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Application of the collocation and difference methods to find the auto-oscillations of differential-difference equations
APPLICATION OF THE COLLOCATION AND DIFFERENCE METHODS TO FIND THE AUTO-OSCILLATIONS OF DIFFERENTIAL-DIFFERENCE EQUATIONS* V. V. STRYGIN and A. 1. TSYGANKOV Kuibyshev (Received
NUMERICAL methods for the approximate convergence
1972)
discovery of auto-oscillations
are presented.
The
of these methods is investigated.
Two features distinguish class of boundary
the problem of the auto-oscillations
of differential
equations
value problems. The first is the absence of complete information
period of the unknown connection
18 December
solution,
the second is the non-isolation
of the auto-oscillations.
PoincarB’s method is usually used to find auto-oscillations.
operators in the problem of auto-oscillations, to use various projection
In this
The passage to integral
proposed by Krasnosel’skii
methods. It is interesting
in the
about the
[l] , makes it possible
to notice that to use a specific projection
method it is necessary to select an integral operator of the auto-oscillation
problem specially adapted
for this method. In [ 1,2] the ideas of Gal&kin’s method were used for the approximate discovery of auto-oscillations. In [4] the collocation method was used in this problem. The present paper is a further extension
of the result of [l-4]
.
1. Fundamental
assumptions
Let Rm bc an m-dimensional Euclidean space, (., .) and II-II the scalar product and the norm in Rm. We assume that the function f (s,, . . . , zk) is defined on R”X . . . XR” and assumes values in R”‘. We assume that f is twice continuously of differential-difference equations
differentiable.
dx/dt=f[s(t--h,),
where O
. . .
dY
dt=
defined by the
system
ckf,
[z’(t - h,), . . . , 5*(t - hk)l Y (t - hi)
(1)
i=l
subspace of w. -periodic solutions.
0
system
. . . , x(t-h,J],
We assume that this system has a cycle F of period o,+O,
solution x*(t). Also, let the variational
have a one-dimensional
We consider an autonomous
.~
be adjoint to (1) for some wo-periodic
Finally, let the system (see [5] )
(2) i=l
solution
rp (t)
*Zh. vjkhisl. Mat. mat. Fiz., 14, 3,691-698, 1974.
149
V. V. Strygin and A. I. Tsygankov
150
We denote by C,, the space of continuously differentiable 2n-periodic functions 5 (7)) which assume values in Rm. We denote by II* llcin the norm in C,,, and by 11.jIc as usual, the norm in the space C of functions continuous in [0,2n] with values in R”. Let x,(a) =J* (roo@n+p) , 0<&2n, OGpG2n. On varyingpE[O, 2n]the function xp(r) defines in the Banach space C,, a closed, continuously differentiable curve L. Let 52(x) , x=Cin, be some continuously differentiable positive functional. We assume that for some point x?,=L the following two conditions are satisfied:
52(x,) =ooD%
CL(XP)k&w
+o.
(3)
For definiteness we will consider that p = 0. By means of the functional S2we will also determine approximations of the collocation method and of the difference method.
2. The collocation method Let T8=2ns/(2n+l), s=O, 1, . . . ,212, be the interpolation nodes. We denote by P, the projection operator which associates with each function XEC In a trigonometric polynomial P,x of degree n, identical with x at the interpolation nodes. The operator P, is defined by the equation 27%
P,x = -?z z (q) D, (z 2n + 1 i=o
zi),
(see [6]). It is easy to show that P,x+x as n-+00 in the norm s=i of the space L2. If x has an absolutely convergent Fourier series, P,,x converges to x uniformly. If d2s/dr2~LZ, then P,,x converges to x in the sense of the space Ci, (see [6] ).
where D,(h)
=I/*+
i cos sh
We will define the approximations of the collocation method as trigonometric polynomials X,(T), OGr<2n, of order n, which are the solutions of the algebraic system with 2ntl unknowns
Z=Z .37
s = 0,1,.
. . ,2n
(that is, they satisfy the system (4) at the points rS). Theorem 1
For sufficiently large values of n the approximations x, of the collocation method exist and converge to x0. The following estimate of the rate of convergence holds for some M > 0:
To prove the theorem we require some auxiliary propositions.
151
Auto-oscillationsof differential-difference equations Lemma 1
where E~+O as n-00.
Let XEC,~. Then II(I-P,)xllc~~,lltllcin The proof easily follows from the inequality IIv -
p?J z IICQ 2 c wn
I 7s If
where the Ys are the Fourier coefficients of the function x. Let
co
a0 +
lE(a, cos sr + b, sin sz)
s=1
be the Fourier series of the function X(T) and m
E-lox=
Xi ’
-
%sinsr
b
+ scos S
ST
s=1
We define the operators T,$nd S by the equations TX=&\
3 (Q dr + Hoz, 0
sx =& (On the subject of theoplrators
T
and S see [7] .)
Lemma 2
If x is an element of a sufficiently small neighbourhood V~‘N, of the point xo , then if and only if
+qx)(x
(T-$-J f...,X(T-A)].
(6)
The proof easily follows from the identity 2x
-$-Hox-_x-~~
5 (z) dz. 0
Lemma
3
Let the sequence X, belong to a fairly small neighbourhood VcC,, the set {P,,Sz,} is compact in C.
of the point x0. Then
Proof: The sequence Sx, is bounded in C,,.This and Lemma 1 imply that for any ~2-0 we can find an 1 such that UP,,SX,--SS,,~~~~E for nBZ. The set {&so} is compact in C. This implies the statement of the lemma. Lemma
4
If x0 is an isolated solution of the equation x=TSx with a non-zero index (see [S] ), then for sufficiently large values of n the system (4) has a solution x, situated in a sufficiently small neighbourhood VcCirr of the point x0 and xn -+x0 asn+monthenormofthespaceC,,.
152
K Y.Shygin end A. L T~ygankov
Roof: We first show that for sufficiently large values of n the field @,x=x-TP,Sx on the =r of sufficiently small radius r does not vanish and is homotopic to the field spheres (Ix-xOllcin x - TSx. Indeed, let us consider the family of vector fields 0 (h) x=x-hTP,Sx(1-A) T&q OGKl, llx-dfc~* =r. Let @I(A,) L,=G for some sequenceh,E [ 0, l]and an x, such that fix,---xo/lcin=r. It is obvious that h,TP,Sx,+ (i-h,) XTSx, forms a compact sequence in C,,,. Therefore the sequence x, is compact in C,,. Let x, -“yin the norm of C,,.Then y=TSy and IIY--xollct,=r. The latter is impossible. Consequently, the fields @ (A) are homotopic on the sphere llx-x&tn =r, if n is sufficiently large. This implies that for sufficiently large IZthe r is non-zero. Therefore, the equation rotation of the field @, on the sphere ~~x--z~~~~~~= x=TP,Sxhas a solution x,. It is obvious that X, is a polynomial of degree n and
It follows from the properties of the projector Pn that X, satisfies the system (4) at the nodes rs. It is obvious that x,-+x0 as n*m. Lem?m 5 Let S’ be some bounded operator acting in C,,. We suppose that the operator I-TS’is invertible. Then for sufficiently large n the operators I-TP,S’ are invertible and sup11(I-TP,S’)-‘ll-e. n It is easy to obtain an indirect proof. Here it is necessary to use the compactness of the operator TS ’ and Lemma 3. We now present the proof of Theorem 1. Let S’ be the Frechkt derivative of the operator S at the point x0 _It is easy to calculate that
We fust show that 1 is not an eigenvalue of the operator Ts’. Let us assume the contrary. Then a non-zero g(z) ECin, exists such that g=TS’g. This implies that g(r) is a 2n-periodic solution of the system
Auto-oscillationsof differential-differenceequations
II,(t) =g (2nt / o u) ISan w. -periodic solution of the system
In this case the function
-a* =
ax*s-k+4
-w-y
at
153
clt
21(.r,,,
iLilx‘ (t -
+
hl), . . f , .r* (t - &)I
?=I
z
Q, (1.0)g : hif ix- it - h,), . . . , x-(It 9 (~0)
X$(t-hi)+
hk)]
ax* (t at
hi)
.
i_-L
The latter is possible only if wg 9,. (50) g \ ([$ 0 y ad (t - hi) / dt
where cp(t) is an wo-periodic Therefore $(t)
{- i h& 7=1 3
[x* (t -
h,), . . . , x* (t -
hk)]
, cp(t)’ at = 0, I
solution of the system adjoint to (1). By (2) we have Q,(sO) g==O.
is an oo-periodic
solution of the system (1) and consequently
it is of the form
(r) / dz and (4) implies that (Y= 0. Consequently, 11,(t) =a&* (t) / dt. But in th’1s case g-adz, g = 0. Therefore, 1 is not an eigenvalue of the operator KS”. Therefore, x0 is an isolated fixed point of the operator TS with non-zero index. By Lemma 4, for sufficiently
large n the system (4) is
solvable and these solutions converge to x0 as n-+00. To prove the estimate (5) we represent the difference x0 - x, in the form
x,-x,=TSx,-TP,Sx,=T{Sx,-P,[Sx,SS’(s,-x,) +o
(xa, CI.,,--50) ] > =T[
-TP,,o
are reversible for sufficiently x0--z,=
-TP,S’(x,--50)
(~0, zn-4,
and E~-+O as n-+m.
where]]@ (x0, s,---ICY) Ilcin~~,,l/~o-~,,IICin I-TP,,S’
SXCP,,SS~]
(I-TP,,S’)
By Lemma 5, the operators
large values of n. Therefore -‘{T[Sz,-P,Ss,]
The estimate (5) follows from these equations
-TP,o
(x0, xn-x0)}.
and the boundedness
of the norms ]I (I-TP,$‘)
-‘II
The theorem is proved.
3. The difference method
In this section we assume for simplicity that hicoo. establish a correspondence
Let n be a positive integer. We
between each set E= {Eo, E,, . . . , E,,}, Er~Rrn, and a polygonal
curve g, defined on [0, 27r], possessing the nodes z,=2ns / n, s=O, 1, . . . , n, and assuming the values z, at the points 5?. We will then consider that this polygonal curve is continuously continued onto the segment [-27r,O] as follows: 2 (r) =Z (r+an)
-C(o>+t;1(2n),
-2nfaGO.
The pointsr,=k / n, s=-n, . . . , n,will be the nodes of the continued polygonal curve. Let x(r) be defined in the interval [-2n, 2n] . Below we will denote by ? a polygonal curve which is defined in the interval
[-27r, 2n] and assumes the values ~(7~) at the nodes t,, s=-n,
is obvious that if the polygonal
. . . , n. It
curve E is close to x0 then Q (g) coo. Therefore the points z,-2nh
154
V. V. Stfygin and A. I. Tsygankov
The difference method of finding cycles is as follows. We seek a set E= {go, El, . . . , En}, go=gnt giERrn, wh’ICh is a solution of the finite-dimensional system
here xi, s=xi, s(g). In other words, the derivative g is replaced by the finite differences (ES+,-&) / (2n/n), and the right side of the system (6) by the set of values at the corresponding interpolation nodes. It happens that the following theorem holds here. l7reorem 2 For sufficiently large n the system (7) has solutions EcR),where pn)*z,, [0,2n] as n-+00.
in the segment
Proofi Let C be the space of functions continuous in [0,2n] with values in Rm. It is known (see [9]) that the problem of 2n-periodic solutions of the system (6) in the neighbourhood of z. (r)is equivalent to the problem of fixed points of the operator Ar=s(27)+5r?(5)lri(~-~)
,...,+-
$--@)]dr,
0
acting in the space C; here 2 is the continuation of the function x from the interval [0,2n] to [-2n, 2n] by the formula 2 (r) =5 (r-tan) fz (0) --IC(2n) . We now consider the subspace E,, of functions of C which vanish at the nodes rS. Then the factor space C/E,, is the aggregate of classes of functions x which assume specified values xS at the nodes rs. It is obvious that C/E, is isomorphic to the space Rmcn+i)and
Ml CjEn=infII&=II4l0 iE3
Let pn be a canonical mapping of CintoC/
E,: P~x={x(O),
x(~i),
. . .‘, x(r,)}.It
is obvious
that Ilpn~llclEn-+ ll~llc asn+m. We now show that the sequence of operators A,, n=l, by the equations
2, . . . , acting in C/E and defined
is a compact approximation of the operator A. We recall that the sequence of continuous operators A, acting in C/E,, ,compactly approximates the completely continuous operator A acting in C, if ]]p,As-A,p,,x]lc I En+O as n-tm for any SEC, and if for any sequence Etn)4/ E,, I]~‘“‘](~~const, n=l, 2, . . . ,elements can be found such that the sequence y, is compact in the space C, see [lo] . !!%=A P),
Auto-oscillationsof differential-difference equations
We notice that p,kz=
Also, A.p,,s={Bo,
155
{CL+al, . . . , a,} is of the form
Pi, . . . , Bn}, where
Here 5oS.j is the value of x”at the node closest on the left to the point z-h, verify that
/ B(f).It
is easy to
Also, let Ecn), n=l, 2, . . . ,be an element of the sphere of the space C/En of radius and centre at pnxo. It is easy to see that the sequence of polygonal curves AT is compact in C. Therefore, the sequence A, is a compact approximation of the operator A. We finally notice that the relation
IlE(n)--Pn41c~E,-+0,
n-+00,
zd,
~(n)=C/E,,
implies the uniform convergence of the polygonal curves tZ) and p% to x. From this it easily follows that
IIAP+L$n~llc
/ En-4
n-too.
To prove the existence of the approximations of the difference method and their convergence, we now use a theorem due to Vainikko [lo] which implies that the approximations EC”)exist and that I(“‘) converges to x e, if the index of the point x0 of the field I - A is non-zero. It is easy to verify that this index is the same as the index of the point x0 of the field I- 7S considered in the space C. Also, it follows from the principle of the invariance of rotations [ 1l] that the index of the point x0 with respect to the field I- TS in C is the same as the index of x0 of the field I- TS in the space C,,. As shown in section 2, the latter is non-zero. The theorem is proved. In conclusion we mention that the construction of the functional satisfying condition (3) is a separate and difficult problem. Information about the region in which the cycle is situated and the value of its period may be extremely useful for its solution. Translated by J. Berry REFERENCES 1.
BOBYLEV, N. A. and KRASNOSEL’SKII, M. A., Functionalization of the parameter and a theorem of affinity for autonomous systems. Differents. Ur-niya,6,11, 1946-1952, 1970.
2.
STOCKS, A., On the approximation of non-linear oscillations. In: Proceedings of the Fifth International Conference on Non-Linear Oscillations,1969 (Tr. V Mezhdunar. Konf. po nelineinym kolebaniyam, 1969), Vol. 2, In-t matem. Akad. Nauk Ukr SSR, Kiev, 1970.
3.
STRYGIN, V. V., Application of Galerkin’s method to the problem of finding auto-oscillations. Prikl. Mat. Mekh., 37,6,1015-1019,1973.
4.
RONTO, N. The problem of auto-oscillations. In: MathematicalMethods in Cybernetic Techniques (Matem. metody v kibernetich. tekhn.), No. 5, 76-84, In-t kibemetiki Akad. Nauk Ukr SSR, Kiev, 1970.
156
V. V.Strygin and A. I. Tsygankov
5.
SHIMANOV, S. N., The theory of differential equations with after effect. Differents. Ur-niya, I, 1, 102, 116,1965.
6.
ZYGMUND, A., Trigonometric series (Trigonometricheskie
7.
ZABREIKO, P. P. and STRYGINA, S. O., Periodic solutions of evolutionary equations. Matem. Zametki, 9,6,651-662, 1971.
8.
KRASNOSEL’SKII, M. A., Topological methods in the theory of non-linear integral equations, Fizmatgiz, Moscow, 1956.
9.
KRASNOSEL’SKII, M. A., An alternative principle of the existence of periodic solutions for differential equations with a delaying argument. Dokl. Akad. Nauk SSSR, 152,4, 801-804, 1963.
ryady), Vols. I, II, “Mir”, Moscow, 1965.
10. KRASNOSEL’SKII, M. A., VAINIKKO, G. M., ZABREIKO, P. P., RUTITSKII, Ya. B. and STETSENKO, V. Ya., The approximate solution of operator equations (Priblizhennoe reshenie operatornykh uravnenii), “Nauka”, Moscow, 1969. 11. ZABREIKO, P. P., KRASNOSEL’SKII, M. A. and STRYGIN, V. V., The principle of invariance of rotation. Izv. Vuzov. Matematika,No, 5 (120), 51-57, 1972.
NUMERICAL methods for the approximate convergence
1972)
discovery of auto-oscillations
are presented.
The
of these methods is investigated.
Two features distinguish class of boundary
the problem of the auto-oscillations
of differential
equations
value problems. The first is the absence of complete information
period of the unknown connection
18 December
solution,
the second is the non-isolation
of the auto-oscillations.
PoincarB’s method is usually used to find auto-oscillations.
operators in the problem of auto-oscillations, to use various projection
In this
The passage to integral
proposed by Krasnosel’skii
methods. It is interesting
in the
about the
[l] , makes it possible
to notice that to use a specific projection
method it is necessary to select an integral operator of the auto-oscillation
problem specially adapted
for this method. In [ 1,2] the ideas of Gal&kin’s method were used for the approximate discovery of auto-oscillations. In [4] the collocation method was used in this problem. The present paper is a further extension
of the result of [l-4]
.
1. Fundamental
assumptions
Let Rm bc an m-dimensional Euclidean space, (., .) and II-II the scalar product and the norm in Rm. We assume that the function f (s,, . . . , zk) is defined on R”X . . . XR” and assumes values in R”‘. We assume that f is twice continuously of differential-difference equations
differentiable.
dx/dt=f[s(t--h,),
where O
. . .
dY
dt=
defined by the
system
ckf,
[z’(t - h,), . . . , 5*(t - hk)l Y (t - hi)
(1)
i=l
subspace of w. -periodic solutions.
0
system
. . . , x(t-h,J],
We assume that this system has a cycle F of period o,+O,
solution x*(t). Also, let the variational
have a one-dimensional
We consider an autonomous
.~
be adjoint to (1) for some wo-periodic
Finally, let the system (see [5] )
(2) i=l
solution
rp (t)
*Zh. vjkhisl. Mat. mat. Fiz., 14, 3,691-698, 1974.
149
V. V. Strygin and A. I. Tsygankov
150
We denote by C,, the space of continuously differentiable 2n-periodic functions 5 (7)) which assume values in Rm. We denote by II* llcin the norm in C,,, and by 11.jIc as usual, the norm in the space C of functions continuous in [0,2n] with values in R”. Let x,(a) =J* (roo@n+p) , 0<&2n, OGpG2n. On varyingpE[O, 2n]the function xp(r) defines in the Banach space C,, a closed, continuously differentiable curve L. Let 52(x) , x=Cin, be some continuously differentiable positive functional. We assume that for some point x?,=L the following two conditions are satisfied:
52(x,) =ooD%
CL(XP)k&w
+o.
(3)
For definiteness we will consider that p = 0. By means of the functional S2we will also determine approximations of the collocation method and of the difference method.
2. The collocation method Let T8=2ns/(2n+l), s=O, 1, . . . ,212, be the interpolation nodes. We denote by P, the projection operator which associates with each function XEC In a trigonometric polynomial P,x of degree n, identical with x at the interpolation nodes. The operator P, is defined by the equation 27%
P,x = -?z z (q) D, (z 2n + 1 i=o
zi),
(see [6]). It is easy to show that P,x+x as n-+00 in the norm s=i of the space L2. If x has an absolutely convergent Fourier series, P,,x converges to x uniformly. If d2s/dr2~LZ, then P,,x converges to x in the sense of the space Ci, (see [6] ).
where D,(h)
=I/*+
i cos sh
We will define the approximations of the collocation method as trigonometric polynomials X,(T), OGr<2n, of order n, which are the solutions of the algebraic system with 2ntl unknowns
Z=Z .37
s = 0,1,.
. . ,2n
(that is, they satisfy the system (4) at the points rS). Theorem 1
For sufficiently large values of n the approximations x, of the collocation method exist and converge to x0. The following estimate of the rate of convergence holds for some M > 0:
To prove the theorem we require some auxiliary propositions.
151
Auto-oscillationsof differential-difference equations Lemma 1
where E~+O as n-00.
Let XEC,~. Then II(I-P,)xllc~~,lltllcin The proof easily follows from the inequality IIv -
p?J z IICQ 2 c wn
I 7s If
where the Ys are the Fourier coefficients of the function x. Let
co
a0 +
lE(a, cos sr + b, sin sz)
s=1
be the Fourier series of the function X(T) and m
E-lox=
Xi ’
-
%sinsr
b
+ scos S
ST
s=1
We define the operators T,$nd S by the equations TX=&\
3 (Q dr + Hoz, 0
sx =& (On the subject of theoplrators
T
and S see [7] .)
Lemma 2
If x is an element of a sufficiently small neighbourhood V~‘N, of the point xo , then if and only if
+qx)(x
(T-$-J f...,X(T-A)].
(6)
The proof easily follows from the identity 2x
-$-Hox-_x-~~
5 (z) dz. 0
Lemma
3
Let the sequence X, belong to a fairly small neighbourhood VcC,, the set {P,,Sz,} is compact in C.
of the point x0. Then
Proof: The sequence Sx, is bounded in C,,.This and Lemma 1 imply that for any ~2-0 we can find an 1 such that UP,,SX,--SS,,~~~~E for nBZ. The set {&so} is compact in C. This implies the statement of the lemma. Lemma
4
If x0 is an isolated solution of the equation x=TSx with a non-zero index (see [S] ), then for sufficiently large values of n the system (4) has a solution x, situated in a sufficiently small neighbourhood VcCirr of the point x0 and xn -+x0 asn+monthenormofthespaceC,,.
152
K Y.Shygin end A. L T~ygankov
Roof: We first show that for sufficiently large values of n the field @,x=x-TP,Sx on the =r of sufficiently small radius r does not vanish and is homotopic to the field spheres (Ix-xOllcin x - TSx. Indeed, let us consider the family of vector fields 0 (h) x=x-hTP,Sx(1-A) T&q OGKl, llx-dfc~* =r. Let @I(A,) L,=G for some sequenceh,E [ 0, l]and an x, such that fix,---xo/lcin=r. It is obvious that h,TP,Sx,+ (i-h,) XTSx, forms a compact sequence in C,,,. Therefore the sequence x, is compact in C,,. Let x, -“yin the norm of C,,.Then y=TSy and IIY--xollct,=r. The latter is impossible. Consequently, the fields @ (A) are homotopic on the sphere llx-x&tn =r, if n is sufficiently large. This implies that for sufficiently large IZthe r is non-zero. Therefore, the equation rotation of the field @, on the sphere ~~x--z~~~~~~= x=TP,Sxhas a solution x,. It is obvious that X, is a polynomial of degree n and
It follows from the properties of the projector Pn that X, satisfies the system (4) at the nodes rs. It is obvious that x,-+x0 as n*m. Lem?m 5 Let S’ be some bounded operator acting in C,,. We suppose that the operator I-TS’is invertible. Then for sufficiently large n the operators I-TP,S’ are invertible and sup11(I-TP,S’)-‘ll-e. n It is easy to obtain an indirect proof. Here it is necessary to use the compactness of the operator TS ’ and Lemma 3. We now present the proof of Theorem 1. Let S’ be the Frechkt derivative of the operator S at the point x0 _It is easy to calculate that
We fust show that 1 is not an eigenvalue of the operator Ts’. Let us assume the contrary. Then a non-zero g(z) ECin, exists such that g=TS’g. This implies that g(r) is a 2n-periodic solution of the system
Auto-oscillationsof differential-differenceequations
II,(t) =g (2nt / o u) ISan w. -periodic solution of the system
In this case the function
-a* =
ax*s-k+4
-w-y
at
153
clt
21(.r,,,
iLilx‘ (t -
+
hl), . . f , .r* (t - &)I
?=I
z
Q, (1.0)g : hif ix- it - h,), . . . , x-(It 9 (~0)
X$(t-hi)+
hk)]
ax* (t at
hi)
.
i_-L
The latter is possible only if wg 9,. (50) g \ ([$ 0 y ad (t - hi) / dt
where cp(t) is an wo-periodic Therefore $(t)
{- i h& 7=1 3
[x* (t -
h,), . . . , x* (t -
hk)]
, cp(t)’ at = 0, I
solution of the system adjoint to (1). By (2) we have Q,(sO) g==O.
is an oo-periodic
solution of the system (1) and consequently
it is of the form
(r) / dz and (4) implies that (Y= 0. Consequently, 11,(t) =a&* (t) / dt. But in th’1s case g-adz, g = 0. Therefore, 1 is not an eigenvalue of the operator KS”. Therefore, x0 is an isolated fixed point of the operator TS with non-zero index. By Lemma 4, for sufficiently
large n the system (4) is
solvable and these solutions converge to x0 as n-+00. To prove the estimate (5) we represent the difference x0 - x, in the form
x,-x,=TSx,-TP,Sx,=T{Sx,-P,[Sx,SS’(s,-x,) +o
(xa, CI.,,--50) ] > =T[
-TP,,o
are reversible for sufficiently x0--z,=
-TP,S’(x,--50)
(~0, zn-4,
and E~-+O as n-+m.
where]]@ (x0, s,---ICY) Ilcin~~,,l/~o-~,,IICin I-TP,,S’
SXCP,,SS~]
(I-TP,,S’)
By Lemma 5, the operators
large values of n. Therefore -‘{T[Sz,-P,Ss,]
The estimate (5) follows from these equations
-TP,o
(x0, xn-x0)}.
and the boundedness
of the norms ]I (I-TP,$‘)
-‘II
The theorem is proved.
3. The difference method
In this section we assume for simplicity that hicoo. establish a correspondence
Let n be a positive integer. We
between each set E= {Eo, E,, . . . , E,,}, Er~Rrn, and a polygonal
curve g, defined on [0, 27r], possessing the nodes z,=2ns / n, s=O, 1, . . . , n, and assuming the values z, at the points 5?. We will then consider that this polygonal curve is continuously continued onto the segment [-27r,O] as follows: 2 (r) =Z (r+an)
-C(o>+t;1(2n),
-2nfaGO.
The pointsr,=k / n, s=-n, . . . , n,will be the nodes of the continued polygonal curve. Let x(r) be defined in the interval [-2n, 2n] . Below we will denote by ? a polygonal curve which is defined in the interval
[-27r, 2n] and assumes the values ~(7~) at the nodes t,, s=-n,
is obvious that if the polygonal
. . . , n. It
curve E is close to x0 then Q (g) coo. Therefore the points z,-2nh
154
V. V. Stfygin and A. I. Tsygankov
The difference method of finding cycles is as follows. We seek a set E= {go, El, . . . , En}, go=gnt giERrn, wh’ICh is a solution of the finite-dimensional system
here xi, s=xi, s(g). In other words, the derivative g is replaced by the finite differences (ES+,-&) / (2n/n), and the right side of the system (6) by the set of values at the corresponding interpolation nodes. It happens that the following theorem holds here. l7reorem 2 For sufficiently large n the system (7) has solutions EcR),where pn)*z,, [0,2n] as n-+00.
in the segment
Proofi Let C be the space of functions continuous in [0,2n] with values in Rm. It is known (see [9]) that the problem of 2n-periodic solutions of the system (6) in the neighbourhood of z. (r)is equivalent to the problem of fixed points of the operator Ar=s(27)+5r?(5)lri(~-~)
,...,+-
$--@)]dr,
0
acting in the space C; here 2 is the continuation of the function x from the interval [0,2n] to [-2n, 2n] by the formula 2 (r) =5 (r-tan) fz (0) --IC(2n) . We now consider the subspace E,, of functions of C which vanish at the nodes rS. Then the factor space C/E,, is the aggregate of classes of functions x which assume specified values xS at the nodes rs. It is obvious that C/E, is isomorphic to the space Rmcn+i)and
Ml CjEn=infII&=II4l0 iE3
Let pn be a canonical mapping of CintoC/
E,: P~x={x(O),
x(~i),
. . .‘, x(r,)}.It
is obvious
that Ilpn~llclEn-+ ll~llc asn+m. We now show that the sequence of operators A,, n=l, by the equations
2, . . . , acting in C/E and defined
is a compact approximation of the operator A. We recall that the sequence of continuous operators A, acting in C/E,, ,compactly approximates the completely continuous operator A acting in C, if ]]p,As-A,p,,x]lc I En+O as n-tm for any SEC, and if for any sequence Etn)4/ E,, I]~‘“‘](~~const, n=l, 2, . . . ,elements can be found such that the sequence y, is compact in the space C, see [lo] . !!%=A P),
Auto-oscillationsof differential-difference equations
We notice that p,kz=
Also, A.p,,s={Bo,
155
{CL+al, . . . , a,} is of the form
Pi, . . . , Bn}, where
Here 5oS.j is the value of x”at the node closest on the left to the point z-h, verify that
/ B(f).It
is easy to
Also, let Ecn), n=l, 2, . . . ,be an element of the sphere of the space C/En of radius and centre at pnxo. It is easy to see that the sequence of polygonal curves AT is compact in C. Therefore, the sequence A, is a compact approximation of the operator A. We finally notice that the relation
IlE(n)--Pn41c~E,-+0,
n-+00,
zd,
~(n)=C/E,,
implies the uniform convergence of the polygonal curves tZ) and p% to x. From this it easily follows that
IIAP+L$n~llc
/ En-4
n-too.
To prove the existence of the approximations of the difference method and their convergence, we now use a theorem due to Vainikko [lo] which implies that the approximations EC”)exist and that I(“‘) converges to x e, if the index of the point x0 of the field I - A is non-zero. It is easy to verify that this index is the same as the index of the point x0 of the field I- 7S considered in the space C. Also, it follows from the principle of the invariance of rotations [ 1l] that the index of the point x0 with respect to the field I- TS in C is the same as the index of x0 of the field I- TS in the space C,,. As shown in section 2, the latter is non-zero. The theorem is proved. In conclusion we mention that the construction of the functional satisfying condition (3) is a separate and difficult problem. Information about the region in which the cycle is situated and the value of its period may be extremely useful for its solution. Translated by J. Berry REFERENCES 1.
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ZABREIKO, P. P. and STRYGINA, S. O., Periodic solutions of evolutionary equations. Matem. Zametki, 9,6,651-662, 1971.
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KRASNOSEL’SKII, M. A., An alternative principle of the existence of periodic solutions for differential equations with a delaying argument. Dokl. Akad. Nauk SSSR, 152,4, 801-804, 1963.
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