CC BY
0DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 3, 2023 Electronic Journal, reg. N &C77-39410 at 15.04.2010 ISSN 1817-2172
http://diffjournMl.spbu.ru/ e-mail: jodiff@mail.ru
Dynamical systems
Methods of nonhyperbolic shadowing
Sergei Yu. Pilyugin Saint Petersburg State University sergeipil47@mail.ru
Abstract. This paper is a survey of some recent results giving sufficient conditions of shadowing for dynamical systems in the absence of hyperbolicity. The main topics of the survey are as follows: method of pairs of Lyapunov type functions, shadowing in a neighborhood of a nonisolated fixed point, conditional multiscale shadowing for sequences of mappings of a Banach space, conditional shadowing for dynamical systems on so-called simple time scales. The paper contains a new result on conditional multiscale shadowing in the case of an infinite family of projections of the phase space.
Keywords: Dynamical system, pseudotrajectory, shadowing, hyperbolicity 1 Introduction
The theory of shadowing of approximate trajectories (pseudotrajectories) is now an intensively developing branch of the global theory of dynamical systems. One can find basic information concerning shadowing in the monographs [1-3]; the survey [4] is devoted to some recent results of the theory.
The main property of dynamical systems studied by the shadowing theory can be stated as follows. Consider a homeomorphism f of a metric space (X, dist). Let d > 0. A sequence {yn £ X} is called a d-pseudotrajectory of f if the inequalities
dist(f(yn),yn+1) <d (1)
hold.
One says that f has the (standard) shadowing property if for any £ > 0 there is a d > 0 such that for any d-pseudotrajectory {yn} of f there is a point x € X for which
dist(fn(x),yn) < £.
In this case, we say that the exact trajectory {fn(x)} £-shadows the pseu-dotrajectory {yn}.
The first sufficient conditions under which a dynamical system has the shadowing property were obtained by D.V. Anosov [5] and R. Bowen [6]. Applying principally different approaches, they showed that, for a diffeomorphism, shadowing is a corollary of hyperbolicity (and, as easilly seen from the proofs in [5] and [6], in a neighborhood of a hyperbolic set, a diffeomorphism has the Lip-schitz shadowing property, i.e., there exists a constant L such that, for small enough d, any d-pseudotrajectory is Ld-shadowed by an exact trajectory).
At present, relations between shadowing and hyperbolicity are well studied (see the book [3] for details).
In the present paper, we give a survey of some recent results giving sufficient conditions of shadowing in the absence of hyperbolicity. This survey is motivated by the talk given by the author at the Joint PDMI - MIRAN Session "Differential Equations and Dynamical Systems" (St. Petersburg, May 12 - 14, 2023).
The paper also contains a new result - we generalize the main theorem of the author's paper [7] to the case of an infinite family of projections (see Section 4 below). This generalization is not straightforward; in fact, we have to work with different Banach spaces etc.
The structure of the paper is as follows. In Section 2, we describe the method of pairs of Lyapunov type functions developed by A.A. Petrov and the author in the paper [8]. This method is applied to a perturbation of the hyperbolic automorphism of the 2-torus studied by J. Lewowicz in [9]. It is noted that, using the described method, one can obtain sufficient conditions of shadowing for nonsmooth systems.
In Section 3, it is shown that the method described in Section 2 can be applied to study shadowing in a neighborhood of a fixed point belonging to a "critical" manifold consisting of fixed points. In this case, not every pseudotra-jectory in such a neighborhood can be shadowed, but it is possible to shadow pseudotrajectories for which "one step errors" are small enough compared to
the distances of points of the pseudotrajectory to the critical manifold.
In Section 4, we study conditional shadowing for sequences of mappings of a Banach space X. It is assumed that there exists a countable family of projections that commute with linear terms of the considered mappings. Conditions of shadowing are obtained in terms of the norms of projections of one step errors of pseudotrajectories and of estimates of Lipschitz constants of projections of the "small nonlinear" terms. As mentioned above, the main theorem of this section generalizes the result of the author's paper [7] in which the case of a finite family of projections has been studied.
Finally, in Section 5, we describe conditions of conditional shadowing for dynamical systems on so-called simple time scales consisting of a family of isolated segments of the positive ray of R; these results are obtained in the author's paper [10].
2 Method of a pair of Lyapunov type functions
Let f be a homeomorphism of a compact metric space (X, dist).
We assume that there exist two continuous nonnegative functions V and W defined in a closed neighborhood of the diagonal of X x X such that
V(p,p) = W(p,p) = 0 for any p E X
and conditions (C1)-(C9) below are satisfied. In what follows, arguments of the functions V and W are assumed to be close enough, so that the functions are defined.
Our conditions are formulated not directly in terms of the functions V and W but in terms of some geometric objects defined via these functions. Our main reasoning for the choice of these form of conditions is as follows.
(1) Precisely these conditions are used in the proofs.
(2) It is relatively easy to check conditions of that kind for particular functions V and W.
Let us introduce the main objects which we work with.
Fix positive numbers a and b and a point p E X. Set
P(a, b,p) = {q E X : V(q,p) < a,W(q,p) < b},
Q(a, b,p) = {q E P(a, b,p) : V(q,p) = a}, T(a, b,p) = {q E P(a, b,p) : V(q,p) = 0}.
Denote by B(£,p) the open £-ball centered at p and set
Int0 P(a, b,p) = {q € P(a, b,p) : V(q,p) < a, W(q,p) < b},
d0 P(a, b,p) = Q(a, b,p) U {q € P(a, b,p) : W(q,p) = b}, Int0 Q(a, b,p) = {q € P(a, b,p) : V(q,p) = a, W(q,p) < b}.
Conditions (C1)-(C4) contain our assumptions on the geometry of the sets introduced above.
(C1) For any £ > 0 there exists a A0 = A0(s) > 0 such that
P(A0, A0,p) C B(e,p) for p € X.
There exists a A1 > 0 such that if p € X, 5^ 52, A < A1 and 52 < A, then there exists a number a = a (5^ 52, A) > 0 such that
(C2) Q(5^52,p) is not a retract of P(5^52,p);
(C3) Q(5b 52,p) is a retract of P(5^ 52,p) \ T(5^ 52,p);
(C4) there exists a retraction a : P(5^ A,p) ^ P, 52,p) such that
V(a(q),p) > aV(q,p) for q € P(5^ A,p).
In the next group of conditions, we state our assumptions on the behavior of the introduced objects and their images under the homeomorphism f.
We assume that for any A < A1 there exist positive numbers 5^52 < A such that the following relations hold for any p € X:
(C5)
f (P(<M2,p)) € Int0 P(A, A, f (p)), f-1(P(<M2,f (p))) € Int0 P(A, A,p);
(C6) f(T(<M2,p)) C Int0 P(^1,^2,f(p));
(C7) f (T(51, A,p)) n Q(5b52,f (p)) = 0;
(C8) f(P(5x,52,p)) n d0P(5x,52,f(p)) c Int0 Q(<M2,f(p));
(C9) f (S(51, A,p)) n P(51,52, f (p)) = 0, where
S(51, A,p) = {q € P(A, A,p) : V(q,p) > 51}.
Theorem 1 [8]. Under conditions (C1)-(C9), f has the finite shadowing property (and hence, the standard shadowing property ) on the space X.
Let us give an example of application of Theorem 1 to a diffeomorphism with nonhyperbolic behavior.
Example 1. Consider a diffeomorphism studied by Lewowicz in [9]. This diffeomorphism is a perturbation of a hyperbolic automorphism of the 2-torus T 2
Fix numbers 0 < a < 1 < 3 and a small r > 0 and define a map F : r2 ^ r2 by
F (x,y) = (ax + Kx)^AУ),3У),
where
nx
X(x) = ((1 - a) - h(s)) ds,
J0
h : r ^ r is a Cfunction such that h(0) = 0, 0 < h(x) < 1, and A(x) = 0 for |x| > r, and n : r ^ r is a Cfunction such that ^(0) = 1, n(y) = n(—y), H is not increasing for y > 0, and n(y) = 0 for |y| > r.
Let A be an integer hyperbolic 2 x 2 matrix with detA = 1. If 0 < a < 1 < 3 are the eigenvalues of A and u1 and u2 are the corresponding eigenvectors, then
A(x,y) = (ax,/y)
in coordinates whose axes are parallel to u1 and u2. The lattice E with vertices
{(n + 1/2)u1, (m + 1/2)u2 : m,n £ z}
is invariant with respect to the action of the map v ^ Av. Let n : r2 ^ r2/E be the corresponding projection of the plane to the 2-torus.
Define f : T2 ^ T2 by f (n(£,n)) = n ◦ F(x,y) (of course, we extend F periodically with repect to the above-mentioned lattice).
It is shown in [9] that if r is small enough, then f is an expansive diffeo-morphism of the 2-torus (see the definition below).
Note that the defined diffeomorphism f is not structurally stable since the eigenvalues of Df at the zero fixed point are 1 and 3.
At the same time, it is shown in [8] that conditions (C1)-(C9) are satisfied for functions V and W defined as follows: If p = (px,py) and q = (qx,qy), then V(p, q) = |py — qy | and W(p, q) = |px — qx| (such functions are properly defined if p and q are close enough). Hence, Theorem 1 is applicable to f, and f has the shadowing property.
Let us relate the obtained result to two classical properties of dynamical systems, namely, to topological stability and expansivity.
Recall the standard definitions.
Denote by H(X) the space of homeomorphisms of a compact metric space (X, dist) endowed with the metric
p(f,g) = maxmax(dist(f (p),g(p)) dist(f-1(p),g-1(p))).
p€X
A homeomorphism f is called topologically stable if for any £ > 0 there exists a neighborhood Y of f in H(X) such that if g € Y, then there exists a continuous map h : X ^ X such that f o h = h o g and
dist(h(p),p) < £, p € X.
A homeomorphism f is called expansive if there exists a positive a such that if
dist(fk(p),fk(q)) < a, k € z,
then p = q.
Walters in [11] proved that if a homeomorphism f is expansive and has the shadowing property, then f is topologically stable.
Thus, we can apply Theorem 1 to establish the topological stability of the diffeomorphism f in Example 1.
In his proof of topological stability of f, Lewowicz reduced the problem to the study of suspension flows, which does not seem natural.
In its turn, Theorem 1 can be applied to nonsmooth homeomorphisms, for example, to perturbations f of a hyperbolic automorphism of the 2-torus T2 corresponding to the map
F (x,y) = (x), ^2(y)),
where and are increasing continuous functions for which there exist numbers r, A € (0,1) such that
(1) |^(x + v) - M1(x)| < A|v| and A-1|v| < |^(y + v) - ^(y)| for |v| < r;
(2) ^i(x) = ax, |x| > r;
(3) M2(y) = ^y, |y| > r.
In this case, one can apply the same Lyapunov type functions V and W as in Example 1.
3 Shadowing near a nonisolated fixed point
The method of Lyapunov type functions can be applied in the case of nonisolated fixed points. Of course, if a fixed point p of a homeomorphism f belongs to a submanifold M consisting of fixed points (as in the case studied below), then f does not have the standard shadowing property in any neighborhood of p. Nevertheless, sometimes it is possible to establish a "conditional" shadowing property for pseudotrajectories {xk} whose points do not belong to M assuming that the size of "one step errors"
dist(xk+1, f (xk))
is small compared to the distances from the points xk to the manifold M. Such an approach (in the case of a nontransverse homoclinic point) had been suggested by S. Tikhomirov.
Let us restrict our consideration to a simple (but nontrivial) example of the 2-dimensional diffeomorphism
' x .2
f (x,y) = (X,y(1 + x2)) . (2)
The origin is a nonisolated fixed point of f (every point of the line M = {(0,y) : y £ r} is a fixed point).
Represent p £ r2 in the form p = (px,py) and consider a finite pseudotra-jectory p0,... ,pn of f such that (pk)x = 0 and
|f(pk) — p*+1| < d(pk)x, k = 0,... ,n — 1, (3)
for some d > 0. Set
K0 = {(x,y) : 0 < |x| < 1}.
In [12], the following result is proved.
Theorem 2. There exists a neighborhood K of the origin and a number c > 0 such that, for any £ > 0 and for any pseudotrajectory p0,... ,pn of f in K n K0 that satisfies conditions (3) with d = c£, there exists a point p satisfying the inequalities
|fk(p) — pk| < £, 0 < k < n.
In the proof of this result, the method of the previous section is applied; the corresponding Lyapunov type functions are
V(q,p) = |py — qy| and W(q,p) = |px qx|
Ы(1 - Ы)'
Analyzing the proof, one can see that the method is applicable not only to diffeomorphism (2) but in more general situations as well.
4 Multiscale conditional shadowing
Let X be a Banach space with norm |.|. Consider a sequence fn, n > 0, of mappings of the space X having the form
fn(x) = Anx + gn(x), (4)
where An are linear mappings.
A sequence xn € X, n > 0, is called a trajectory of (4) if
xn+1 = fn (xn), n > 0.
Let n > m > 0; set
$(n,n) = Id
and
$(n, m) = An-1 • • • Am if n > m.
We fix a family of projections Pk,n of the space X, where k € K and n > 0; here K is a countable index set having the form K = KS U KU.
It is assumed that the projections Pk,n have the following properties:
Pi,nPj,n = 0 if i = j; ^ Pk,n = Id for n > 0;
k€K
and
AnPk,n = Pk,n+1An for k € K and n > 0. (5)
Of course, property (5) implies that
$(n,m)Pk,m = Pk,n$(n,m) for k € K and n > m. (6)
Let k € K and n > 0; denote by Xk,n the image Pk,nX.
We assume, in addition, that the restrictions of An to Xk,n are invertible for k € KU; this allows us to define for k € KU the operators
^(n,m) = A-1 ••• Am-1 |Xfc,m
acting from Xk,m to Xk,n with n < m. We agree that
^(n,n + 1) = A-1|Xfc,„+1.
Formally, we have to write (n,m) instead of ^(n,m), but the present short form will lead to no misunderstanding.
Of course, an analog of property (6) is valid; if k € KU and n < m, then
^(n,m)Pk,m = Pk,n^(n,m). We make the following
Main assumption. There exists a number M > 0 and two sequences of positive numbers ak,n and ^k,n with the following properties.
The inequalities
n
^^ ||$(n,1)Pk,i|Ki < M, n > 0, (7)
keKS 1=1
and
4
^ ^||$(n,/)Pk,i||AM < 4, n > 0, (8)
kGKS 1=1
hold.
The inequalities
œ
^ ^ ||tt(n, /)Pk,i|K,i < M, n > 0, (9)
kGKU 1=n+1
and
œ i
Y, E !^(n,1)Pk,i||Am < 1, n > 0, (10)
4
kGKU 1=n+1
hold.
Of course, the choice of sequences ak,n and is arbitrary to a large extent; our future estimates essentially depend on this choice.
Fix a sequence
V = {vn G X : n > 0};
set Wk,n = Pk,nVn for k G K,
||Vn| =Y |Wk,n|, n > 0,
k€K
and
IV= sup ||vn||.
n0
(Of course, we work with sequences V for which the above values are finite). We emphasize that the value ||vn|| depends on the index n.
Let yn be a sequence of points of X with known "errors"
Sn+1 = fn(yn) — yn+1, n > 0.
Introduce functions
Yn(v) = gn(yn + v) — gn(yn).
Note that Yn(0) = 0.
Our goal is to find a trajectory xn of (4) for which we can estimate the values
|Xn — yn|, n > 0,
in terms of the values Sn (to be exact, in terms of the values |Pk,nSn| and of Lipschitz constants of the functions Pk,nYn—1).
Represent xn in the form
xn yn + vn;
then it follows from the equalities
xn+1 fn(xn) Anxn + gn(xn)
that the sequence vn must satisfy the nonlinear difference equation
vn+1 = Anvn + Sn+1 + Yn(vn). (11)
Our main result in this section is the following statement.
Theorem 3. Let conditions (7)-(10) be satisfied. Fix a positive d and assume that
|Pk,iS| < akiid, k £ K,l > 0. (12)
Assume, in addition, that Lipschitz constants of the projections Pk,iYi—1(v) satisfy the estimates
Lip Pk,iYi—1(v) < 3k,i, k £ K, l > 0. (13)
Then there exists a solution V of Eq. (11) such that
||V||TO < 4Md.
Proof. The set
V = {V : ||V||TO < 4Md}
is a subset of the Banach space of sequences V with norm ||V||TO. Define on V the operator T which takes a sequence
V = {v0,...,vn,... } £ V
to a sequence
Z = T(V) = {zn £ X : n > 0}, where zn are determined by their projections
Zk,n = Pk,nZn, k £ K,n > 0. If k £ KS, set Zk,0 = 0 and
(14)
Zk,n = ^ $(n, l)Pk,i(Si + Yi-i(vi-i)), n> 0
1=1
If k G KU, set
TO
(k,n = - ^ ^(n, l)Pk,i(Si + Yi-1(vi-1)), n > 0.
l=n+1
Recall that wk;i = Pk;ivi and note that the inclusion v G V implies the estimates
|vi | =
Pk,iVi
keK
keK
< ||vi|| < 4Md, l > 0.
(15)
Fix an index k G KS and represent
Zk,n = Pk,n^ $(n,l)(Si + Yi-1(vi-1))
i=1
^ Pk,n^(n, l)(Si + Yi-1(vi-1)) = ^ $(n, l)Pk,i(Si + Yi-1(vi-1)), n> 0.
i=1
i=1
Here we refer to property (6) of the projections. Conditions (7) and (12) imply that
£
keKS
£ $(n,/)Pk,i S
1=1
£
keKS
£ $(n,/)Pk,i PkA
1=1
<
< £ £ ||$(n,/)Pk,i|Kid < Md.
kGKS 1=1
Condition (13) combined with inequalities (15) implies that
|Pk,iYi-1(vi-1)| < Ak,i|vi-11 < 4&.1 Md, l < n. Hence, it follows from (8) that
£
kGKS
£$(n,1)Pk,i Yi-1(vi-1)
i=1
Md,
which, combined with (16), gives us the estimates
£ |Zk,n| < 2Md, n > 0.
kGKS
(16)
(17)
Similar reasoning based on inequalities (9) and (10) shows that analogs of estimates (17) are valid for k € KU as well. Thus, T maps V into itself.
Now we note that the same reasoning as in the proof of estimate (15) shows that if V, V' € V, then
|vi-1 - 1| < ||vi-1 - J < ||V - V'|
00 )
and it follows from condition (13) that
|Pk,i (Yi-1(vi-1) - 7i-1(vi-1))| <
< Ak,i|vi-1 - V-11 < AM||V - v'||
Thus,
| Ck,n Ck.n| =
£ $(n,/)Pk,i(Yi-1(vi-1) - Yi-1(v'-1))
l=1
<
<^||$(n,/)Pk>/ ||ami|v - V'
| TO 1=1
for k E KS and a similar estimate is valid for k E KU. Hence, conditions (8) and (10) imply the estimate
IIT(V) - T(V1||V - V'|U
for V, V' E V.
Thus, T is a contraction on V; hence, T has a unique fixed point V in V. Projections of coordinates of this point satisfy the equalities wk,0 = 0 and
n
Wk,n = ^ /)Pm№ + 7z-1(vz-1)), n> 0, 1=1
for k E KS and
TO
Wk,n = - ^(n, /)Pm№ + Yi-1 (vz-1)), n > 0,
1=n+1
for k E KU.
Standard calculations from the theory of Perron series show that the sequence (14) with
Vn = ^ Wk,n kEK
is a solution of Eq. (11). □
Example 2. To avoid unnecessary complications, we give an example of application of Theorem 3 with a finite index set K; the reasoning used in this example can be easily extended to the general case.
Let X = r4 and K = {1,..., 4}; fix the projections Pk;1x = (x1, 0,0, 0),
... , pkAx = (0,0,0,X4).
Assume that the matrices An in mappings (4) are constant diagonal matrices
An = diag (0,1/2,1, 2).
Take KS = {1,...,3} and KU = {4}. Fix arbitrary positive numbers
m1,..., m4.
Let k = 1; then ||$(n,n)P1jn|| = 1 and ||$(n,/)Pi,/1| = 0 for n = l. Thus, if we take a1?n = m1 for all n, then
n
£ ||$(n, 1)P1,1 < m1, n > 0. 1=1
Let k = 2; then ||$(n,1)P2)1|| = 21-n. Thus, if we take a2,n = m2/2 for all n, then
n
£ ||$(n, 1)P2,i|a2,i < m2, n > 0. l=1
If k = 3, then ||$(n, 1)P3,/| = 1, and if we take a sequence a3,n for which
n
< m3, n > 1,
l=1
then
n
£ ||$(n,1)P3,/||«3,1 < m3. l=1
Thus, we get estimate (7) in the form
n
£ £ ||$(n, 1)Pk,i||ak,i < M = m1 + m2 + m3, n > 0,
kGKS 1=1
and one can repeat a similar procedure in the general case of infinite set KS with an arbitrary series mk, k £ KS.
Finally, if k = 4, then ||^(n,/)P4j11| = 2n-1 for n < l, and, as for k = 2, we may take the constant sequence a4,n = M to get estimate (9).
After that, we can similarly select the sequences (we leave details to the reader).
5 Perturbations of dynamical systems on simple time scales
Let us consider a simple variant of a time scale T that is a subset of [0, to) and consists of isolated segments Tn, where n = 1, 2,..., Tn = [1n,rn], and 0 < 11 < r1 < 12 <____
The phase space is a Banach space X with norm | • |. We denote by || • || the operator norm of a linear operator.
The system on Tn is generated by a differential equation
x = An(t)x + an(t, x), t E Tn. (18)
We assume that the operators An(t) are continuous and bounded on Tn. The functions an(t,x) are assumed to be continuous and Lipschitz continuous in x on Tn x X (with small Lipschitz constants).
Denote by 0n(t, t0, x0) the solution of the Cauchy problem (t0, x0) for system (18).
Thus, for any t0 E Tn and for any x0 E X there exists a solution 0n(t, t0, x0) defined on the whole segment Tn (in what follows, we work with such solutions).
Let $n(t) and ^n(t) be the fundamental matrices of the system
x An(t)x
such that $n(ln) = E and ^n(rn) = E, respectively, where E is the identity map of X.
For any index n = 1, 2,... we fix a map of X taking a point x to Bnx+bn(x), where Bn is a linear operator and bn(x) is a continuous function (it is not assumed, in general, that any operator Bn is an isomorphism of the space X).
The trajectory x(t), t E T, of the appearing system starting at a point x0 E X at the time moment l1 (the left-hand end of the segment T1) is defined as follows:
• x(t) = 01(t,/1,x0), t E T1,
• x(l2) = B1x(f1) + b1(x(f1)),
• x(t) = fa(t,l2,x(k)), t E T2,
• x(l3) = B2x(r2) + b2(x(r2)), and so on.
We fix countable sets of indices KS and KU and assume that there exist families of continuous projections Pk(t), t E T, of the space X indexed by k E K = KS{J KU and having the following properties (19)-(23).
Let
P +(t) = ^ Pk (t) and P-(t) = ^ Pk (t).
kEKS kEKU
We assume that
P-(t) + P+(t) = E, t eT; (19)
P- (t)P+(t) = 0, t eT; (20)
$n(t)Pk(In) = Pk(t)$n(t) and $-1(t)Pk(t) = Pk(/n)^-1 (t),
t £ Tn; k £ KS; (21)
*n(t)Pk(rn) = Pk(t)^n(t) and ^-1(t)Pk(t) = Pk(rn)^-1(t),
t £ Tn; k £ KU; (22)
BnPk(rn) = Pk(/n+1)Bn, n > 1, k £ K. (23)
Concerning the projections Pk, k £ KU, we assume, in addition, that the following property holds: the restriction of any map Bn, n > 1, to the subspace Pk(rn)X, k £ KU, is an isomorphism of the subspace Pk(rn)X to the space
Pk (/n+1)X.
For a trajectory x(t), we denote yk(t) = Pk(t)x(t), k £ KS, and zk(t) = Pk(t)x(t), k £ KU.
Let us write down the analog of the Perron operator for the functions yk (t) and zk (t) on an interval Tn.
First we write the term of the "direct" operator which is the finite sum of summands including a1,..., an and b1,..., bn-1:
• including a1:
$n(t) f Bn-1$n-1(rn-1) ••• B2$2(r2)p1 / 1 ^1(r1)^-1(s)Pk (s)^(s, x(s)) ds) ;
r11
including a2:
$n(t) (Bn-1$n-1(rn-1) • • • $3(r3)B^ 2 $2(r2)^2-1(s)Pk(s^s, x(s)) ds^) ;
including an:
$n(t) $-1(s)Pk(s)fln(s,x(s)) ds^ ;
including b1:
$n(t)Bn-1$n-1 (rn-1) • • • B2$2(r2))Pk(/2)61 (®(r1));
• including bn-1:
$n(t)Pk (ln )bn-1(x(rn-1)).
The corresponding term of the "inverse " operator representing zk (t) for t E Tn is the infinite sum of summands including an, an+1,... and bn, bn+1,...:
• including an:
^n(t) i ^-1(s)Pk(s)an(s,x(s)) ds;
n
• including an+1:
fln+ 1
^n(t)B-1^n+1(ln+1) / ^-+1 (s)Pk(s)an+1(s, x(s)) ds;
Jrn+1
• including an+2:
Wn+2
^n(t)B-1^n+1(ln+1)B-|1^n+2(ln+2W ^-^2(s)Pk(s)an+2(s,x(s)) ds;
«^rn+2
including bn:
-^n(t)B-1Pk (ln+1)bn(x(rn));
including bn+1:
-^n(t)B-1^n+1(ln+1 )B-11Pk (ln+2)bn+1(x(rn+1));
Perturbations. The natural statement of the perturbation problem is as follows.
We replace systems (18) on the segments Tn by systems
x = Cn(t)x + cn(t, x)
(assuming that the operators Cn(t) and the functions cn(t,x) have properties similar to those of An(t) and an(t,x)) and the maps Bnx + bn(x) by similar maps Dnx + dn(x), take a trajectory £(t) of the new system and look for a close trajectory x(t) of the original system.
As usual, we are looking for functions v(t) on Tn with values in X such that
x(t)= £(t)+ v(t), t £ Tn.
From the relations
x = An(t)x + fln(t,x) = An(t)(£ + v)+ fln(t,£ + v) = £ + v = Cn(t)£ + Cn(t,£)+ V
we deduce the equations for v:
v = An(t)v + <(t,£,v), t £ Tn, (24)
where
<(t, £, v) = An(t)£ + fln(t, £ + v) - Cn(t)£ - cn(t, £),
which we represent in the form
<(t, £, v) = An(t)£ + an(t, £ + v) - an(t, £) - Cn(t)£ + a„(t, £) - cn(t, £),
and the summands of the right-hand side of the above formula have the following properties: an(t,£ + v) - an(t,£) vanishes for v = 0 and has small Lipschitz constant in v for small |v| (of course, if we impose a similar condition on ak(t,x)) and an(t,£) - cn(t,£) is small (if the perturbed system is close to the nonperturbed one).
Now we look at the "transition rule." From the equalities
x(/n+1) = Bnx(rn) + 6n(x(rn)) = Bn£ (rn) + Bnv(rn) + 6n(£ + v(rj) = £ (/n+1) + v (/n+1) = Dn£(rn) + dn(£ (rn)) + v(/n+1)
we deduce the relations
v(/n+1) = Bnv(rn) + bn(£(rn), v(rn)), n > 1, (25)
where
bn(£ (rn),v) = (Bn-Dn)£ (rn)+6n (£ (rn)+v(rn))-6n(£(rn))+bn(£ (rn))-dn(£ (rn)).
Thus, for v(t) we get system (24)-(25) similar to the original one (with the same An(t) and Bn but, of course, with different "small" nonlinear terms).
We solve this system in a standard way.
Let V be the space of continuous functions on T with values in X and with the norm
||v|| = sup max |v(t)|.
n>1 t£Tn
Clearly, V is a complete metric space with the metric p(v,w) = ||v — w||.
Our goal is to indicate conditions under which the "Perron operator" corresponding to system (24) and (25) has a fixed point in V whose norm we can control.
Main assumption. We make the following main assumption.
There exist sequences of positive numbers an,k and a number M > 0 such that
||$n(t)H (ai,k||Bn—A—i(rn—1) • • • B2$2(r2)Bi Jh ^i(ri)^——1(s)Pk(s) ds|| +
r r2
+«2,k ||Bn—l$n—l(rn—l) • • • $3(rs)BW $2^21(s)Pk(s) ds|| +
Jl 2
+ • • • + an,k || / 1(s)Pk (s) ds|| + A,k ||Bn—l$n—l(rn—l) • • • B2$2(T2))Pk (k)|| +
J ln
+ ••• + Pn,k||Pk(Wl)||) < M, t G Tn, n > 1, k G KS, (26)
and ^
||^n(t)| (an,k|| / 1(s)Pk(s) ds|| + V Jrn
Wn+l
+an+l,k||B—1^n+l(1n+l) +l(s)Pk(s) ds|| +
«^rn+l
fln+2
+«n+2,k |B-1^n+l(1n+l)B-|l^n+2(1n+2W +2(s)Pk (s) ds|| + • • • +
J rn+2
+^n,k |B—lPk (1n+1))| + ^n+1,k ||B—1^n+1 (1n+1)Bn+L1Pk (1n+2))|| +
+ ...^ < M, t G Tn, n > 1, k G KU. (27)
Theorem 4 [10]. Let conditions (26) and (27) be satisfied. Fix a positive d and assume that if |v| < 2Md, then the following estimates hold for t G Tn, n > 1, and k G K, where Lipv is a Lipschitz constant in variable v:
|Pk(t)<(t,£(t),v)| < an,kd,
|Pk(1n+l)bn(£(rn),v)| < Pnkd,
Lip, (Pk(t)<(t,£(t), v)) < ,
Lip, (Pk(i„+1)6;(£(r„),v)) < .
Then for the trajectory £(t) of the perturbed system there exists a trajectory x(t) of the unperturbed system such that
|x(t) - £(t)| < 2Md, t £ T.
Acknowledgements
The research of the author was supported by the Russian Science Foundation, grant no. 23-21-00025, https://rscf.ru/project/23-21-00025/
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