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Period bounds for generalized Rayleigh equation
hr. J. Non-Linear
Mechunics.
Vol. 6, pp. 271-277.
Pergamon
Press 1971.
Printed in Great Britain
PERIOD BOUNDS FOR GENERALIZED
RAYLEIGH
EQUATION
A. SMITH
RUSSELL
Department of Mathematics, University of Durham, Durham, England AhstrnetSimple upper and lower bounds are obtained for the least period T of any non-constant solution x(t) of the differential equation x” - F(x’) + g(x) = 0.
periodic
1. INTRODUCTION
THE paper concerns the differential equation x” - F(x’) + g(x) = 0,
(1)
where x(t) is the unknown, a prime denotes differentiation with respect to t, and F(y), g(x) are known functions with continuous first derivatives. All variables are real. Several physical occurrences of this equation are listed by Minorsky in the foreword of [l]. Sufficient conditions for the existence of a non-constant periodic solution were given by Reissig [2]. Throughout the present paper it is assumed that (1) has at least one nonconstant periodic solution x(t) with least period T. In the special case of weak non-linearities it is possible to study T by small parameter methods. But in the general case, very little is known about T and it is the aim of this paper to give upper and lower bounds for T which are simple and explicit. The lower bound is discussed in section 4. For the upper bound the following assumptions are made :
dg
for--cocx
vG-
dx
.fw z 09
for-b
(3) (4)
where p, v are positive constants, f(y) = dF/dy and f+(y) = max (0, f(y)). In section 2 the following is proved: THEOREM
1. Zf (2)-(4) hold and (1) has a periodic solution x(t) with least period T then +p-+vT
G
(2 + vmlp)(J1 + J2) + (1 + v-‘~1)’ (1 + max(JJ,)},
where J1 = (bJv)-’
r
f+(y)
dy,
s” f+(y)
dy.
-KZ
0
Since
JZ = (bJv)F’
(5)
f+(y) dy is equal to the positive variation of F(y) in the interval (0, co), J1 can be 0
271
272
RUSSELLA.SMI+H 0
computed in practice without integration.
Similarly,
1 f+(y)dy is the positive variation -.a) of F(y) in the interval (- cc, 0). An unexpected feature of Theorem 1 is the weakness of the assumptions made about F(y). In section 3 the inequality (5) is improved at the cost of further restrictions on F(y) and g(x). At the end of section 3, some comments are made about the order of accuracy to be expected from (5). When ,U= v, g(x) is linear and (1) can be differentiated to give y” - ylf(y) + vy = 0,
(6)
where y = x’. This Lienard equation is a special case of the equation studied by the author in [3]. Since the period bound obtained in [3] coincides with (5) for this special equation, theorem 1 can be regarded as an extension of this earlier work to a different class of equations. 2. PROOF OF THEOREM I It can be assumed without loss of generality that F(0) = 0,
g(0) = 0.
(7)
If this is not already so it can be achieved by absorbing the constant F(0) in g(x) and then shifting the origin of x to the unique zero of g(x) which exists because of (2). These changes do not affect (2)-(4). Then (2), (7) give vx2 d xg(x) < 1x2, for all x. Equation (1) is equivalent to the plane autonomous x’ = y,
(8) system
Y’ = F(Y) - g(x),
(9)
whose only singular point is at the origin of the (x, y) plane. The non-constant solution x(t) of (1) corresponds to a closed trajectory r in the (x, y) plane which encircles the origin and is described in the clockwise sense. LEMMA1. r crosses both lines y = f 6. Pro05 If G(x) = Ig(t) dt then (8) shows that G(x) is strictly decreasing in (- cc, 0) and
s strictly increasing in (0, cc). The equation y2 + 2G(x) = b2 defines a simple closed curve Q in the (x, y) plane which encircles the origin and touches the lines y = f 6. Along any trajectory of (9), $ {y’ + 2G(x)} = 2yy’ + 2g(x) x’ = 2yF(y).
(10)
Since f(y) = dF/dy is continuous, (3) shows that f(y) is of constant sign in (- 6, 6). Since F(0) = 0, it follows that yF(y) is of constant sign in the closed interval [ - 6, b], except for its zero at y = 0. In the case when yF(y) is positive in [ - 6, b], (10) shows that 52is crossed in the outward direction only, by every trajectory of (9) which meets it. In the case when yF(y) is negative in [ - 6, b], (10) shows that Q is crossed in the inward direction only, by
every trajectory of (9) which meets it. In both cases s2 cannot meet a closed trajectory of (9) because this would have to cross it both inwards and outwards. Furthermore no trajectory inside 51can be closed because (10) shows that y2 + 2G(x) is strictly monotonic along these trajectories. All closed trajectories of (9) therefore lie outside a. Since all closed trajectories
Period bounds for generalized
273
Rayleigh equation
encircle the origin, they must encircle i(l,and therefore meet both lines y = f b. This fmishes the proof of lemma 1. The curve with equation F(y) = g(x) passes through the origin by (7) and meets each line parallel to the x axis in exactly one point, because of (2). Since y = x’, r crosses the curve F(y) = g(x) once only in each of the half planes y > 0, y < 0. Also, (9) shows that x’, y’ have constant signs along each of the four arcs PQ, QR, RS, SP of r, (see figure).
FIG. 1.
Therefore y and g(x) are monotonic functions oft along each of these arcs. By considering the total variation of y and g(x) round r we obtain
Since v < dg/dx, these give j
(U>’
+ /W213dt
6
267~ + [YR\)
+ 2+@(gQ
+ ~~s;sB~
(11)
On r, (9) shows that (y’)’ + &z’)~ = W(x, Y)~,where ~(X,Y~
=
&Y2
4
El;(Y) -
SOW)’
(12)
Clearly W(x, y) > 0 for (x, y) # (0,O). If w is the minimum value of W(x, y) on the closed trajectory r then, replacing (Y’)~-t r_~(x’)~ by o2 in (1 l), we get oT G 2(Y, + (YRJ)+ 2v-‘Cl%o + Is& = z/KVV, + W,) + 2v-’ P(WQ -t Ws),
(13)
where WP, We, W,, W, are the values of W(x, y) at the points P, Q, R, S, respectively. From (9) and (12), W2 = py2 + (y’)’ along I’. Differentiate this with respect to t and use (9) to get
RUSSELLASMITH
274
Since & - dg/dx) y’y < 0 along the arcs PQ and RS, (14) gives w’ < W-‘(Jq2f
<
w-‘(_Yy f+ d fy'~f+(y)l
(15)
along PQ and RS. Integrate this along PQ and RS to get W, - w, C
(16)
IY’(f+(Y) dr = ‘f+(Y) dY G ~1, % d = ; f,cv)dY d 012,
w, - w, < ! [Y’lf.+Odt where I, = o- ’ 1 f+ dy, 1, = ,‘I
7 f+ dy. Al& -Co
(17)
QR and SP we have
0 ,< (P - dg,'dx) y’y d (P - v) y’y. This and (14) give w’ ,< w- ‘((y’)’
f+ + (j4 - V)Y’Y) s U+(Y) + (p - v)O>
IY’I,
(18)
along QR and SP. Integrate this along QR to get K - w, G [U+(Y) + (CL- v)P-+) IY’Idr, = R I f+cV)dY -+ (P - V)P-+ ly”l. Since WR = ,u* 1y, 1by (12), this gives j&-rvwa - u;, G ?f,(Y)dY R
(19)
G 01,.
Integrate (18) along Sp to get, similarly, #u-‘VW, - Ws < j f+dv)dy < ol,. s
(20)
Let M be the point on r at which W(x, y) takes its minimum value CU.We now estimate W, + W, + p- ‘v( WR + Wp) by discussing separately the four possible cases when M lies on each of the arcs PQ, QR, RS, SP. In the case N E PQ, integrate (15) along the arc MQ to get W, - w G ts f+(Y)lY’(dt d ;f+odY M Q
4 of,.
This and (17), (19), (20) give w, + ws + l*-‘v(W, + Wp) - 2(1 + v-‘j_A)w= (~-lvWp - W,) + 2(W, - W,) + (1 + 2v-‘~)(jPvw,
- WQ)+ 2(1 + V_‘&(WQ - 0) G (3 + 2v- r/J) o(l, + I,).
In the case M E QR, integrate (18) along the arc MR to get w, - 0 d s” (j-+ + fcl M
V)/A-+)
/y’j
dt ,( Uf, + (ji -
V)j.i-’
WR.
(21)
Period bow& for generalized Rayleigh equation
212
Hence, ~1-t VW, - w G wl,. This and (16), (17), (20) give Wo+ W,+jC’v(W,+ W-p)-(1 +v-‘~)%G(WoWp) + (1 + v-‘/A)@“- ‘VW, - W,) + (2 + v- ‘p)(W, - IV,) +(l +v-’ cc)2(P-r VW, - 0) < (2 + v-1 ~)@(I, + I,) + (1 + v-‘/@oZ2.
(22)
In the case M E RS, integrate (15) along the arc MS to get W, - o < ml,. This and (16), (19), (20) give w, + w, + p-1 v(W, + Wp) - 2(1 + v-1 /A)0 = (jf-’ VW, - WQ) + 2(Wo - Wp) + (1 + 2v-’ &(p-’
VW, - W,) + 2(1 + v-l p)(W, - 0) < (3 + zv-’ p)ofZ, + 1,).
(23)
In the case M E SP, integrate (18) along the arc MP to get
wp-co,< p{f+ if
+oL-
v)p-f}
\y’(dt < wZ, + 01 - V)P-’ I+‘,.
Hence, p-1 VW, - w < ml,. This and (16), (17), (19) give wo+
w,+/.l-’
v(W, + Wp) - (1 + v-l Zqo
+ (1 + v-l p)(p-‘v Inall fourcases,(21)
= (W, - W,)
w, - wo, + (2 + v-l p)(W, - Wp) + (1 + v-l /L)2(/P VW, - 0) < (2 + v- 1 p) o(Z, + 12) + (1 + v- 1 j.ly oz,. (24) - (24)give
Wo + K + 3%
+ Wp) <
2+ f (
o(Z, + I,) + >
(
1 + F 2 ~(1 + max (I,, I,)}. >
Substitute this in the right-hand side of (13) to get $/A-+ VT 6 (2+ v-lp)(I1 + Z2) + (1 + v - ’ j.i)’ { 1 + max (I,, I,)}. Theorem 1 follows from this by replacing I,, I, by the numbers .Zl, .Zz which satisfy J, = oZ,/b,/v, J, = oZ,/b Jv. The following lemma shows that .Z, > I,, J, >, I,: LJMMA 2. v*b < o. Proof: If the point M at which W attains its minimum o on Z lies outside the strip < y < b then (12) gives UJ >, p*lyl 2 vj b, which satisfies the lemma. It remains to consider separately the four possible cases when M lies on each of the arcs AQ, QB, CS, SD, (see figure). Since y’ # 0 on these four arcs and IV’ = 0 at M, (14) gives -b
signf = - sign (y’y),
at M
(25)
In the case M EAQ, (25) and (3) give f > 0 along AQ because y’y < 0 on AQ. Then (14) gives W W’ > (p - dg/dx) y’y 2 @ - v) y’y along AQ and W2 - W$=j2WW’draj2(p A
A
- v) y’y dt = @ - v) (y: - b2).
Since W: 2 ,ub2 by (12), this gives o2 2 vb2 which satisfies lemma 2. In the case ME QB, (25) and (3) give f < 0 along QB. Then (14) gives
276
RUSSELLA. SMITH
w&o*=
- Y&J.
B*WWldt~12(8-Y)ylydt=(II-Y)(b* 5
Since Wjj > pb* by (12), this gives W* 2 vb* which satisfies lemma 2. In the case ME CS, (25) and (3) give f > 0 along CS. Then (14) gives M AA w2 - Wf = j 2WW’dt 2 f 2(p - v)y’ydt = (,u - v)(yh - bz). C
C
Since Wz 2 pb* by (12), this gives o* 2 vb* which satisfies lemma 2. In the case M E SD. (25) and (3) give f < 0 along SD. Then (14) gives D
W;-w*=;2WW’dt$ M
L
2(p - v) y’y dt = 01 - v) (b* - y;).
Since Wi 2 ,ub* by (12), this gives w* 2 vb* which satisfies lemma 2. This finishes the proof of both lemma 2 and theorem 1. 3. IMPROVEDUPPER BOUND FOR T The following result is analogous to theorem 3 of [3] and coincides with it in the special case (6). THEOREM2. Ifg(x), F(y) satisfy (2), (3), (4) and F(--y)
CA-X) = -g(x),
= -Kv),
(26)
fir all x, y then (5) can be replaced by the sharper inequality +p-* VT < (2 + v-l jL)(.Jl + J*) + 2(1 + v-l cl).
(27)
Proof. If x(t), y(t) is a solution of (9) which describes the closed trajectory r then (26) shows that -x(t), -y(t) is also a solution of (9) which describes a trajectory r* got by reflecting r in the origin. Since r, r* are trajectories of (9) which intersect, they must coincide. That is, r is symmetrical with respect to the origin. If M* is the reflection of the point M at which W(x, y) attains its minimum value o on r then W(x, y) = o at M* also, because W(x, y) = W( - x. - y) by (12) and (26). It can be assumed that either M E PQ and M* E RSor M E QR and M* E SP.Thisenables W, + Ws + p- ’ v(W, + W,) tobeestimated as follows. In the case ME PQ, M* E RS both WQ- w G 01, and Ws - w < WI, hold as in theorem 1. These and (19), (20) give
3w(Z, + I,) b 2(WQ - w) + 2(Ws - 0) + (‘K 7
- WQ) + (‘K T
= wQ+
- wq,
w,+p-‘v(wR+
Wp)-40.
(28)
In the case M E QR, M* E SP both p-’ VW, - w < WI, and p-’ VW, - w < WI, hold as in theorem 1. These and (16), (17) give wo+
v(w, + w,) - 2(1 + v-r p)w = (WQ - w,) + (ws - w,) ws+p-r +(l+v-‘/A)~-‘vwR-w)+ (1+v_‘p)@-‘VW,-w), < (2 + v-i /J)o(Z, + I,).
(29)
Clearly (29) is true in both cases since (28) implies (29). Substitute (29) in the right-hand side of(13) and then use lemma 2 to obtain (27). This finishes the proof of theorem 2.
Period bout& for generalized
Rayleigh eguorion
211
The Rayleigh equation is the special case of (1) when F(y) = k(y - +y3), g(x) = x. The corresponding equation (6) is the Van der Pol equation. For this case, (5) gives T < 8 + (4Ok/3) and (27) gives T < 8(1 + k), which can be compared with the Poincare formula, T + 27~as k + 0, and the Lienard formula, T * 1.614k as k + + 00. This gives some idea of the order of accuracy to be expected from the estimates (5), (27) in the case p = v. It is possible that (S), (27) may become less accurate when p/v is large, because then the estimation leading to (19), (20) may be less precise. 4. LOWER BOUND FOR T The
following proof is an adaptation of that used by GraRi [4] for the Lienard equation.
THEOREM3. lf dg/dx < p for all x then the least period T of any non-constant periodic solution x(t) of(l) satisfies T 2 Zn/Jp. Pro05 Since y = x’ is a periodic function oft with zero mean value, Wirtinger’s inequality gives 4n2 ’ y2 dt < T2 ’ (y’)2 dt, d d
(30)
(see [S], p. 185). If Q(y) = i F(q) d rtthen (9) gives
$ {YY) - Y&4 =Y'fQY) - g(x))-
YX$
= (Y/l2 -
Y2
2.
Since @(y) - yg(x) is a periodic function oft, integration over (0, T) gives ~+‘)2dr-jy2~dt+‘)2dt-yilTy.dt. 0
0
This and (30) give 4n2 < T2p, which establishes theorem 3. REFERENCES [I] N. MINORSKY, Inrroduction to Nonlinear Mechanics, Edwards, Ann Arbor (1947). [2] R. Rmasro, Selbsterregung eines einfachen Schwingers, Moth. Nachr. 15, 191-196 (1956). [3] R. A. SMITH, Period bound for autonomous LiCnard oscillations, Q. appl. Mafh. 27.516522 (1970). [4] D. GRAFFI,Sopra alcune equazioni differenziali della radiotechnica, Mem. Accad. Sci. 1st. Bologna CI. Sci. Fix (9) 9, 145-153 (1942). [S] G. H. HARDY, J.E. LITTL~W~OD and G. P~LYA, Inequalities.
2nd Edition. Cambridge University Press (1952).
(Received 7 May 1970)
Rksum&On obtient des limites sup&ieures et inferieures simples de la plus petite p&ode de n’importe quelle solution .x(t)periodique. non constante, de I’equation differentielle x” - F(Y) + g(x) = 0. Zlaammeafnssung-EinfacheobereunduntereGrenzenfiirdiekleinstePeriodeTeinerbeliebigennichtperiodischen Losung x(r) der Differentialgleichung x” - F(x’) + g(x) = 0 werden erhalten. Ammq~-~Ony~amTCU npocme eepxme nso6oro rlIenOCTORHHOP0~ nepuoRu9ecHoro x"-F(x')+g(x) = 0.
u HHH(H~I~ rpaiiu &JW HauMeHbutero pexuetiau x(l) auq14iepeuuuanbtioro
nepuo;la T ypaeHeHkifl
Mechunics.
Vol. 6, pp. 271-277.
Pergamon
Press 1971.
Printed in Great Britain
PERIOD BOUNDS FOR GENERALIZED
RAYLEIGH
EQUATION
A. SMITH
RUSSELL
Department of Mathematics, University of Durham, Durham, England AhstrnetSimple upper and lower bounds are obtained for the least period T of any non-constant solution x(t) of the differential equation x” - F(x’) + g(x) = 0.
periodic
1. INTRODUCTION
THE paper concerns the differential equation x” - F(x’) + g(x) = 0,
(1)
where x(t) is the unknown, a prime denotes differentiation with respect to t, and F(y), g(x) are known functions with continuous first derivatives. All variables are real. Several physical occurrences of this equation are listed by Minorsky in the foreword of [l]. Sufficient conditions for the existence of a non-constant periodic solution were given by Reissig [2]. Throughout the present paper it is assumed that (1) has at least one nonconstant periodic solution x(t) with least period T. In the special case of weak non-linearities it is possible to study T by small parameter methods. But in the general case, very little is known about T and it is the aim of this paper to give upper and lower bounds for T which are simple and explicit. The lower bound is discussed in section 4. For the upper bound the following assumptions are made :
dg
for--cocx
vG-
dx
.fw z 09
for-b
(3) (4)
where p, v are positive constants, f(y) = dF/dy and f+(y) = max (0, f(y)). In section 2 the following is proved: THEOREM
1. Zf (2)-(4) hold and (1) has a periodic solution x(t) with least period T then +p-+vT
G
(2 + vmlp)(J1 + J2) + (1 + v-‘~1)’ (1 + max(JJ,)},
where J1 = (bJv)-’
r
f+(y)
dy,
s” f+(y)
dy.
-KZ
0
Since
JZ = (bJv)F’
(5)
f+(y) dy is equal to the positive variation of F(y) in the interval (0, co), J1 can be 0
271
272
RUSSELLA.SMI+H 0
computed in practice without integration.
Similarly,
1 f+(y)dy is the positive variation -.a) of F(y) in the interval (- cc, 0). An unexpected feature of Theorem 1 is the weakness of the assumptions made about F(y). In section 3 the inequality (5) is improved at the cost of further restrictions on F(y) and g(x). At the end of section 3, some comments are made about the order of accuracy to be expected from (5). When ,U= v, g(x) is linear and (1) can be differentiated to give y” - ylf(y) + vy = 0,
(6)
where y = x’. This Lienard equation is a special case of the equation studied by the author in [3]. Since the period bound obtained in [3] coincides with (5) for this special equation, theorem 1 can be regarded as an extension of this earlier work to a different class of equations. 2. PROOF OF THEOREM I It can be assumed without loss of generality that F(0) = 0,
g(0) = 0.
(7)
If this is not already so it can be achieved by absorbing the constant F(0) in g(x) and then shifting the origin of x to the unique zero of g(x) which exists because of (2). These changes do not affect (2)-(4). Then (2), (7) give vx2 d xg(x) < 1x2, for all x. Equation (1) is equivalent to the plane autonomous x’ = y,
(8) system
Y’ = F(Y) - g(x),
(9)
whose only singular point is at the origin of the (x, y) plane. The non-constant solution x(t) of (1) corresponds to a closed trajectory r in the (x, y) plane which encircles the origin and is described in the clockwise sense. LEMMA1. r crosses both lines y = f 6. Pro05 If G(x) = Ig(t) dt then (8) shows that G(x) is strictly decreasing in (- cc, 0) and
s strictly increasing in (0, cc). The equation y2 + 2G(x) = b2 defines a simple closed curve Q in the (x, y) plane which encircles the origin and touches the lines y = f 6. Along any trajectory of (9), $ {y’ + 2G(x)} = 2yy’ + 2g(x) x’ = 2yF(y).
(10)
Since f(y) = dF/dy is continuous, (3) shows that f(y) is of constant sign in (- 6, 6). Since F(0) = 0, it follows that yF(y) is of constant sign in the closed interval [ - 6, b], except for its zero at y = 0. In the case when yF(y) is positive in [ - 6, b], (10) shows that 52is crossed in the outward direction only, by every trajectory of (9) which meets it. In the case when yF(y) is negative in [ - 6, b], (10) shows that Q is crossed in the inward direction only, by
every trajectory of (9) which meets it. In both cases s2 cannot meet a closed trajectory of (9) because this would have to cross it both inwards and outwards. Furthermore no trajectory inside 51can be closed because (10) shows that y2 + 2G(x) is strictly monotonic along these trajectories. All closed trajectories of (9) therefore lie outside a. Since all closed trajectories
Period bounds for generalized
273
Rayleigh equation
encircle the origin, they must encircle i(l,and therefore meet both lines y = f b. This fmishes the proof of lemma 1. The curve with equation F(y) = g(x) passes through the origin by (7) and meets each line parallel to the x axis in exactly one point, because of (2). Since y = x’, r crosses the curve F(y) = g(x) once only in each of the half planes y > 0, y < 0. Also, (9) shows that x’, y’ have constant signs along each of the four arcs PQ, QR, RS, SP of r, (see figure).
FIG. 1.
Therefore y and g(x) are monotonic functions oft along each of these arcs. By considering the total variation of y and g(x) round r we obtain
Since v < dg/dx, these give j
(U>’
+ /W213dt
6
267~ + [YR\)
+ 2+@(gQ
+ ~~s;sB~
(11)
On r, (9) shows that (y’)’ + &z’)~ = W(x, Y)~,where ~(X,Y~
=
&Y2
4
El;(Y) -
SOW)’
(12)
Clearly W(x, y) > 0 for (x, y) # (0,O). If w is the minimum value of W(x, y) on the closed trajectory r then, replacing (Y’)~-t r_~(x’)~ by o2 in (1 l), we get oT G 2(Y, + (YRJ)+ 2v-‘Cl%o + Is& = z/KVV, + W,) + 2v-’ P(WQ -t Ws),
(13)
where WP, We, W,, W, are the values of W(x, y) at the points P, Q, R, S, respectively. From (9) and (12), W2 = py2 + (y’)’ along I’. Differentiate this with respect to t and use (9) to get
RUSSELLASMITH
274
Since & - dg/dx) y’y < 0 along the arcs PQ and RS, (14) gives w’ < W-‘(Jq2f
<
w-‘(_Yy f+ d fy'~f+(y)l
(15)
along PQ and RS. Integrate this along PQ and RS to get W, - w, C
(16)
IY’(f+(Y) dr = ‘f+(Y) dY G ~1, % d = ; f,cv)dY d 012,
w, - w, < ! [Y’lf.+Odt where I, = o- ’ 1 f+ dy, 1, = ,‘I
7 f+ dy. Al& -Co
(17)
QR and SP we have
0 ,< (P - dg,'dx) y’y d (P - v) y’y. This and (14) give w’ ,< w- ‘((y’)’
f+ + (j4 - V)Y’Y) s U+(Y) + (p - v)O>
IY’I,
(18)
along QR and SP. Integrate this along QR to get K - w, G [U+(Y) + (CL- v)P-+) IY’Idr, = R I f+cV)dY -+ (P - V)P-+ ly”l. Since WR = ,u* 1y, 1by (12), this gives j&-rvwa - u;, G ?f,(Y)dY R
(19)
G 01,.
Integrate (18) along Sp to get, similarly, #u-‘VW, - Ws < j f+dv)dy < ol,. s
(20)
Let M be the point on r at which W(x, y) takes its minimum value CU.We now estimate W, + W, + p- ‘v( WR + Wp) by discussing separately the four possible cases when M lies on each of the arcs PQ, QR, RS, SP. In the case N E PQ, integrate (15) along the arc MQ to get W, - w G ts f+(Y)lY’(dt d ;f+odY M Q
4 of,.
This and (17), (19), (20) give w, + ws + l*-‘v(W, + Wp) - 2(1 + v-‘j_A)w= (~-lvWp - W,) + 2(W, - W,) + (1 + 2v-‘~)(jPvw,
- WQ)+ 2(1 + V_‘&(WQ - 0) G (3 + 2v- r/J) o(l, + I,).
In the case M E QR, integrate (18) along the arc MR to get w, - 0 d s” (j-+ + fcl M
V)/A-+)
/y’j
dt ,( Uf, + (ji -
V)j.i-’
WR.
(21)
Period bow& for generalized Rayleigh equation
212
Hence, ~1-t VW, - w G wl,. This and (16), (17), (20) give Wo+ W,+jC’v(W,+ W-p)-(1 +v-‘~)%G(WoWp) + (1 + v-‘/A)@“- ‘VW, - W,) + (2 + v- ‘p)(W, - IV,) +(l +v-’ cc)2(P-r VW, - 0) < (2 + v-1 ~)@(I, + I,) + (1 + v-‘/@oZ2.
(22)
In the case M E RS, integrate (15) along the arc MS to get W, - o < ml,. This and (16), (19), (20) give w, + w, + p-1 v(W, + Wp) - 2(1 + v-1 /A)0 = (jf-’ VW, - WQ) + 2(Wo - Wp) + (1 + 2v-’ &(p-’
VW, - W,) + 2(1 + v-l p)(W, - 0) < (3 + zv-’ p)ofZ, + 1,).
(23)
In the case M E SP, integrate (18) along the arc MP to get
wp-co,< p{f+ if
+oL-
v)p-f}
\y’(dt < wZ, + 01 - V)P-’ I+‘,.
Hence, p-1 VW, - w < ml,. This and (16), (17), (19) give wo+
w,+/.l-’
v(W, + Wp) - (1 + v-l Zqo
+ (1 + v-l p)(p-‘v Inall fourcases,(21)
= (W, - W,)
w, - wo, + (2 + v-l p)(W, - Wp) + (1 + v-l /L)2(/P VW, - 0) < (2 + v- 1 p) o(Z, + 12) + (1 + v- 1 j.ly oz,. (24) - (24)give
Wo + K + 3%
+ Wp) <
2+ f (
o(Z, + I,) + >
(
1 + F 2 ~(1 + max (I,, I,)}. >
Substitute this in the right-hand side of (13) to get $/A-+ VT 6 (2+ v-lp)(I1 + Z2) + (1 + v - ’ j.i)’ { 1 + max (I,, I,)}. Theorem 1 follows from this by replacing I,, I, by the numbers .Zl, .Zz which satisfy J, = oZ,/b,/v, J, = oZ,/b Jv. The following lemma shows that .Z, > I,, J, >, I,: LJMMA 2. v*b < o. Proof: If the point M at which W attains its minimum o on Z lies outside the strip < y < b then (12) gives UJ >, p*lyl 2 vj b, which satisfies the lemma. It remains to consider separately the four possible cases when M lies on each of the arcs AQ, QB, CS, SD, (see figure). Since y’ # 0 on these four arcs and IV’ = 0 at M, (14) gives -b
signf = - sign (y’y),
at M
(25)
In the case M EAQ, (25) and (3) give f > 0 along AQ because y’y < 0 on AQ. Then (14) gives W W’ > (p - dg/dx) y’y 2 @ - v) y’y along AQ and W2 - W$=j2WW’draj2(p A
A
- v) y’y dt = @ - v) (y: - b2).
Since W: 2 ,ub2 by (12), this gives o2 2 vb2 which satisfies lemma 2. In the case ME QB, (25) and (3) give f < 0 along QB. Then (14) gives
276
RUSSELLA. SMITH
w&o*=
- Y&J.
B*WWldt~12(8-Y)ylydt=(II-Y)(b* 5
Since Wjj > pb* by (12), this gives W* 2 vb* which satisfies lemma 2. In the case ME CS, (25) and (3) give f > 0 along CS. Then (14) gives M AA w2 - Wf = j 2WW’dt 2 f 2(p - v)y’ydt = (,u - v)(yh - bz). C
C
Since Wz 2 pb* by (12), this gives o* 2 vb* which satisfies lemma 2. In the case M E SD. (25) and (3) give f < 0 along SD. Then (14) gives D
W;-w*=;2WW’dt$ M
L
2(p - v) y’y dt = 01 - v) (b* - y;).
Since Wi 2 ,ub* by (12), this gives w* 2 vb* which satisfies lemma 2. This finishes the proof of both lemma 2 and theorem 1. 3. IMPROVEDUPPER BOUND FOR T The following result is analogous to theorem 3 of [3] and coincides with it in the special case (6). THEOREM2. Ifg(x), F(y) satisfy (2), (3), (4) and F(--y)
CA-X) = -g(x),
= -Kv),
(26)
fir all x, y then (5) can be replaced by the sharper inequality +p-* VT < (2 + v-l jL)(.Jl + J*) + 2(1 + v-l cl).
(27)
Proof. If x(t), y(t) is a solution of (9) which describes the closed trajectory r then (26) shows that -x(t), -y(t) is also a solution of (9) which describes a trajectory r* got by reflecting r in the origin. Since r, r* are trajectories of (9) which intersect, they must coincide. That is, r is symmetrical with respect to the origin. If M* is the reflection of the point M at which W(x, y) attains its minimum value o on r then W(x, y) = o at M* also, because W(x, y) = W( - x. - y) by (12) and (26). It can be assumed that either M E PQ and M* E RSor M E QR and M* E SP.Thisenables W, + Ws + p- ’ v(W, + W,) tobeestimated as follows. In the case ME PQ, M* E RS both WQ- w G 01, and Ws - w < WI, hold as in theorem 1. These and (19), (20) give
3w(Z, + I,) b 2(WQ - w) + 2(Ws - 0) + (‘K 7
- WQ) + (‘K T
= wQ+
- wq,
w,+p-‘v(wR+
Wp)-40.
(28)
In the case M E QR, M* E SP both p-’ VW, - w < WI, and p-’ VW, - w < WI, hold as in theorem 1. These and (16), (17) give wo+
v(w, + w,) - 2(1 + v-r p)w = (WQ - w,) + (ws - w,) ws+p-r +(l+v-‘/A)~-‘vwR-w)+ (1+v_‘p)@-‘VW,-w), < (2 + v-i /J)o(Z, + I,).
(29)
Clearly (29) is true in both cases since (28) implies (29). Substitute (29) in the right-hand side of(13) and then use lemma 2 to obtain (27). This finishes the proof of theorem 2.
Period bout& for generalized
Rayleigh eguorion
211
The Rayleigh equation is the special case of (1) when F(y) = k(y - +y3), g(x) = x. The corresponding equation (6) is the Van der Pol equation. For this case, (5) gives T < 8 + (4Ok/3) and (27) gives T < 8(1 + k), which can be compared with the Poincare formula, T + 27~as k + 0, and the Lienard formula, T * 1.614k as k + + 00. This gives some idea of the order of accuracy to be expected from the estimates (5), (27) in the case p = v. It is possible that (S), (27) may become less accurate when p/v is large, because then the estimation leading to (19), (20) may be less precise. 4. LOWER BOUND FOR T The
following proof is an adaptation of that used by GraRi [4] for the Lienard equation.
THEOREM3. lf dg/dx < p for all x then the least period T of any non-constant periodic solution x(t) of(l) satisfies T 2 Zn/Jp. Pro05 Since y = x’ is a periodic function oft with zero mean value, Wirtinger’s inequality gives 4n2 ’ y2 dt < T2 ’ (y’)2 dt, d d
(30)
(see [S], p. 185). If Q(y) = i F(q) d rtthen (9) gives
$ {YY) - Y&4 =Y'fQY) - g(x))-
YX$
= (Y/l2 -
Y2
2.
Since @(y) - yg(x) is a periodic function oft, integration over (0, T) gives ~+‘)2dr-jy2~dt+‘)2dt-yilTy.dt. 0
0
This and (30) give 4n2 < T2p, which establishes theorem 3. REFERENCES [I] N. MINORSKY, Inrroduction to Nonlinear Mechanics, Edwards, Ann Arbor (1947). [2] R. Rmasro, Selbsterregung eines einfachen Schwingers, Moth. Nachr. 15, 191-196 (1956). [3] R. A. SMITH, Period bound for autonomous LiCnard oscillations, Q. appl. Mafh. 27.516522 (1970). [4] D. GRAFFI,Sopra alcune equazioni differenziali della radiotechnica, Mem. Accad. Sci. 1st. Bologna CI. Sci. Fix (9) 9, 145-153 (1942). [S] G. H. HARDY, J.E. LITTL~W~OD and G. P~LYA, Inequalities.
2nd Edition. Cambridge University Press (1952).
(Received 7 May 1970)
Rksum&On obtient des limites sup&ieures et inferieures simples de la plus petite p&ode de n’importe quelle solution .x(t)periodique. non constante, de I’equation differentielle x” - F(Y) + g(x) = 0. Zlaammeafnssung-EinfacheobereunduntereGrenzenfiirdiekleinstePeriodeTeinerbeliebigennichtperiodischen Losung x(r) der Differentialgleichung x” - F(x’) + g(x) = 0 werden erhalten. Ammq~-~Ony~amTCU npocme eepxme nso6oro rlIenOCTORHHOP0~ nepuoRu9ecHoro x"-F(x')+g(x) = 0.
u HHH(H~I~ rpaiiu &JW HauMeHbutero pexuetiau x(l) auq14iepeuuuanbtioro
nepuo;la T ypaeHeHkifl