flcquet theory tells us p1p2 = 1 ard pi, i = 1, 2 satisfies the characteristic equation 2 x - ( p +p )x + 1 = 0. (4 1 1 2 PT+P2
-
Define A(a,B) = 7System (3) is stable if \ A \< 1, that is, if p2 = 01 and p1 $1; and (3) is unstable if \ A /> 1, that is, if o < p1 < 1 < p2. fie buridary seFating the stability regions in the (a,B) plane is thus given by p1 = ?I. On the bourdary p1 = 1 there are periodic solutions of period 2n,arrrl if p1 = 1 has sinple divisors for a particular ( a , @ ) then all solutions of (3) are of period 2n for this particular (a,B). Similar statments hold when p1 = -1 except m the solutions are of period 4 ~ .
z
Lbmrical studies (with the assistance of B. Asner) lead to the follcwimg observations in the stability amlysis of ( 3 ) : Observations:
(a) The region of instability, I ( ~ , B can) be written as
izlwi, rn
1 =
where each Wi is the union of a closed unbounded set Ui with a mnernpty interior and a one dimensional disjoint curve C.. bbreover W. n W . = 0 when i#j. On Ci 1 1 1 p1 = 1,whereas on the baundary of Ui,pl = -1. (b) The region of stability, S, can be written as co
s
=
g1oi,
where each Oi is apen, and each Oi can be written as Oi = Oil J Oi2 such that - Oil, Oi2 are each open ard Oil n Oi2 = Ci. which is periodic in ( a , 8 ) of (c) "he r q e of the m i n g ( p l ( a , B ) , p 2 ( a , f 3 ) ) , period 2 n , is the set
where a is approxirrately -021 ard b i s a rminately 46.54. (d) For each constant c E { [-b,-11 u (zTz/=l) 3 there exists a function Bc(a) such that pl(a,BC(a)) z c for all a . %ere exists in addition, a family $,(a) orthogonal to the family Bc(a) such that the range of (p1(a,Qc(a)),
69
Linear Second Order Periodic Differential Equations p2(a,$c(a)) is the set T f o r each c. Remarks:
.
For c=l we can shm that B1(a) = a in (d)
Indeed i n this case we can
inteyrate (3) obtaining
d &x
cos(t+a)) = x
leading t o s=xcos(t+a) + K where K is a constant. p1 = p 2 = 1.
The solutions are 2~ periodic i f and only i f K = 0.
Hence
I n subsequent work with B. Asner details of this note w i l l be qiven as ell as other developwnts i n the study of two dimensional systems depxling on t w o r rmre paramters. Ackrmledgement: The authors m u l d l i k e to thank Professor B. Asner of the University of Dallas f o r both his important rnrmerical contributions as w e l l as for several interesting conversations. BIBLIrnHY
[l] Stoker, J.J. [2]
, Nonlinear V i b r a t i o n s ,
Interscience (New York, 1957).
Yakubovich, V.A. and V.M. Starzkinskii, Linear Differential Fqations w i t h Pericdic Coefficients (Et-qlish translation--two volumes) (Wiley, New Yak, 1975).
The final (detailed) version of this paper will be submitted for publication elsewhere.
67
STABILITY REGIONS OF CERTAIN LINEAR SECOND OFtDER PERIODIC DIFFERENTIAL EQUATICNS
Bernfeld and M. Pandian
S.R.
Department of L%thematics University of Texas a t Arlington Arlhg-ton, Texas 76019 USA
I m m I o N The difficulty in fb%llg the Flcquetmultipliers of H i l l ' s equation x
(1)
+ p(t)ic + q ( t ) x =
0
is w e l l known (see [l] or [21), where p ( t ) and q ( t ) are 2n-periodic. Consequently the s t a b i l i t y properties of (1)are very d i f f i c u l t to ascertain i n genzral unless p ( t ) and q ( t ) are constant. Recall that (1) is stable i f ard only i f a l l solutions are bowrded and is unstable i f there e x i s t s an sol tion. I f the two m l t i p l i e r s pl, p2 of (1) are within the u n i t c i r q + 0 2< 1, then (pl( > 1 o r I p 2 ( > 1 then (1) is unstable. If both m l t i p l i e r s have sinple elementary divisors then (1)is
(1) is stable whereas if either p12
+ p22
= 1
stable: i f one of the multipliers does not have a simple elewntary divisor than (1) is unstable.
equations have heen analyzed i n various contexts. Fbr example i f OIE wants to study the s t a b i l i t y properties of periodic solutions of autonmxls second order mnlinear equations then a useful technique is to analyze the s t a b i l i t y properties of the corresponding variational equation w h i c h is of the form (1). k@reuverin problems of mathemtical physics, such as i n the theory of elastic vibrations, p a r t i a l a r cases of (1) such as the Mathieu equation ( p ( t ) z 0, q ( t ) = a + b cos t) a d other forms of H i l l ' s equation are used [l]. Second order periodic
In scIty3 problerns in ecology and mnlkar &mica1 kinetics, for example, one i s often led to (1) when certain parameters are m t identically constant but rather fluctuate periodically around a c o n s d t value.
As irdicated above p ( t ) a d q ( t ) may deped on parameters. Consequently Iuxmledge of the d e ~ + & ~ =of the multipliers on these p a r m t e r s yields information on the dependence of the s t a b i l i t y regions i n the p a r a t e r space. Such regions have been depicted for the Mathieu q u a t i o n [2] ard other forms of H i l l ' s equation [2]. bre0ve.r i n problms i n bifurcation theory it is also important to have infonmtion on changes of the s t a b i l i t y behavior of (1)as paramters vary, again leading to an analysis of the depdeme of the m l t i p l i e r s on parmters. I n this preliminary rpte we begin with a study of the aepenaenCe of m l t i p l i e r s on parameters for a particular case of (1).
S.R. Bernfeld and M. Pandian
68
(2)
x + (A + B sin(t+a))jc
+ (c + D cos(t+B))x
= 0,
where A, B, C, D, a,B, are real constants. (the case A = B = 0, 8 = 0 is the Mathieu equation) We will look at the critical case A = C = Q B = D = 1 a d later study a neighborhood in parameter space of the critical case. Thus consider
x
(3)
+
sin(t+a)l
+ cos(t+B)x
= 0;
we will attempt to determine the behavior of the +SKI multipliers p l ( a , B ) , p2(a,B) on a ard 6. In fact we may restrict a,B to S = [0,2n] x [0,2n] by observing (x,v) = (y,w) if ard only if x z y n-cd 2n and v z w n-cd 2n, is an equivalence relationship. Flcquet theory tells us p1p2 = 1 ard pi, i = 1, 2 satisfies the characteristic equation 2 x - ( p +p )x + 1 = 0. (4 1 1 2 PT+P2
-
Define A(a,B) = 7System (3) is stable if \ A \< 1, that is, if p2 = 01 and p1 $1; and (3) is unstable if \ A /> 1, that is, if o < p1 < 1 < p2. fie buridary seFating the stability regions in the (a,B) plane is thus given by p1 = ?I. On the bourdary p1 = 1 there are periodic solutions of period 2n,arrrl if p1 = 1 has sinple divisors for a particular ( a , @ ) then all solutions of (3) are of period 2n for this particular (a,B). Similar statments hold when p1 = -1 except m the solutions are of period 4 ~ .
z
Lbmrical studies (with the assistance of B. Asner) lead to the follcwimg observations in the stability amlysis of ( 3 ) : Observations:
(a) The region of instability, I ( ~ , B can) be written as
izlwi, rn
1 =
where each Wi is the union of a closed unbounded set Ui with a mnernpty interior and a one dimensional disjoint curve C.. bbreover W. n W . = 0 when i#j. On Ci 1 1 1 p1 = 1,whereas on the baundary of Ui,pl = -1. (b) The region of stability, S, can be written as co
s
=
g1oi,
where each Oi is apen, and each Oi can be written as Oi = Oil J Oi2 such that - Oil, Oi2 are each open ard Oil n Oi2 = Ci. which is periodic in ( a , 8 ) of (c) "he r q e of the m i n g ( p l ( a , B ) , p 2 ( a , f 3 ) ) , period 2 n , is the set
where a is approxirrately -021 ard b i s a rminately 46.54. (d) For each constant c E { [-b,-11 u (zTz/=l) 3 there exists a function Bc(a) such that pl(a,BC(a)) z c for all a . %ere exists in addition, a family $,(a) orthogonal to the family Bc(a) such that the range of (p1(a,Qc(a)),
69
Linear Second Order Periodic Differential Equations p2(a,$c(a)) is the set T f o r each c. Remarks:
.
For c=l we can shm that B1(a) = a in (d)
Indeed i n this case we can
inteyrate (3) obtaining
d &x
cos(t+a)) = x
leading t o s=xcos(t+a) + K where K is a constant. p1 = p 2 = 1.
The solutions are 2~ periodic i f and only i f K = 0.
Hence
I n subsequent work with B. Asner details of this note w i l l be qiven as ell as other developwnts i n the study of two dimensional systems depxling on t w o r rmre paramters. Ackrmledgement: The authors m u l d l i k e to thank Professor B. Asner of the University of Dallas f o r both his important rnrmerical contributions as w e l l as for several interesting conversations. BIBLIrnHY
[l] Stoker, J.J. [2]
, Nonlinear V i b r a t i o n s ,
Interscience (New York, 1957).
Yakubovich, V.A. and V.M. Starzkinskii, Linear Differential Fqations w i t h Pericdic Coefficients (Et-qlish translation--two volumes) (Wiley, New Yak, 1975).
The final (detailed) version of this paper will be submitted for publication elsewhere.