CC BY
3DIFFERENTIAL EQUATIONS AND
CONTROL PROCESSES N. 4, 2022 Electronic Journal, reg. N &C77-39410 at 15.04.2010 ISSN 1817-2172
http://diffjournal, spbu. ru/ e-mail: jodiff@mail.ru
Delay differential equations
Nonlinear semigroups for delay equations in Hilbert spaces, inertial manifolds and dimension
estimates
Mikhail Anikushin Saint Petersburg State University demolishka@yandex.ru.
Abstract. We study the well-posedness of nonautonomous nonlinear delay equations in Rn as evolutionary equations in a proper Hilbert space. We present a construction of solving operators (nonautonomous case) or nonlinear semigroups (autonomous case) for a large class of such equations. The main idea can be easily extended for certain PDEs with delay. Our approach has lesser limitations and much more elementary than some previously known constructions of such semigroups and solving operators based on the theory of accretive operators. In the autonomous case we also study differentiability properties of these semigroups in order to apply various dimension estimates using the Hilbert space geometry. However, obtaining effective dimension estimates for delay equations is a nontrivial problem and we explain it by means of a scalar delay equation. We also discuss our adjacent results concerned with inertial manifolds and their construction for delay equations.
Keywords: Delay equations, Nonlinear semigroups, Inertial manifolds, Dimension estimates.
1 Introduction
In this paper we consider the following class of nonlinear nonautonomous delay differential equations in Rn:
±(t) = Ax + Bf(t, C7xt), (1.1)
where xt(6) := x(t + 6), 6 £ [-t, 0], denotes the history segment; t > 0 is a constant; A: C([-t, 0]; Rn) — Rn, B: Rm — Rn and C: C([-t, 0]; Rn) — Rr are bounded linear operators and F: R x Rr — Rm is a nonlinear continuous function such that for some constant A = A(t) > 0, which is bounded in t from compact intervals, we have
|F(t,yi) - F(t,y2)| < A(t)|yi - y2| for all yi,y2 £ Rr,t £ R. (1.2)
Here and below by | • | we denote the Euclidean norm in R7 for any j > 0. For
t > s we put AS := sups<0<t A(6).
From the classical theory (that is the application of the Banach fixed point theorem), it follows that for any 00 £ C([-t, 0]; Rn) and to £ R there exists a unique classical solution x(-) = x(^,t0,00): [t0 — t, — Rn, i. e. such that xto = 00, #(•) £ C 1([t0, Rn) and x(-) satisfies (1.1) fort > t0. We define the family of solving operators U(t, s): C([—t, 0]; Rn) — C([—t, 0]; Rn), where t > s, by U(t, s)00 := s, 00), where xt(6, s, 00) = x(t + 6, s, 00) for 6 £ [—t, 0].
Consider the Hilbert space H := Rn x L2(-t, 0; Rn) with the usual product norm, which we denote by | • |H, i. e. for (x, 0) £ H we have
r 0
l(x,0)|H = M2 + J J0(6)l2d6. (1.3)
Consider the operator A: D(A) C H — H given by
(x,0) A (A^d^), (1-4)
where (x,0) £ D(A) := {(x,0) £ H | 0(0) = x,0 £ W 1'2(-t,0;Rn)}. From the monograph of A. Batkai and S. Piazzera [10] we have that A is a generator
C0 H
B: Rm —> H is defined as B^ := (B£, 0) and we define the unbounded linear operator C: H — Rr as C(x,0) := C0 for 0 £ C([-t,0];Rn). Now (1.1) can be
H
v(t) = Av(t) + BF(t, Cv(t)), (1.5)
for which we will study the question of well-posedness. In our Theorem 1 below we precisely state in what sense solutions to this evolution equation can be understood.
We put E := C([-t, 0]; Rn) and consider the embedding E c H given by 0 ^ (0(0), 0). We identify the elements of E and H under such an embedding. For any T > 0 let HT be the set of all continuous functions v: [0, T] ^ H for which there exists a continuous function x: [—t, T] ^ Rn such that v(t) = (x(t), xt) for all t > 0. We will also make the use of the following property, which is, in fact, the main ingredient of our proofs in the first part.
(MES) There is a constant MC > 0 such that for all T > 0 the inequality
T
f |Cv(t)|dt < Mc (|v(0)|h + ||v(-)Hl1(G,T;H)) . (1.6)
J 0
is satisfied for all v(-) G HT.
To explain it, consider, for example, the case of n = 1 and r = 1. Let v(t) = (x(t),xt) and (70 := 0(—t). Then we have (for convenience, let T > t)
(1.7)
"T pT i>0 pT—t
|Cv(t)|dt = |x(t — t)|dt = |x(t)|dt w |x(t)|dt <
) J 0 J—T J 0
< (l^VT) (|v(0)|e + ||v(-)M
In fact, we can use the density of Junctionals in the weak-* topology of C([—t, 0]; R) and approximations by such functional given by the Riesz representation theorem to show that (MES) holds for any operator C (with similar arguments in the case n > 1, r > 1, see Lemma 2.1 below). This simple observation besides the results contained in the present paper also led to a version of the Frequency Theorem for delay equations [2]. The L2-version of (MES) helps to prove differentiability properties of constructed semigroups (see Section 3).
One of our main results is the following.
Theorem 1. Suppose for (1.1) that F satisfies {1.2). Then for any tG G R and vG G H there is a unique generalized solution v(t) = v(t,tG,vG) to (1.5), which is a continuous function [tG, ^ H. This solution is uniquely determined by the property that the family of solving operators U(t,s)vG := v(t,s,vG); t > s7 in H agrees with the family U(t,s) on E. Moreover, the following properties are satisfied
(ULIP) For any T > 0 md s £ R there is a constant M1 = M1(T, A^+T) > 0 such that for all t £ [s, s + T] and v1, v2 £ H we have
|U(t,s)V1 - U(t,s)V2|H < M1IV1 - V2|h. (1.8)
Moreover, for allt > s + t we have U(t, s)H C E and for any T > t, s £ R and t £ [s + t, s + T] we have
||U(t,s)v1 - U(t, s)v2||e < M1|V1 - V2|h. (1.9)
(COM) The map U(t,s): E — E ¿5 compact for t > s + t and ¿/¿e map U(t, s): H — H is compact for t > s + 2t.
(REG) For v0 £ D(A) the generalized solution v(t) = v(t,t0, v0)7 t > t0; ¿5 a classical solution to (1.5), e. we have v(-) £ C 1([t0, +w);H) R C ([t0, E^ v (t) £ D(A^ dv (t) satisfies {l.ro) fo r allt > t0.
(VAR) For a// v0 £ E and t0 £ R we have the variation of constants formula satisfied for v(t) = v(t; t0, v0) and t > t0 as
v(t) = G(t - t0)v0 +/ G(t - s)BF(s,Cv(s))ds, (1.10)
to
where G(t) ¿5 th e C0-semigroup genera ted by A in H.
Remark 1. One can interpret the variation of constants formula from (1.10) for v0 £ H
The proof of Theorem 1 is given in the next section. Note that Lp-versions of (MES) (see Lemma 2.1 below) can be used to show the well-posedness in Rn x Lp(-t, 0; Rn).
A( t)
t
depend only on T (or t - s), but not on s.
Our approach for Theorem 1 is very simple and it is based on three steps. The
H
estimates and the variation of constants formula. Such results for (partial) delay equations are given in [10]. The second step is the existence of classical solutions
E
H
E is well-known fo delay equations in Rn (see, for example, J.K. Hale [21]) as well
as for some partial delay equations (see, for example, I. Chueshov [17]). The third step is the derivation of elementary a priori estimates for the norm of classical H
linear problem, (1.2) and (MES). This allows us to obtain generalized solutions by continuity. Thus, the conclusion of Theorem 1 can be easily extended to some partial differential equations with delay.
The well-posedness of nonlinear autonomous and nonautonomous (partial) delay equations in Banach spaces was studied in several papers, for example, [19, 43, 42]. The main approach in these papers is based on applications of the theory of accretive operators. Besides the fact that the theory itself is nonelementary and its applications require pages and pages of various estimates, in applications to delay equations one faces with several restrictions. For example, to apply results of G. F. Webb [43] and G.F. Webb and M. Badii [42] to ODEs in Rn with delay,
n
that is unnatural. In [42] there is also assumed some smoothness of the right-t
norms to obtain the accretiveness condition. In the paper of M. Faheem and M.R.M. Rao [19] only the case of nonautonomous delay differential equations in Rn is considered. Their main restriction is posed on the nonlinear part that must
H
t
considered a nonautonomous linear part and showed the well-posedness for the linear problem, but their assumptions on the linear part in our context allow to consider it as a nonlinear part of (1.1), i. e. (1.2) and (MES) are satisfied. Thus, our Theorem 1 covers in most and largely extends the final result of [19] as well as results from [43, 42]. Moreover, if we put A = 0 in (1.1), then the generation^ a CG-semigroup for the operator A is easier to obtain. Now, if we make F(t, Oxt) to be a linear function of xt for each t, then Theorem 1 can be considered as a
H
are other papers on the question of well-posedness for delay equations, which use more concrete approaches (see D. Breda [12] and links therein), but none of them entirely covers the result of Theorem 1 and cannot compete with the simplicity of its proof.
We also note that the studying of delay equations in the Hilbert space setting is rarely seen in works on dynamical systems. For example, recent monographs on infinite-dimensional dynamical systems (see A.N. Carvalho, J.A. Langa and J.C. Robinson [13]; I. Chueshov [17]) treat delay equations in the space of contin-
uous functions. Below we will justify advantages of the Hilbert space geometry for understanding of the dynamics of such equations.
Namely, the obtained variation of constants formula (VAR) is useful for applications of the Frequency Theorem [2, 28] (a theorem that allows to construct quadratic Lyapunov functionals, which we use to construct inertial manifolds) since it allows to pass from the nonlinear equation to a linear inhomogeneous system, which is studied in the context of the theorem*. It is also convenient for studying of almost periodic in time equations, where, as it is well-known, one should compactify the equation by considering the so-called limiting equations and check the continuous dependence with respect to perturbations of the right-hand side. We refer to our works [1, 5] for applications to such and other equations in the infinite-dimensional context. In finite-dimensions the Frequency Theorem (also known as the Kalman-Yakubovich-Popov lemma) has already proved to be useful for studying of dimensional-like properties of autonomous and almost periodic ODEs (see, for example, M.M. Anikushin [9, 7]; M.M. Anikushin, V. Reitmann and A.O. Romanov [8]; N.V. Kuznetsov and V. Reitmann [24] and references therein for a range of applications).
Theorem 1 is also convenient for studying of differentiability properties of semigroups (see Section 3) and inertial manifolds (see Section 5). We do not explicitly state these results in this introduction because strict formulations require long preparations. Along with Theorem 1 these results form a basis for parts of our adjacent works [1, 5, 3] concerned with delay equations.
A part of the present work is motivated by our previous paper [6] on the existence of invariant topological manifolds for cocycles in Hilbert spaces and especially Problem 2 posed therein, which asks for extensions of the theory for delay equations with discrete delays. At Appendix A we present such an extension of the theory, which also gives a solution to Problem 1 from [6] asking for conditions to provide continuous dependence of the inertial manifold fibres. In fact, under some natural additional assumptions these manifolds possess classical properties such as exponential tracking and normal hyperbolicity, which are proved and discussed within a more general context in our adjacent work [1]. In the present paper we discuss these extensions for delay equations in Section 5.
Concerning inertial manifolds for delay equations, here we should mention the paper of C. Chicone [15], which extends classical results of Yu.A. Ryabov and R.D. Driver, devoted to inertial manifolds for delay equations. The theory
*See [4, 2] for discussions on developments of the Frequency Theorem starting from the first infinite-dimensional version proved by V.A. Yakubovich and AX. Likhtarnikov [28].
n
equations in Rn with small delays. For comparison, our approach has some deli-cateness in applications, where the Frequency Theorem for delay equations [2] is used. In particular, it allows to construct inertial manifolds with the dimension
n
[15] can be deduced (with relaxed in some cases conditions) in most from the Frequency Theorem and our general theory from [1] (which, as we have mentioned, uses results of the present paper) after simple computations.
Our results from [1, 4, 2] allow to consider various inertial manifolds theories within a general geometric context based on the use of quadratic Lyapunov functionals. In particular, it is shown by the author in [4] that the Spectral Gap Condition, used by C. Foias, G.R. Sell and R. Temam [40] (see also the survey of S. Zelik [44]) is a particular case of some frequency inequality arising in various versions of the Frequency Theorem, which provides the existence of quadratic functionals. This frequency inequality was also used by R.A. Smith [35, 34] for his developments of the Poincare-Bendixson theory for delay equations in Rn and certain reaction-diffusion equations"!". Moreover, in the surveys of A. Kostianko et al. [23] or S. Zelik [44] it is shown that the Spatial Averaging Principle, which was proposed by J. Mallet-Paret and G. R. Sell to relax the Spectral Gap Condition in some cases, also lead to the existence of certain quadratic functionals. For more discussions in this direction see [1, 4, 2]. Note also that our geometric approach can be also applied to construct stable/unstable and local center manifolds along invariant sets.
Besides the above mentioned construction of quadratic Lyapunov functionals, an advantage may be given by possible applications of dimension estimates, which use the Hilbert space geometry, and approximation of dimension-like characteristics and spectra (see D. Breda and E. Van Vleck [11]). In this direction we prove in Section 3 the C^differentiability property for semigroups in H generated by (1.1) with F G C1 and F' globally bounded. This allows to apply well-known dimension estimates for the fractal dimension (see V.V. Chepyzhov and A.A. Ilyin [14]), the Hausdorff dimension (see R. Temam [40], J. Mallet-Paret [31]) and the topological entropy (see N.V. Kuznetsov and V. Reitmann [24]) of compact invariant sets. However, obtaining effective dimension estimates for delay equations seems to be a nontrivial problem and we try to discuss its nontriviality in Section 3. This can be seen also from rare papers on the topic, where the
+ .\oi o that R.A. Smith did not use quadratic Lyapunov functionals in the infinite-dimensional case, probably due to his inability to show their existence. Our extensions of the Frequency Theorem presented in [2, 4] show that under Smith's frequency inequality such functionals do exist.
pioneering paper of J. Mallet-Paret [31] and a paper of J.W-H. So and J. Wu [38] should be mentioned. Both of the papers represent an approach, which somehow utilizes compactness of the differentials to show finite-dimensionality (as well as in the mentioned monographs [13, 17, 21]) and does not provide any effective estimates. It seems that the authors of [38] were the first to mention some non-triviality of the problem, but they did not provide any discussions on it (referring to subsequent works that never appeared). Moreover, in [29] J. Mallet-Paret and R.D. Nussbaum studied compound processes arising from linear nonhomogeneous scalar delay equations with monotone feedback (such equations may appear after linearization). Their nontrivial results show that such compound processes for certain powers (corresponding either to odd or even-dimensional volumes growth) preserves a convex reproducing normal cone of rank 1. This allows us to treat the problem in the general context of spectral theory. Here the variational principle for subadditive functions proved by P. Thieullen [41], which allows to describe the largest uniform Lyapunov exponent of the compound cocycle as a Lyapunov exponent over some ergodic measure, is crucial for understanding of the problem. In Section 4, by means of the Suarez-Schopf model for ENSO [39], we use this approach to obtain effective conditions for the nonexistence of periodic orbits and homoclinics. To the best of our knowledge, this is the first time when the criterion of R.A. Smith [33] and its developments by M.Y. Li and J.S. Muldowney [27] are effectively applied for equations with delay.
This paper is organized as follows. In Section 2 we prove Theorem 1. In Section 3 we prove a theorem on C^differentiability of the semigroup given by (1.5) in the autonomous case (Theorems 2 and 3). Here we also consider the problem of obtaining effective dimension estimates (see Problem 1) and discuss its nontriviality. In Section 4 we consider the Suarez-Schopf model for El Niño given by a scalar delay equation, pose a problem linked with dimension estimates and give a partial solution. In Section 5 we discuss inertial manifolds and their properties, including C^differentiability (see Theorem 11), normal hyperbolic-ity and exponential tracking. Moreover, we obtain dimension estimates using some Riemannian metric, which naturally arises from the construction of inertial manifolds (see Theorem 12). In Section 6 we continue our investigation of the Suarez-Schopf model and obtain conditions for the existence of one-dimensional and two-dimensional inertial manifolds (see Theorems 13 and 14). At Appendix A we present a generalization of our main result from [6], which is, in particular, convenient for delay equations.
2 Construction of delay semigroups
At first we state here a lemma from [2], which in particular shows that (MES) is satisfied for any operator C Let T > 0 and consider the subspace ST C C([0, T]; C([-t, 0]; Rn) of all continuous functions 0: [0,T] ^ C([—t, 0]; Rn) such that there exists a continuous function x: [—t, T] ^ Rn with the property 0(t) = xt for all t > 0. The following lemma is Lemma 8 from [2].
Lemma 2.1. Let C: C([—t,0];Rn) ^ Rr be a bounded linear operator. Then there exists a constant M = M(O) > 0 such that for all T > 0 P > 1 and any 0 E ST we have
(j^ < MC ■ (||0(°)IIL,<—t,0;R.) + |x(-}ILf(0,r;K„)) 1/P . (2.1)
Using Lemma 2.1 with p = 1, we get that (MES) with MC := Mj;(1 + yT)
C CJ
Remark 3. From the Riesz representation theorem, there exists a r x n-matrix-valued function of bounded variation y(0), where 6 E [—t, 0], such that
¿70 = J (6)0(6) for any 0 E C([—t,0];Rn). (2.2)
It can be shown that M^j < Var 7, where Var is the total variation on [—t, 0] (see
Remark 4. Let us consider the space Hp of all functions 0(-) E C([0, T]; Lp(—t, 0); Rn) such to for some x(-) E Lp(—t, 0; Rn) we have 0(t) = xt for all t > 0. It is convenient to endow %T with the norm
( ) 1/p
||0(')|HT := (J|0(0)||Lp(- r>0;R") + |x(-)|LPp(0,T;M«^ . (2'3)
Now Lemma 2.1 shows that the operator %T 3 0(-) ^ IC(0(-)) E Lp(0,T; Rr), where
Ic(0(-))(s) = C0(s) for 0 e st and s E [0,T] (2.4)
is well-defined. Moreover, its norm is independent of T and p. This is the sense in which the variation of constants formula from (1.10) can be understood for any vo E H.
The following lemma is Theorem 3.23 from [10].
Lemma 2.2. The operator A given hy (I A) is the generator of a Co-semigroup G(t); t > 0, in H. In particular, there are constants Ma, k0 > 0 such that
|G(£Mh < MAeKot|vo|H for all t > 0 md vo G H. (2.5)
The following lemma is a particular case of Lemma 3.6 from [10].
Lemma 2.3. Let x: [—t, to) ^ Rn be a function such that x G
12 i" W/o'C ([—T, to); Rn). Define the history function) 0(t) := xt for t > 0. Then
0G C 1([0, +to); L2(—t, 0; Rn)) and for allt > 0 we hare^(t) G W 1'2(-t, 0; Rn)
and
d
0(t) = l2(-T, 0; Rn). (2-6)
An immediate consequence of Lemma 2.3 is the following.
Corollary 1. Let x(t) = x(t,to,0o) be the classical solution to (1.1) such that 0o G W 1,2(—t, 0; Rn). Then the function v(t) = (x(t),xt), t > t0, is a classical solution to (1.5) with v0 := v(t0) = 0o (or, more precisely, v(t0) = (0o(0),0o)). Moreover, if we put £(t) := F(Cv(t)), then v(-) also satisfies the inhomogeneous equation
V(t) = Av(t) + (t) (2.7)
t > to
v(t) = G(t - to)vo + / G(t - s)BF(Cv(s))ds (2.8)
ft 0
is valid.
Remark 5. An analog of Lemma 2.2 for parabolic or hyperbolic equations can also be proved (see, for example, Theorem 3.29 in [10]). To proceed from Lemma 2.3 to Corollary 1 we have to establish the well-posedness (=existence of classical solutions) in the space of continuous functions. This is also well-studied for partial differential equations with delay. See, for example, [17].
Now we can give a proof of Theorem 1.
Proof Let xj(t) = xj(t,to,0o,j), where t > t0 and j = 1,2, be two classical solutions to (1.1) and put Vj(t) := (x(t),xt). If we suppose that G W1'2(—t,0;Rn), then vj(•) satisfies (2.8) due to Corollary 1. From this, (2.5),
*Here, as in the previous section, xt(0) := x(t + 0) for 0 g [—t, 0].
t
(1.2) and (MES) we get
|vi(t) — V2(t)|e <
< MAeK0(t—to)|vi(to) — V2(to)|H + MaA;0+T||B|| f eKo(t-s)|C(vi(s) — V2(s))|ds <
J t0
< (Ma + MaMcAJ+t||B||)eKo(t-to)|vi(to) — V2(to)|H+
+MaMcAt;+T||B||eKo(t-to) / |vi(s) — V2(s)|Mds.
J to
(2.9)
Mi =
Mi(T, A£+t) we get
|vi(t) — V2(t)|e < Mi|vi(to) — V2(to)|h for all t E [to,to + T]. (2.10)
From (2.10) for any to E R, vo E H we can define a generalized solution v(t, to, vo), t > to, by the continuity and density of Wi,2(—t, 0; Rn) in E and H. Indeed, let a sequence vo,k E Wi,2(—t, 0; Rn), where k = 1, 2,..., tod to vo in H as k ^ Then (2.10) shows that the sequence vk(t) = vk(t,to,vo>k) t E [to,to + T], is fundamental in H for any T > 0. Its limit is the generalized solution v(t,to, vo), t > t^, which ^s independent on the choice ofvo,k. This proves the initial statement of Theorem 1 and (ULIP). Moreover for T > t > to + t we have
||vi(t) — V2(t)||E < sup |vi (t + 6) — V2 (t + 0)|h < Mi|vi(to) — V2(to)|H. (2.11)
This proves that, in fact, the sequence vk(t), t E [to,to + T], defined above is fundamental in E for t > to + t and, consequently, v(t,to, vo) E E. This proves the smoothing property stated in (ULIP). In particular, the map U(t, s) for t > s + t takes bounded sets in H into bounded sets in E, where U(t, s) coincides with U(t, s). From the Arzela-Ascoli theorem, the map U(t, s) for t > s + t takes bounded sets in E into precompact sets in E. Consequently, U(t,s): H ^ E is compact for t > s + 2t. This shows (COM). Summarizing the above, it is clear that (REG) is also satisfied. From Corollary 1 and continuity arguments we get (VAR). The proof is finished. □
3 Differentiability of delay semigroups
In this section we suppose that (1.1) is autonomous, i. e. F is independent of t. We suppose also that F E Ci(Rr;Rm). Note that from (1.2) it immediately
follows that F' is bounded. Let the assumptions of Theorem 1 hold. Then there is a semiflow ^: H ^ H given by equation (1.5), i. e. it is defined as ^t(v0) := v(t, 0, v0) for all t > 0 and v0 G H. For v0 G E one can formally write from (1.5) the linearized along the trajectory ^ (v0) equation
Putting F(t, y) := F'(C^t(u0))^, we see that (3.1) is of the form (1.5) with F changed to F. Thus, we have the following well-posedness result for (3.1), which follows^ from Theorem 1.
Lemma 3.1. For any v0 G H and £0 G H equation (3.1) has a unique generalized solution V(t) = V(t; £0; v0) such that V(0) = £0. For £0 G D(A) the solution V(t) = V(t; £0; v0) is a classical solution, i. e. V(•) G C 1([0, +to); H) Pi C([0, +to); E), V(t) G D(A) and V(t) satisfies (3.1) for all t > 0.
There is, in fact, continuous dependence of solutions to (3.1) onv0 G E that can be seen from the variation of constants formula (VAR) (see Lemma 5.1).
Let K C H be an invariant compact set, i. e. ^(K) = K for a 11 t > 0. From the smoothing property in (ULIP) and the fact that a compact invariant set consists of complete trajectories we have the inclusion K C D(A). Let us formulate the following property.
(MES*) There is a constant M£ > 0 such that for all T > 0
for all v(-) G Ht-
From Lemma 2.1 with p = 2 we get that (MES*) with M£ := (M^)2 is satisfied C CF
For the proof of differentiability properties below we need a simple observation given by the following lemma.
Lemma 3.2. Let Q: Rr ^ Rm be a globally bounded, continuous function and fix some T > 0. Then the operator ^ Q(^,(-)) is a continuous map from
§ Although formally Theorem 1 is proved for nonlinearities F (t, y), which are define d for t G R, it clearly remains true (with obvious modifications) when we have t G [0, +œ) only.
l>(i) = [A + BF"(CV (vo))C]V (i)
(3.1)
(3.2)
¿2(0,T; Rr) to ¿2(0, T; Rm)
Proof. One can prove the continuity at every point by contradiction, combining the facts that any convergent L2 sequence contains a subsequence, which converges almost everywhere, and the Dominated Convergence Theorem. □
Theorem 2. Suppose that F E Ci(Rr; Rm) and F' is globally bounded. Then for any vo E H, any T > 0 and any bounded in H subset B we have
lim |((vo + h£) — ((vo) — hV(t; vo;£)|h ^ 0 (3 3)
h-o h ,
where the limit is uniform in t E [0, T] and £ E B.
Proof. Put £(t) = ((vo + h£) — ((vo) — V(t; vo; h£). We apply the variation of constants formulas (see Remark 4) to get the representations
(/(vo + h£) = G(t)(vo + h£) + / G(t — s)BF(C(s(vo + h£))ds (3.4)
Jo
and for V(t) = V(t; vo; h£)
V(t) = hG(t)£ + / G(t — s)BF'(C(s(vo))CV(s)ds. (3.5)
o
Putting A(s) := (s(vo + h£) — (s(vo) and using the Newton-Liebniz formula, we get
F (C(s(vo + h£)) — F (CVsvo) = = (/ F' [C(«('(vo + h£) + (1 — a)((vo))] da^ CA(s). ^
Putting ^(s) = ^(s; vo, a, h, £) := C(a(s(vo + h£) + (1 — a)(s(vo)) and ^o(s) = ^o(s; vo) = C(>s(vo), we may write £(t) as
= / G(t - s)B
F'(^(s))daCA(s) - F'(C((vo))CV(s)
ds. (3.7)
Since V(s) = A(s) — £(s), we have
J(t) = G(t — s)B Jo
/ (FV(s)) — F'(Mo(s))) da 'o
+ / G(t — s)BF'(C((vo))C£(s)ds o
C A(s)ds+
(3.8)
l
i
o
l
t
For the first integral using Lemma 2.2 and the Hoelder inequality, for anyt G [0, T] we have
G(i - s)B
(FV(s)) - F'(Mc(s))) da
C A(s)
<
< MAeKoT • ||By • ^ ^ |F'(M(S)) - F'(^o(s)|2dad^ x (3.9)
if T \ !/2
2
x
|C A(s)|2ds
From Lemma 2.1 we have that ^ ^0(s) in L2(0, T; Rr) as h ^ 0+ uniformly in £ G B. Now using the Tonelli theorem, we may apply Lemma 3.2 to conclude that the limit
R(h; £,Vo):= ^ ^ |F'(m(s)) - F'(^o(s)|2dad^ ^ 0 as h ^ 0 (3.10)
is uniform in £ G B.
Since A(-) G HT (see Remark 4), from (MES*) and (ULIP) we get a constant M' = M'(T) > 0 such that
. T x
( i |CA(s)|2ds ) < M'|h£|h. (3.11)
For the second integral in (3.8) we have
G(i - s)BF'(C^s(vo))C^(s)ds
< MAeKoT • ||B|| • C^(s)ds. (3.12)
Since 6(0) = 0, from (MES) we get that /J |C6(s)|ds < MC/J |6(s)|Mds. Combining this and (3.9), (3.11), (3.12) with (3.8), we get for some constant M'' = M''(B,T,vo) > 0 that
16(t)|h < M'' • R(h; £,vo) • |h£|h + M''• i |6(s)|Mds. (3.13)
Now applying the Gronwall inequality, we finish the proof.
□
Let K be a compact invariant set. As in [40, 14], we say that the family ^ is quasi-differentiable on K w. r. t. the family of quasi-differentials L(t; v) G L(H), t 0 v t 0
sup
V1,V2 GK |vi-V2|e<£
I^M - ^t(vi) - L(i; vi)(v2 - vi)|
H
|vi - V2I
—> 0 as £ —> 0 +
(3.14)
H
i
t
t
and the following properties are satisfied
(QD1) sup ||L(t; v)|| < to; ie[0,i]
vGK
(QD2) L(t + s; v) = L(t; f s(v))L(s; v) for all t, s > 0 and v GK.
Theorem 3. Suppose that F is independent oft, F G C 1(Rr; Rm) anrf F' ¿5 globally bounded. Consider a compact set K C H7 which is invariant w. r. t. the semiflow f\ t > 07 given by equation (1.5). Then the family f is quasi-differentiable w. r. t. the family of quasi-differentials L(t; v) given by
L(t; v)£ := V(t; v;£), (3.15)
where V(t) = V(t; v0; £) ¿5 ¿/¿e solution to (3.1) with V(0) = £. Moreover, the map v0 ^ L(t, v0) ¿5 continuous as a map from % to L(H) for a 111 > 0.
Proof. If the linear operator L(t; v): H ^ H is defined as in (3.15), Theorem 2 guarantees that (3.14) is satisfied for all t > 0 since it shows that L(t; v0) is the Frechet differential of (at v0. From (ULIP) of Theorem 1 it follows that L(t; v) is a bounded linear operator and (QD1) is satisfied. The property in (QD2) is equivalent to
V(t + s; v; £) = V(t; fs(v); V(s; v; £)) for all t,s > 0,v G K,£ G H, (3.16)
but this is just the uniqueness of solutions to (3.1) given by Lemma 3.1. To show that the map H 9 v0 ^ L(t; v0) G L(H) is continuous for every t > 0 we define ¿(t) := L(t; v)£ — L(t; v0)£ = V(t; v; £) — V(t; v0; £) for some fixed v0 G H and arbitrary v G H. Let also £ G H be such that |£|H < 1. Then £(•) satisfies the equation
¿(t) = r G(t — s)B(F'(Cf s(v))CV(t; v; £) — F'(Cf sM)CV(t; v0; £))ds = ./0
= / G(t — s)B(F'(Cfs(v)) — F'(Cfs(v0)))CV(t; v0; £))ds+ ./0
+ / G(t — s)BF'(Cfs(v))C£(s)ds. 0
(3.17)
Using the Gronwall lemma argument as in Theorem 2, one can show thatv ^ v0 implies that ¿(t) ^ 0 uniformly in £ with |£|H < 1. But this gives the desired continuity. The proof is finished. □
The continuity of the map v0 ^ L(t; v0) plays an important role in dimension estimates. It is shown by V.V. Chepyzhov and A.A. Ilyin [14] that this property is the only missing ingredient that makes the Hausdorff dimension estimate obtained by P. Constantin, C. Foias and R. Temam [40] hold for the fractal dimension also. This property is also essential for Theorem 4 below.
KE the topologies of E and H on K coincide. Moreover, from (ULIP) we get that K is an image of itself under the Lipschitz map ^>2t. Therefore, the Hausdorff and fractal dimensions of K w. r. t. the metrics of H and E coincide. Applying a theorem of J. Mallet-Paret [31] (see also Section 4.6 in [24]) to the compact Frechet differentiable map ^>2t we get the following theorem.
K
is finite.
Moreover, from [14] we can show finiteness of the fractal dimension that is not less than the Hausdorff dimension (see Remark 3.6 in [24]). For this we need some preparations.
For a compact linear operator L: H ^ H let a1 > a2 > a3 > ... be its singular values. For d = k + s, where k is a non-negative integer and s G (0,1],
L
From Theorem 2.1 in [14] applied to the map ^ for some fixed t > 2t we get the following theorem.
Theorem 5. Under the hypotheses of Theorem 3 suppose that for somet > 2t d > 0
Then for the fractal dimension ofKin H we have dimF K < d.
The main result of [14] and the above definitions can be also formulated for
L
L(t; v) for t > 2t allows us to stay in the compact context, which is simpler. Since L(t; v) is a compact operator for t > 2t, its singular values tend to zero and therefore wd(L(t; v)) < 1 is satisfied for all sufficiently large d > 0. Using the compactness of K and the continuity of L(t,v) (and, consequently, wd(L(t; v))) w. r. t. v G K, one can show that for every t > 2t the inequality in (3.19)
:= ai • ... • • ak+i = ^ ^+1.
(3.18)
supv)) < 1.
vGK
(3.19)
is satisfied for some sufficiently large d > 0. Thus, we immediately have the following theorem which strengthens Theorem 4.
K
finite.
In particular, from Theorem 6 and the Lipschitz property of the semiflow we immediately have finiteness of the topological entropy of (restricted to K (see, for example, [24]).
It is interesting to obtain effective dimension estimates and this seems to be a nontrivial problem. Let us start a discussion by formulating the following analog of the Liouville trace formula for (3.1).
Lemma 3.3. Under the hypotheses of Theorem 3 let k > 0 be an integer and v0 G H be fixed. Then for any £j G D(A) and Vj(t) = V(t; v0,£j\ where j = 1, . . . , k t > 0
|Vi(t) A ... A Vk (t)|f h =
if* \ (3'2°) = l£i A ... A £k|fH exp ^ Tr((A + BF'(C(M)C) ◦ n(s))dsj ,
where n(s) is the orthogonal projector onto Span(V1(s),... , Vk(s)).
Here /\k H denotes the k-th exterior power of H endowed with the inner product | • lfk H which is given for n A ... A nk G /\k H by the k-dimensional volume of the parallelepiped with edges n1,..., nk (f°r a more precise treatment see, for example, [40]). Since £j G we have that Vj(•) is a classical solution of
(3.1) and, in particular, Vj (t) G D(A) fo all t > 0. Thus, the operator under Tr s
particular, its trace is well-defined. After these explanations Lemma 3.3 can be proved in a standard way (see, for example, Subsection 2.3 of Chapter V from [40]).
As above, let L: H ^ H be a compact operator. It is well-known (see Propo-
k>0
have
^k(L) = sup |L£1 A ... A L£k |fk H = suP |L£1 A ... A L£k |fk H ,
iCj |e<1Vj iCj |e<1Vj
(3.21)
where the last inequality is due to the density of D(A) in H.
Using (3.21) and Lemma 3.3, one usually provides some estimates for the trace of the operator in (3.20), which makes it possible to apply Theorem 5. In the case of parabolic problems there is a general estimate for the trace of a positive self-adjoint operator (see Chapter VI in [40] or Lemma 4.21 in [13]), which in some cases (for example, for reaction-diffusion equations) overrides the effect
F
dimension estimates. Linear operators corresponding to delay equations are not self-adjoint and thus it is not obvious how one may derive effective dimension estimates with the use of Theorem 5 in the general case. As we will see below, a naive approach concerned with the study of the symmetrized operator (A + A*)/2 does not work here, although it has proved to be effective for ODEs (see [24]). We are familiar only with the paper of J.W-H. So and J. Wu [38], where dimension estimates for delay equations through the trace estimates are considered (in [38] a class of parabolic equations with delay is studied). However, [38] ends with an abstract application of the Constantin-Foias-Temam estimate (in the same way we did in Theorem 6) and, as it is said in the introduction, its concrete realizations are left for forthcoming papers, but none of such papers seems to be appeared. Another example is the monograph of A.N. Carvalho, J.A. Langa and J.C. Robinson [13], which in particular treats dimension estimates, contains a chapter devoted to delay equations, but there are no discussions concerned with effective dimension estimates for such equations. Thus, we consider it important to draw attention to the following problem.
Problem 1. How to obtain effective dimension estimates for delay equations?
A partial solution to the above problem can be given by construction of inertial manifolds [1, 6, 15, 40]. However, the existence of an inertial manifold only gives an integer estimate given by its dimension and such an estimate is too rough to apply, for example, some criteria of non-existence of periodic orbits based on dimension estimates (see R.A. Smith [33]; M.Y. Li and J.S. Muldowney [27]). Such criteria can be mixed with various developments of the Poincare-Bendixson theory for infinite-dimensional dynamical systems (see [1, 30]) to obtain convergence properties. On the other hand, the existence of smooth inertial manifolds at least theoretically leads to a certain ODE given by a smooth vector field (the so-called inertial form), which describes dynamics on the manifold. The linearized vector field can be symmetrized to obtain effective dimension estimates. To the best of our knowledge, the only case, to which this approach can be efficiently applied, is given by delay equations with small delays, for which C. Chicone in [15] obtained an expansion for the vector field by the delay value considered as a
small parameter.
Note that in the case of ODEs there is an effective approach called the Leonov method (see N.V. Kuznetsov [25]; G.A. Leonov et al. [26]; N.V. Kuznetsov and V. Reitmann [24]) concerned with changes of the metric tensor via special Lyapunov-like functions. Sometimes it allows to obtain sharp (or even exact [26]) estimates for the Lyapunov dimension (and, consequently, for the Hausdorff and fractal dimensions) on the global attractor. To the best of our knowledge, there is still no efficient applications of the Leonov method for infinite-dimensional systems. Its development with the synthesis of the above mentioned criteria for the nonexistence of periodic orbits [33, 27] has particular interest.
Let A: D(A) C H ^ H be a closed linear operator. Let L c D(A) be a finite dimensional subspace and nL be the orthogonal (in H) projector onto L. The trace of A on L is the value Tr(A o nL). Let us define the trace numbers A (A), A2(A),... by induction from the relations
A(A) + ... + Ak (A) = sup Tr(A o nL), (3.22)
LcD(A) dim L=k
where k = 1, 2,.... In Theorem 7 below the trace numbers are related to the eigenvalues of a self-adjoint extension S of the symmetrized operator S0 := (A + A*)/2. In the case of delay equations such operators as S are defined everywhere H
cases, one can see that the trace numbers do not depend on the delay value and their sum can never be negative that makes them inappropriate for studying delay equations.
Let us illustrate the problem by means of the following example. We consider for a G (0,1) and t > 0 the operator A in H = R x L2(-t, 0; R) given by
(x,0) ^ (V(0) - a0(-T(3-23)
Its domain D(A) consists of all (x,0) G H such that 0 G W 1,2(-t, 0; R) and 0(0) = x. Such an operator arises in the example from Section 4.
Lemma 3.4. For the operator A given by (3.23) we have A (A) = anc^ Ak (A) = 0/or all k > 2
Proof. Let L c D(A) be any k-dimensional subspace. Let e = (0«(0), 0«(-)) G
D(A), where i = 1, 2,..., k, be an orthonormal basis for L. Then we have
k k /3 1 \
Tr(A o nL) = ^(Aei, ei)M = ^ i ^0*(0)2 - a0*(-T)0i(0) - ^0i(-T)2 I .
¿=1 ¿=1 ^ '
(3.24)
Let us consider the case k = 1. From (3.24) we have
ßi(A) = sup (30(Ö)2 - a0(-T)0(ö) - 10(-t= 2 /3 0(-t ) 1
= sup 0(0) • - — a
2 0(0) 2
(3.25)
where the first supremum is taken over all 0 £ W 1,2(-t, 0; R) such that |0(0),0(^)|H = 1 and the second has the additional constraint 0(0) = 0. Since 0(0)2 < 1 and the maximum value of the quadratic polynomial -1/2x2 — ax+3/2 is (3 + a2)/2, we have A (A) < (3 + a2)/2. But it is easy to construct a sequence of functions 0(-) showing that (3 + a2)/2 is indeed the value of A (A).
In the case k = 2 let us observe that the orthonormal basis e1, e2 for L can always be chosen such that 01(0) = 0. Then from (3.24) we have
A (A) + &(A) = sup(-201(-T)2 + 202(0)2 - a02(-T)02(0) - 202(-T)2) ,
(3.26)
where the supremum is taken over all 01, 02 £ W1,2(-t, 0; R) such that 01(0) = 0, |0«(0), 0«(-)|m = 1 and f°T 01(^)02(0)d0 = 0. It is clear that the supremum do (3 + a2)/2
(assuming also that 01(-t) = 0), which shows that it is, in fact, equal to (3 + a2)/2. Thus, &(A) = 0.
If k > 2 then we can choose the orthono rmal basis e1,..., ek in such a way that 0j(0) = 0 for i = 1,..., k - 1. From similar as above arguments we have that ^(A) = 0 for k > 2. □
A
eigenspace, it can be seen from the proof of Lemma 3.4 that for the operators corresponding to delay equations the supremum in (3.22) is not achieved on any nice subspace and the limiting subspace consists of discontinuous functions.
Remark 6. Below, we prove Theorem 7, which shows that "the limiting subspace" is an eigenspace for a self-adjoint extension of the symmetrized operator So :=
(A + A*)/2. Let us illustrate this for A from Lemma 3.4. In this case So can be extended to a bounded self-adjoint operator S in H given by
It is clear that the trace numbers Ak(A) calculated in Lemma 3.4 coincide with
S
Note also that there are two leading real eigenvalues A1 > 0 and A2 < 0 of the A
A1 + A2 < 0
the space of parameters (t, a) where two-dimensional volumes are squeezed. But it seems impossible to reveal this region from the trace formula (3.20) without directly referring to the spectral decomposition.
Thus, it seems that the trace numbers Ak(A), which are useful for studying parabolic problems and ODEs, are not appropriate for delay equations.
We finish this section by proving the following theorem.
Theorem 7. Let a closed operator A be such that D(A) R D(A*) is dense in H. Suppose the operator S0 = (A + A*)/2, which is defined at I east onV(A) RD(A* )7
S
D(S) D D(A) and such that its eigenvalues ak(S) can be ordered by non-increasing. Then for any k = 1, 2,... we have Ak(A) = ak(S).
Proof. For breviety, let (•, •) denote the inner pro duct in H. Let us firstly prove that we have the equality
Indeed, the equality in (3.28) holds for any e G D(A) R D(A*). But then we have an analogous identity for the corresponding bilinear forms as
Now take instead of v a sequence vm G D(A) R D(A*) converging to some v G D(A) in H as m ^ Using the equalities (Avm, w) = (vm, A*w), (Svm, w) = (vm, Sw) and D(S) D D(A), one can show that (3.29) holds with v G D(A) and, as a consequence of the symmetry, w G D(A). Now taking v = w = e G D(A) in (3.29) yields (3.28).
(3.27)
(Ae, e) = (Se, e) for all e G D(A).
(3.28)
(Av,w) + (v, Aw) = 2(Sv,w) for all G D(A) R D(A*). (3.29)
By definition, ^i(A) + ... + (A) is given by the supremum over all linear k-dimensional subs paces L c D(A) of the value Tr(A o nL). Le t L be fixed and let e1,..., ek be its orthonormal basis. In virtue of (3.28) we have
k k k Tr(A o nL) = ^(Aei, ei) = ^ (Sei, ei) < ^ ai(S), (3.30)
¿=i i=i i=i
where the last inequality is due to the variational principle for self-adjoint operators as in Lemma 2.1, Chapter VI from [40]. This inequality becomes an equality if L is the eigenspace corresponding to a1(S),..., ak(S). The proof is finished. □
It is, however, interesting, whether the sum from (3.22) can be negative for some k in some nontrivial cases arising from delay equations. We do not know of any such example.
4 Example
We consider the Suarez-Schopf model [39] for El Nino-Southern Oscillation (ENSO), which is given by the following scalar delay equation:
X(t) = x(t) - ax(t - t) - x3(t), (4.1)
where a £ (0,1) and t > 0 are parameters. Let us put 7 := y/1 + a and define the sets CR := {0 £ C([-t, 0]; R) | ||0||TO < 7 + R} for R > O We also use CR to denote the interior of CR. If a semiflow in C([-t, 0]; R) ^s given, by w(0o) we denote the w-limit set of 0o £ C([-t, 0]; R) w. r. t. this semiflow.
Let us show that (4.1) is dissipative and generates a semiflow in C([-t, 0]; R). This is contained in the following lemma.
Lemma 4.1. The set CR is positively invariant w. r. t. solutions of (4.1). In particular, (4.1) generates a semiflow ^ t > 0, in C([-t,0];R). Moreover, w(0o) c Co for all 0o £ C([-t, 0]; R).
Proof Since the closure of a positively invariant set is positively invariant and the intersection of positively invariants sets is also positively invariant, it is sufficient
0
to show that CR is positively invariant for all R > 0. Suppose the opposite, i. e.
0
there exist an initial condition 0o £ CR and a time t > 0 such that for the classical
0
solution x(-;0,0o) we have ^s(0o) = £ CR for all s £ [0,t) and |x(t)| = 7 + R.
From (4.1) we have
x(t) < y + R + a (7 + R) - (7 + R)3 < 0, if x(t) = R + 7, x(t) > -(7 + R) - a(7 + R) + (7 + R)3 > 0, if x(t) = -(R + 7).
that leads to a contradiction.
Since any solution does not leave CR for some R > 0, it remains bounded and therefore can be extended for times up to (see Theorem 3.1, Section 2.3 in [21]). Thus, (4.1) generates a semiflow in C([-t, 0]; R).
Now suppose there is 00 G C([-t, 0]; R) such that u(00) C C0. Then there is R > 0 such th at u(00) C CR and u (00) C CR-e for all 0 < £ < R. Since u(00) cannot entirely lie on the boundary of CR (due to similar as in (4.2) arguments), we can find 0 < £1 < R and 01,02 G u(00) such th at 01 G CR_ei and 02 G CR with ||02|| = y + R. Then there is a time moment t0 > 0 for which (0(00) is close to 0;l , namely, (0 (00) G CR-£o for some £0 < s^ But since CR-£o is positively invariant we have that ft(00) is separated from 02 for a 111 > t0 that contradicts to 02 G u(00). Thus, u(00) C C0 and the proof is finished. □
It should be noted that from a theorem of J. Mallet-Paret and G.R. Sell [30] it follows that for (4.1) the u-limit set of any point 00 satisfies the Poincare-Bendixson trichotomy, i. e. it can be either a stationary point, either a periodic orbit or a union of a set of stationary points and homoclinic and heteroclinic orbits connecting them. However, numerical experiments in [39, 3] show that there is a region, for which there are no periodic orbits (see below).
An elementary analysis shows that there are three stationary states 0+ = VT-a, 00 = 0 and 0- = -y/1 - a. There is also a one-dimensional unstable manifold for 00 for any parameters a G (0,1) and t > 0. When a and t are relatively small, the stationary states 0+ and 0- are asymptotically stable. These parameters correspond to the region of linear stability in [39], for which no periodic orbits were observed. However, in our paper [3] we numerically found hidden and unstable periodic orbits in (4.1), arising after homoclinic bifurcations, for parameters from the region of linear stability which are close to its boundary (called the neutral curve in [39]). Moreover, in [3] we justified that these parameters (unlike the parameters from the region of linear instability) are related to the irregular nature of ENSO since the discovered multistability is sensible to exterior periodic forces or noise.
Since the semiflow generated by (4.1) is dissipative, there is a global attractor K C C0. To make the developed theory applicable, let g: R ^ R be a function
which coincides with x3 on [-(7 + R),7 + R] for some R > 0 and it is smoothly extended outside of [-(7 + R), 7 + R] in such a way that g £ C 1(R; R) and g' is globally bounded. Clearly, the set K will be also invariant for the semiflow ^ generated by (4.1) with x3 changed to g(x). As before, we consider K as a subset of H = R x L2(-t, 0; R) and K is also invariant w. r. t. the corresponding semiflow in H given by Theorem 1 which we also denote by
Let vo £ K, where vo = (yo,0o). Let y(t) = x(t;0,xo) be the classical solution of (4.1) with y(0) = yo. For £ £ D(A), where £ = (zo,0o) such that 0o £ W 1,2(-t, 0; R) and 0o(0) = zo, we consider the solution V(t) = V(t,vo,£) of (3.1), which in our case can be described as a solution to
Here V(t) = (z(t),0(t)) and V(0) = (zo,0o).
Let L(t; v), where t > 0 and v £ K, be the quasi-differentials given by Theorem 3 for It is clear that the zero stationary point 0o belongs to K. From a naive look at (4.3) one may suggest that (due to the presence of the term -3y2(t)z(t)) the squeezing of d-dimensional volumes at 0o (where y(t) = 0) is
K
presence of delay makes it not so obvious and we can only conjecture this (see below).
The roots of the linear part of (4.1) are given by
It can be verified that if a £ (0,1) and t > 0 then there are always one positive real root A1? one negative real root A2 and the others roots are located to the left from A2. From the dichotomy of autonomous linear systems (see Theorem 4.1, Chapter 7 in [21]) it follows that A1 + A2 < 0 indicates the squeezing of 2-dimensional volumes at the zero stationary state 0°. From the above observations we pose the following problem.
Problem 2. Let A1 = A1(a,T) and A2 = A2(a,T), where a £ (0,1) and t > 0, be the positive and the negative real roots of (4.4). Is it true that the condition A1 + A2 < 0 excludes the presence of periodic orbits and homoclinics in (4.1)?
Remark 7. It can be shown that A1 + A2 < 0 is equivalent to
¿(t) = z(t) - az(t - t) - 3y2(t)z(t),
d
<Kt) = ^(t).
(4.3)
1 - ae-Tp - p = 0.
(4.4)
(4.5)
A1 + A2 < 0
(a,T), which is included (but strictly smaller) in the region of linear stability
A1 + A2 > 0
A1 + A2 < 0
(but strictly smaller) in the region of true stability (with a gradient-like behavior) numerically obtained in [3]. Thus, [3] provides numerical evidence that the answer to Problem 2 should be positive.
At the end of Section 3 we justified that straightforward applications of the Liouville trace formula (3.20) are not appropriate to study delay equations.
Let us present another viewpoint onto the above problem. For this we consider the 2nd compound cocycle, say S(2) = {S(2)(q, •)}, where t > 0 and q G K,
associated with (4.3). This cocycle S(2) acts in the 2-nd exterior power /\ E of E = C ([—t, 0]; R) and it is deter mined by the formu la /\ E 9 A £2 ^ S(2)(q,6 A £2) := L(t; q)6 A L(t; q)& G A2 E for 6,6 G E. Note that A2E can be identified with the anti-symmetric functions in C([-t,0]2) (see [29] for details). Let us consider the largest uniform Lyapunov exponent ofS(2) over K given by
lnsupgGK|^(2)(q, •)| b1 :=limsup-—-. (4.6)
t^+TO t
Then for every £ > 0 there exists a constant M > 0 such that
sup ||S(2)(q, •)! < Me(6l+e)t for t > 0. (4.7)
qGK ( )
Clearly, b1 < 0 indicates the uniform squeezing of two-dimensional volumes over K. Thus, one may ask how to estimate b1. This can be done via the variational principle for subadditive functions as in the following theorem.
Theorem 8. In the above introduced notation, the largest uniform Lyapunov exponent b1 of the cocycle S(2) over the global attractor K can be realized as
61 = ln |M№|, (4.8)
a
where ^ and are the first two Floquet multipliers 0ver a certain a-periodic
1
Proof. Below, when referring to certain results for discrete-time systems, we consider the discrete cocycle A(q, n) := S[2)(q, •) in A C([-t, 0]; R) over the home-omorphism' f (•) := (•^n K. Note that the operators A(q,n) are compact.
^For the uniqueness of backward extensions of solutions (which gives the homeomorphism property of ^ restricted to the global attractor) see Theorem 4.1, Section 3.4 in [21].
Let ^ be an ergodic Borel measure for f on K. Since recurrent points lying in the support of any Borel invariant measure are typical (see item 1, Proposition 4.1.18 in [22]), the support of ^ must contain stationary or periodic points of the semiflow ^ due to the Poincare-Bendixson trichotomy (Theorem 2.1 in [30]).
By Lemma 2.3.5 in [41], one can take ^ such that b1 is realized as a Lyapunov exponent for Let qo = 0o be any stationary or periodic point in the support at which the Lyapunov exponent exists. Without loss of generality, we may assume that qo is a-periodic for some a > 0 If and are the first two Floquet multipliers (their existence is guaranteed by Theorem 5.1 in [29]), then ab1 < ln But it is clear that aln < b1. This finishes the proof. □
Thus, Theorem 8 reduces Problem 2 to that any periodic orbit must be asymptotically stable (in the sense < 1) in the considered region.
For a real number r we consider the roots A1r) = A1r)(a,T), A2r) = A2r)(a,T),
...
ity of the equation
1 - 3r2 - ae-Tp - p = 0. (4.9)
The following theorem provides an estimate for It highly relates on
the monotonicity results for S(2) from [29].
Theorem 9. Let yo(-) be a a-periodic solution of (4.1) and let ^ an d be the first two Floquet multipliers. Let ro := maxt£[o,a] |yo(t) |. Then for any r > ro we have
1 Ca
6 • - (r2 - y0(s)) ds + Re Alr) + Re A2r)
ln IM1M2I < a
ao
where A1r) and A2r) are the first two of roots of (4.9).
(4.10)
Proof Let qo £ K be the point corresponding to yo(-). Let Mq0 :=
S[2)(qo, •) ^e the monodromy operator over qo for the 2nd compound cocycle S(2). Note that the linearization along the orbit is given by
z(t) = (1 - 3yo2(t))z(t) - az(t - t). (4.11)
We will compare Mq0 with the time-a map for the 2nd compound process, say Mr, generated by
Z(t) = (1 - 3r2)z(t) - az(t - t). (4.12)
After the changes of variables given by w(t) = exp ^J0(s) — 1)dsj z(t) and
w(t) = exp ^/Qt(3r — 1)ds^ z(t) equations (4.11) and (4.12) transfer respectively into
w(t) = —bi(t)w(t — T ), w(t) = — b2(t)w(t — T ).
Note that bi(t) = a0 /t_T(3yQ(s) — 1)ds and b2(t) = a0 //_T(3r2 — 1)ds. Let Mq0
and Mr denote the time-a maps of the corresponding 2nd compound processes in f\ C([—T,0];R). Since b1 < b2, by repeating arguments from the proof of Proposition 5.3 in [29], we have that 0 < Mq0 < Mr in the sense of the partial ordering given by a closed convex reproducing normal cone (the cone K2 from formula (4.7) in [29]). By Proposition 5.7 in [29] we have that p(Mq0) < p(Mr), where p(-) denotes the spectral radius.
Now note that p(Mq0) = exp (2 /Q7(3y2(s) — 1)ds) p(Mq0) and p(Mr
exp (2 f0(3r2 — 1)ds) p(Mr). Since p(Mq0) = and
p(Mr ) = ea(Re A1r)+Re A2r)), (4.14)
we obtain the desired conclusion. □
Remark 8. One can also obtain an estimate similar to (4.10) for any even number m of the Floquet multipliers ..., as
ln • ... • Mm| < J
1 ra m
3m-y (r2 - y0(s))ds Akr)
(4.15)
From the point of view given by Lemma 4.10, it is required to provide a bound for periodic orbits. It turns out that the bound y/1 + a given by Lemma 4.1 is not appropriate due to its roughness. The following lemma provides sharper estimates for the region containing the global attractor K for certain parameters.
Lemma 4.2. Suppose that ar < 1/2 and put
C(a<T):=r—ari(1—a)-V1—a' (4-16)
Let R1 be the unique positive root of — p3 + (1 — a)p + C(a; t) = 0 and put
R := C(a, t) • max{(1 — aT)—1, (aT)—1}. (4.17)
Then the global attractor K lies in the set C(R1; R2) defined by
C(R1; R):= {0 G C1 ([ T, 0]; R) | ||0||« < R1,||0'IU < R}. (4.18)
Remark 9. Note that for the radius R1 from Lemma 4.2 we always have R1 > \/1 - a. Moreover, in the region aT < 1/2 the estimate for the radius of dis-sipativity given by the lemma significantly improves the estimate \/1 + a from Lemma 4.1, especially for larger a since R1 ^ y/1 - a as a ^ 1-.
Proof The proof is similar to Lemma 4.1, although contains more calculations.
Let us consider the set C(R1; R2) as in (4.18) for certain R1 and R2 (not necessarily
0
the ones specified in the statement of the lemma). Let C(R1; R2) be the set
determined by strict inequalities and let us check its "positive invariance" for
0
certain initial conditions and values R1 and R2. Suppose that 0o £ C(R1; R2).
We suppose that the solution x(t) = x(t; 0o) satisfies x(0) = 0o(0) and there
0
exists to > 0 such that xt £ C(R1; R2) for all t £ [0,to) and xto £ C(R1; R2). There are only two possible cases (when one of the strict inequalities is violated).
Case 1: |x(to)| = R1. It is sufficient to assume that x(to) = R1 and |x'(t)| < R2 for t £ [0,to], From (4.1) we have
x(to) = R1 - R3 - ax(to - t). (4.19)
Note that x(to - t) > R1 - tR2. Thus, to get the impossibility of our situation it is sufficient to require that
(1 - a)R1 - R? + aTR2 < 0. (4.20)
Case 2: |x' (to)| = R^. It ^s sufficient to assu me that x'(to) = R2 and |x(t)| < R1 for t £ [0,to], From (4.1) we have
rto
R2 = x(to) - ax(to -t) - x3(to) = (1 - a)x(to) + a x(s)ds - x3(to). (4.21)
Jt o-t
In particular, we have R2 < (1 - a)x(to) - x3(to) + aTR2. For x(to) > 0 we have that _
(1 - a)x(to) - x3(to) < (1 - a) • • 3 (422)
and, consequently, it is sufficient to require
R2 > (aT)-1 • C(a; t) (4.23)
to get the desired impossibility. On the other hand, if x(to) < 0, it is sufficient to require that
R2 > (1 - aT)-1 • (-(1 - a)R1 + R3). (4.24)
Let us take R0 > 0 and R2 := (aT) 1 • C(a,T) + R0. Then (4.20) takes the form
(1 — a)R1 — R1 + C(a, t) + aTRo < 0. (4.25)
Moreover, for R2 := (1 — aT)—1(—(1 — a)R1 + R3) + R0 (4.20) takes the form (1 — a)R1 — R3 + aT • (—(1 — a)R1 + R3) + aTRo < 0 (4.26)
1 — aT
Note that aT < (1 — aT) is equivalent to aT < 1/2. Thus, for any R0 > 0
there exists R1 = R1(R0) such that both (4.25) and (4.26) are satisfied and the
0
corresponding region C(R1,R2) will be positively invariant in the above given sense if we take R2 = max{(aT)—1 • C(a, t), (1 — aT)—1(—(1 — a)R1 + R3)} + R0. Note that as R0 ^ 0+ the value R1(R0) can be taken such that it tends to the value defined in the statement of the lemma.
Now the final statement about the global attractor can be shown analogously to the corresponding part of Lemma 4.1. □
Now we use the obtained in Theorem 4.2 sharper bounds for the global at-K
parameters.
Theorem 10. Suppose that aT < 1/2 and for the radius R1 from Lemma, 4-2 we have
6 • R1 + Re A1Rl) + Re a2R) < 0. (4.27)
Then b1 < 0 and there are no periodic orbits and homoclinics in (4.1). Moreover, any point tends to one of the stationary states 0070+ 0—.
Proof. Indeed, (4.27) and (4.10) imply that for any periodic orbit the first two Floquet multipliers satisfy < 1. Note that aT < 1/2 implies that A1 + A2 < 0 for the roots of (4.4) and that 0+ and 0— are asymptotically stable. Thus, Theorem 8 implies that b1 < 0 and we have the uniform decay for two-dimensional
K
Due to the smoothing estimate in (ULIP) from Theorem 1, one can show that an analog of (4.7) holds also in the norm of H and from (3.21) we get that (3.19) is satisfied for d = 2 and sufficiently large t > 0. From Theorem 5 it follows that dimF K < 2. Then Corollary 2 from [27] implies that there are no periodic
K
2.1 in [30]) guarantees that the w-limit set of any point must be a single stationary point. The proof is finished. □
Fig. 1 shows the region determined by (4.27), however it is much smaller than the expected region considered in Problem 2.
Figure 1: A numerically obtained region in the space of parameters (t, a) of system (4,1), where
(4,27) is satisfied (orange) and (4,5) is satisfied (orange and blue),
5 Inertial manifolds
For v e E and t > 0 let us défi ne ^(v, £ ) := L(t, v)£, where L(t, v)£, as in Section 3, is a solution to the linearized equation (3.1). From Theorem 2.2, Chapter 2 in 211, we immediately have the following lemma.
Lemma 5.1. Let F e C1(Mr;Mm) and F' be globally bounded. Then the map (t, v, £) ^ i^(v, £) is continuous as a map from M+ x E x E to E.
Now let A c E be an invariant w. r. t. the semiflow p from Section 3 finite-dimensional topological manifold such that the semiflow is invertible on A and, consequently, in virtue of the Brouwer theorem on invariance of domain, it defines a flow on A. We also put ^(v) := p^Cv) for t e M, v e A mid Q := A. Then (Q, $) is a flow. It is obvious that we have the cocycle property satisfied as
= ^(r(v),^s(v,£)), for all t,s > 0, v e A,£ e E. (5.1)
From this and Lemma 5.1 it follows that ^ is a cocycle in E over the base flow $ = p in Q = A. This allows us to apply results from Appendix A.
Now we suppose that F e C1 with F' globally bounded and the semi-p
from Appendix A, i. e. there exists an operator P e L(E; E*) such that for
V(v) := (v, Pv) and some 6 > 0, v > 0 we have for all v^ v2 E E and r > 0 (in terms of (H3) here we also put tv = 0 for convenience) that
pr
e2vrV(pr(vi) - pr(v2)) - V(vi - v2) < -6 / e2vs|ps(vi) - pr(v2)|Hds (5.2)
Jo
and there exists a splitting E = E+ © E- with dim E- = j and such that P is positive on E+ and negative on E-. Moreover, E+ and E- can be assumed to be V-orthogonal in the sense that V(v) = V(v+) + V(v-) for all v E E, where v = v+ + v- is the unique decomposition with v+ E E+ and v- E E- (see
P
Theorem (see [2] for a proof and detailed discussions which especially concern delay equations).
Remark 10. In this section we discuss inertial manifolds in the classical phase space E (of course, we can work directly in H). However, it seems crucial (at least from the point of view given by the Frequency Theorem [2]) to consider the weaker norm | • |H in the right-hand side of (5.2).
We will use the V-orthogonal projector n: E ^ E- defined by Pv := v-.
From Theorem 1 we have properties (ULIP) and (COM) satisfied for p and
j
manifold A C E, on which the semiflow is invertible". Our aim is to show that we may pass to the limit in (5.2) to get the corresponding inequality for the cocycle (0, $) given by the linearization of ^ on A. Below we always assume that F E C 1(Rr; Rm) and F' is globally bounded.
Lemma 5.2. Let the semiflow ip satisfy (H1),(H2) and (H3) as in (5.2). Then for every £ E E; v E Am d r > t we have
r
e2vrV(L(r; v)£) - V(£) < -W e2vs|L(s; v)£||ds. (5.3)
Jo
Proof. Let £ E E and v E A ^e Consider (5.2) with v1 := v + and
v2 := v, divide both sides by h2 and take it to the limit as h ^ 0 to get (5.3). Such a passage is justified by Theorem 2. □
Remark 11. In fact, the restriction r > t for (5.3) is unnecessary if we use differentiability results for in E (see Theorem 4.1, Section 2.4 in [21]).
"This follows from the fact that an amenable trajectory passing through a given point on A is unique [6, 1]. Note that the continuity of the inverse to ^ can be also shown without any appealing to the Brouwer theorem
A
Thus, the cocycle which is obtained through the linearization of p on A, also satisfies the hypotheses of Theorem A.l, which gives us a family of amenable sets for v e A, which we denote by A'(v). Clearly, A'(v) is a j-dimensional subspace of E. Let us consider the maps $: E- ^ E and $: A x E- ^ E, where $(nv) = v for all v e A and $(v, n£) = £ for all v e A and^ e A'(v). Clearly, $(v, Z) is linear in Z so we will usually write $(v)Z instead jDf $(v, Z). One should think of A' (v) as a tangent space to A at v and think of $ (v) as the differential of $ at Z = nv. We shall not give a proof of this, referring the interested reader
to [1]. From this one can obtain the following theorem.
p
$: E- ^ E is C1 -differentiable and $'(Z) = $($(Z)) for all Z e E-. Moreover, A is a C1 - differentiable subman if old in E; its tangent space at an y point v e A is given by A'(v); the flow p on A is C^smoo^ and the differential of pt at v e A
is given by the map L(t; v): A'(v) ^ A'(pt(v)) for a 111 e M.
C1 A
under the classical Spectral Gap Condition (which is included in our theory as it is shown in [4]) for semilinear parabolic equations and to obtain the differentiability of higher orders one needs a more restrictive condition (see S.-N. Chow and G.R. Sell [16]; R. Rosa and R. Temam [32]). Results in the theory of normally
A
Cfc-smooth if (H3) is satisfied for two parameters > v1 (with possibly different constants, operators and spaces, but with the same j) such that v2/v1 > k. However, such large spectral gaps are rarely seen in nonlocal applications of the theory.
A
property.
(HYP) There exists 0 < v' < v such that the semiflow p satisfies (HI) with a possibly different operator P, (H2) with the same j and (H3) with v changed to v' and possibly different constants 5 and tv.
P
v
A
v
and it is normally hyperbolic under (HYP). This, in particular, answers the questions of R.A. Smith [34] on the differentiability and normal hyperbolicity
posed for reaction-diffusion equations (the geometric context is also applicable for such problems [4]). Let us give brief details of the constructions. Namely, for each vo E A there is the stable fib re over vo given by
r +
Ast(vo) := {v E E I e2vs|ps(v) - ps(vo)|H < (5.4)
o
Let n+ := Id -n be the complementary projector. Then one can show that n+: Ast(vo) ^ E+ is a homeomorphism and Ast(vo) is a C^differentiate E+-submanifold in E. These stable fibres foliate E and, consequently, for every v E E there exists a unique nc(v) E A such that v E Ast(nc(v)). The continuous nonlinear map nc( • ): E ^ A is called the central projector and the triple (E, A, nc) is a fiber bundle. Analogously, for the linearization cocycle 0 one can define the vo E A
r
AJf(vo):= {£ E E I e2vs|L(s; vo)£|H < (5.5)
o
It can be verified that E = A^(v) © A'(v) fo any v E A. Such a decomposi-
A
projectors nv, where RannV = A'(v) and KernV = A^f(v). These constructions does not depend on the exponents from (HYP) and it can be shown that for some constant M > 0 we have for all t > 0 £ E E the inequalities
||L(t; v)(Id -nV)£||e < Me-vt||£||E,
||L(-1; v)nV£||e < Mev't|£|E. '
For j = 2 these properties allow to extend results of R.A. Smith concerned with the Poincare-Bendixson theory [35, 34], isolated periodic orbits [36] and the Poincare index theorem [37].
From the V-orthogonal projector n we obtain a chart on A. However, to
A
A
projector na in E admissible if for any nonzero v E Ran na we have V(v) < 0
and for any nonzero v E Kerna we have V(v) > 0. Under the conditions of
Theorem 11 it turns out that na: A ^ Ranna is also a homeomorphism and
na: A'(v) ^ Ranna ^s an isomorphism for any v E A. Consequently, due to
the C^differentiability of A and linearity of na, the inverse map is also C1-
differentiable and globally Lipschitz. Moreover, transition maps between such
Ci
The following lemma describes the dynamics of p on A and its linearization L(t; v) on A'(v) by ordinary differential equations in E-. Equation (5.7) is called the inertial form, of p on A. We give a sketch of its proof based on the above introduced arguments.
Lemma 5.3. Let the semi flow p satisfy (H1),(H2) and (H3) as in (5.2). Suppose there exists a spectral projector for A which is admissible. Then
1. There is a one-to-one correspondence between the trajectories of the flow p A E-
C(t) = n [A$(C(t)) + BF(C$(C(t)))] =: f (C(t)), (5.7)
where the vector field f: E- ^ E- is C1 -differentiate. This correspondence is given by the identities $(£(t; (o)) = p*($(Co)) and npt($((o)) = ((t; (o) for all t E Ran d(o E E-.
2. There is a one-to-one correspondence between the trajectories of the cocycle 0 and solutions of the following ODE in E-:
T)(t) = n[A + BF'(Cpt(v))C]$(pt(v))n(t) =: Ai(pt(v))n(t), (5.8)
where the linear operator AL(v): E- ^ E- depend continuously on v E A and norms of AL(v) are uniformly bounded inv E A. This correspondence is given by the identities $(pt(v))n(t; no; v) = L(t; v)<£(v)no and nL(t; v)$(v)no = n(t; no; v) for a,lit E R v E A and no E E-.
f
the projector n has nothing to do with the operators A and B and,
f
given by n. Due to this, it is convenient to use different charts on A given by other projectors.
By the hypothesis, there exists a spectral projector na for A which is admissible. Note that such a projector can be defined everywhere in H (since A is an operator in H) and commutes with A. Thus, naA = Ana and naB are bounded linear maps and in the chart given by na the conclusions of Lemma 5.3 are obvious. Then the arguments before lemma show that the same holds in any
V
n. □
The following theorem is a generalization of Corollary 2.2 from [33], where the case of ODEs in Rn with j = n is considered. The main idea is to use the
quadratic form -V(•) restricted to each tangent space A'(v) as a Riemannian A
L(t; v): A'(v) ^ A'(pt(v)) in this metric. This lower bound can be used to
estimate the product of the first l < j singular values through the product of j
independent of metric changes.
Theorem 12. Under the conditions of Lemma 5.3 suppose thatK is an invariant compact and for some d e [0, j] we have
(j - d)v + Tr(AL(v)) < 0 for all v e K, (5.9)
where Al(-) is defined in (5.8). Then dimF K < d.
Proof It is sufficient to estimate the fractal dimension of nK, which is an invariant
E-
5.3 it follows that the vector field f: E- ^ E- from (5.7) is C^smooth and its derivative at Z e E- is given by AL($(Z))• For any t > 0 and Zo e E- put L(t; Zo) := nL(t; $(Zo))$ '(Zo) Clearly, L(t; Zo): E- ^ E- is the differential at Zo of the map E- 9 Zo ^ Z(t; Zo) e E-. Let us endow the space E- with any scalar product. Let a1(t; Zo) > ... > (t; Zo) denote the singular values of L(t; Zo) w. r. t. this scalar product. From the Liouville trace formula for (5.8), we have
^(t; Zo) • ... • *j(t; Zo) < exP Tr AL(ps($(Zo)))d^ . (5.10)
t > 0 Zo e EE- defined trough the quadratic forms (n,n)1 := -V( $'(Zo)n) and (n,n)2 := -V( $'(Z(t; Zo))n) for n e E-. Let E1 and E2 be the space E- endowed with scalar products (•, •)1 and (•, ^)2 respectively. Let a'(t; Zo) > ... > aj(t; Zo) denote the singular values of L(t; Zo): E1 ^ E2. By the Courant-Fischer-Weyl min-max principle, we have for k = 1,..., j that
( > u t ^^2 • f (L(t; Zo)n,L(t; Zo)n)2 ^^ (t; Zo)) = sup inf -----, (5.11)
k JJ wfc cE- newfc ,n=o (n,n)1
where the supremum is taken over all k-dimensional subs paces Wk of E-. From (5.3) with £ := $'(Zo)n r := t and v := $(Zo) we have
(L(t; Zo)n,L(t; Zo)n)2 ^ —2vt
> e-2vt. (5.12)
(5.13)
From this and (5.11) we have ak(t; (o) > e-vt. Moreover, from (5.11) and the minmax principle for ak it is also clear that there exists a constant C = C (t; (o) > 0 such that ak(t; (o) > C(t; (o)a'(t; (o) for all t > 0 and (o E E-. Moreover, since K is compact and $'((o) depend continuously on (o, there exists a constant CK > 0 such that ak(t; (o) > CKak(t; (o) > CKe-vt for a 111 > 0 and (o E nK. From this and (5.10) for any k = 1,..., j we have
O (t; Z ) O (t; Z ) a1(t; Co)- ...'aj(t; Co) ^ (t; Co) = ak+1(i; Co)- ...-Oj (t; « <
< C-(j"k) exp QT [(j - k)v + Tr(Ai(ps($(Co))))] ds) .
Now let d = k + 5, where k is a nonnegative integer and 5 E (0,1]. Since
a1(t; Co) ■... ■ °k(t; Co) ■ ak+1 = = (a1(t; Co) ■... ■ °k(t; Co))1-* ■ (°1(t; Co) ■... ■ °k+1(t; Co))<5,
from (5.13) we have
a1(t; Co) ■... ■ °k(t; Co) ■ (°k+1(t; Co))<5 <
< C-(j-d) exp f f [(j - d)v + Tr(AL(ps($((o))))] d^ .
(5.14)
(5.15)
If (5.9) is satisfied then the right-hand side of (5.15) tends to0 as t ^ But this implies that dimF nK < d (see Theorem 2.1 from [14]) and since K = $(nK) and $ is a Lipschitz map, we get dimF K < d. □
6 Example (continued)
Let us consider again the Suarez-Schopf model (4.1). Here we will provide conditions for the existence of one-dimensional and two-dimensional inertial manifolds in the model. These manifolds are actually perturbations of the eigenspaces corresponding to the leading characteristic roots A1 and A2 at the zero equilibrium 0°. Conditions for persistence of such eigenspaces are given in terms of the so-called transfer function of the linear part (see [2]) and take the form of an inequality (called frequency inequality), which generalizes the Spectral Gap Condition [4]. In the case of (4.1) written in the form (1.5) the transfer function W(p) = C(A - pi)-1B is given by
W (p) = ----. (6.1)
J 1 - ae-TP - p v ;
Here we assumed that A0 = 0(0) -a0(-t), B = -1, C$0 = 0(0) and F(y) = y3.
Let A be the Lipschitz constant of x3 on [-R1,R1^ where R1 is defined in Lemma 4.2, that is A = 3R2.
Theorem 13. Suppose that aT < 1/2 and for so me v e (0, - A2) the frequency inequality
Re W(-v + iw) + A-1 > 0 for a 11 w e M (6.2)
K
C1
The proof is standard: we truncate** the nonlinearity x3 outside a small K
P
K
v>0
K
K
0o
K
of parameters of interest lies in the ball of radius 1. Let A3 and A4 be the first pair of complex-conjugate roots of (4.4).
Theorem 14. Suppose for some v e (-A2, - Re A3) the frequency inequality
Re W(-v + iw) + 1/3 > 0 for a 11 w e M (6.3)
1
C1
Fig. 2 shows a region where the conditions of Theorem 13 or Theorem 14 are satisfied. Note that a part of the blue region can be obtained also using the rough estimate \/1 + a from Lemma 4.1 to bound the global attractor K. This allows to rigorously justify the existence of two-dimensional inertial manifolds containing K
periodic orbits can be observed. We expect that the relative simplicity of the Suarez-Schopf model may allow to construct sharper regions and, consequently, rigorously justify the existence of inertial manifolds for the interesting parameters.
**Note that it is important to preserve the condition g'(x) > o.
ft For this we consider the quadratic form (in terms of [2]) given by F (v, £) = £ • (ACv - £), wher e £ e S := R, v = (^(o),^)and Cv = ¿(o).
0.25 0.50 0.75 1.00^- 1.25 1.50 1.75 2.00
Figure 2: A numerically obtained region in the space of parameters (t, a) of system (4,1) for which the conditions of Theorem 13 (orange) or Theorem 14 (blue) are satisfied.
Funding
The reported study was funded by the Russian Science Foundation (Project 2211-00172).
A An addition to the reduction principle
Let Q be a metric space and f&t: Q ^ Q, where t G R be a flow on Q. Let E be a real Banach space. We say that the family of maps ^(q, •): E ^ E, where q G Q and t > 0 is a cocycle in E over the driving system (Q, $) if
1. ^°(q,v) = v and ^t+s(q,v) = ^t(^s(q), ^s(q, v)) for all t, s > 0 v G E and q G Q.
2. The map (t, q, v) ^ ^t(q, v) is continuous as a map R+ x Q x E ^ E.
E
H which is assumed to be Hilbert for convenience We identify elements of E and H under such an embedding. Let (v, f) denote the dual pairing between v G E and f G E*.
Let P: E ^ E* be a bounded linear operator, i. e. P G L(E; E*). We say that P is symmetric if (vi,Pv2) = (v2,Pvi) for all vi,v2 G E. For a subspace
L c E we say that P is positive (respectively, negative) on L if (v,Pv) > 0 (respectively, (v, Pv) < 0) for a 11 v e L with v = 0.
(HI) There is P e L(E; E*), which is symmetric and such that E splits into the direct sum of some subspaces E+ and E-, i. e. E = E+ © E-, such th at P is positive on E+ and negative on E-.
(H2) For some integer j > 0 we have dim E- = j.
(H3) For V(v) := (v, Pv) and some numbers 5 > 0 v > 0 tv > 0 we have e2vrV(0r(q,v1) - 0r(q,v2)) - e2v1 ^ (q,^) - (q,v2)) <
/r
e2vs|0s(q,v1) - 0s(q,v2)|Hds (A.l)
satisfied for every v1, v2 e E, q e Q and 0 < l < r such that r - l > tv.
Note that due to the cocycle property it is sufficient to require that (H3) holds for l = 0 and any r > tv. A more general case when P and v depends on q e Q is considered in [1].
E-
dimensional and V(•) is of constant sign on it, one can find the V-orthogonal E- E
form (v1, -Pv2) defines an inner product in E-. Every v e E gives rise to a continuous linear functional on E- as (-,Pv). By the Riesz representation theorem, there exists a unique element nv e E- such that (w,Pv) = (w,Pnv) for all w e E-. Clearly, n e L(E;E). Put := Ker(1 - n). It is easy to verify that E = E±,V © E- and P ^s positive on E±,V. Thus, we can always assume that the subspaces E+ and E- from (HI) are V-orthogonal in the sense that V(v) = V(v+) + V(v-), where v = v+ + v- is the unique decomposition with v+ e E+ and v- e E- We say that n is the V-orthogonal projector onto E- E = H V
the use of the continuous functional calculus for bounded self-adjoint operators considered in [6].
Let v(-): M ^ E be a continuous function. We say that v(-) is a complete trajectory of the cocycle if there exists q e Q such th at v(t + s) = 0t(^s(q), v(s)) for all t > 0 and s e M ^n ^^^s case we say that v(-) is passing thro ugh v(0^t q (or simple, v(-) is a complete trajectory at q). Under (H3) a complete trajectory
v(-) is called amenable if
r0
/ e2vs|v(s)|Hds< (A.2)
J — TO
Let us consider the following assumption. (S) There is a number tS > 0 and there is a constant CS > 0 such that
||0temb(q,vi) — 0temb(q,v2)||E < CS|vi — v2|H for all q G Q and vi,v2 G E.
(A.3)
Note that in the case E = H considered in [6] we have (S) automatically satisfied with tS = 0 and CS = 1.
In the construction of invariant manifolds by the methods of [6] a compactness assumption is required as follows.
(COM) There exists tcom > 0 such that the map 0tcom(q, •): E ^ E is compact for all q G Q.
Let A(q) be the set of all amenable trajectories at q. A generalization of Theorem 1 from [6] can be given as follows.
Theorem A.l. Let the cocycle 0 in E satisfy (H1),(H2), (H3), (S) and (COM). Let n be the V-orthogonal projector onto E— (see Remark 12). Then for any q G Q either A(q) is empty or the map nq := n]^^ : A(q) ^ E— is a homeomorphism.
A proof can be given following the same arguments as in [6] and using the above given remarks. A proof for the more general case, where P and v depends q G Q
a number tv > 0 in (HI) (in [6] there was no such a number), which may be convenient sometimes. Convergence theorems for periodic cocycles given in [6] can be also generalized to this setting (see [1]). Along with the Frequency Theorem from [2], which covers delay equations and allows to construct operators as in (H3), this satisfactorily solves Problem 2 from [6].
Let the conditions of Theorem A.l be satisfied and the sets A(q) be not empty for all q G Q. Then we can consider the map $ : Q x E— ^ A(q) defined as $(q, Z) := $q((), where $q : E— ^ A(q) is the inverse to nq = n|A(q) : A(q) ^ E—. For every fixed q the map $(q, •) is a homeomorphism between E— and
A(q) due to Theorem A.l. In [6] it was posed a problem (see Problem 1 therein), whether the map $: QxE- ^ E is continuous. Here we present an answer, which extends the case of semiflows and periodic cocycles studied in [6] and covers many interesting cases arising in practice (see Remark 13 below).
(BA) For any q e Q there is a bounded in the past complete trajectory w*(-) at q and there exists a constant Mb > 0 such that supt<0 ||w*(t)||E < M for all q e Q. <
We say that $: Q x E- ^ E is uniformly compact if $(C, B) is precompact in E for any precompa ct set C C Q and bounded set Be E-.
Theorem A.2. Under the hypotheses of Theorem A.l suppose in addition that (BA) holds and the map $ is uniformly compact. Then $: Q x E- ^ E is continuous.
Proof. Let qk, q e Q and Zk, Z e E-, where k = 1, 2,..., and qk ^ q ^ Z as k ^ Let vk(^),v(^) be amenable trajectories at qk and q respectively and
such that $(qk, Zk) = vk(0) and $(q, Z) = v(0). Then from (H3) for l < —tv we have
f o
V(vk(0)-wqk(0))-e2vZ V(vk(l)-wqk(l)) e2vs|vk(s)-wqk(s)||ds. (A.4)
If k is fixed, a subsequence lm, m = 1, 2,..., lm ^ -to as m ^ can be chosen^ such that e2vZmV(vk(lm) - wqk(lm)) ^ 0 as m ^ Putting l = lm in (A.4) and taking it to the limit asm ^ we get
f o
-¿-1V(vk(0) - wqk(0)) > e2vs|vk(s) - wqk(s)|Hds (A.5)
J -TO
Since E+ and E- are V-orthogonal, we have -V(vk(0) - wqk(0)) < ||n|| • ||Zk -nwqk (0)|E- Using the uniform boundedness of wqk we have
n0 \ 1/2 / n0 \ 1/2
I ^ e2vsk(s) - wqfc> (y ^ e2vs|vk(s)|Hdsj / r0 \1/2 / r0 \1/2 M
I ^ e2v>qk(s)|Hd^ >(J ^ e2>k(s)|HdsJ - -.
(A.6)
^Since vk(•) and (•) are amenable, we have f e2vs||vk(s) - ( s)||j|ds < To get Zm one may
apply the mean value formula to the previous integral on [Z - 1, Z] for Z = -1, 2,....
From this, (A.5) and (S) there exists a constant M (independent of k) such that
— r°
M > e2vs||vk(s)llEds. (A.7)
J — TO
Now suppose that <«(qk, Zk) = vk(0) does not converge in E to v(0) = <«(q,()• Then for a subsequence (we keep the same index) we have that |vk (0) — v(0)|E is separated from zero. Let us for every k consider the integral in (A.7) on segments [1 — 1, /], where I = —1, —2,.... Using the mean value theorem, we get a family of numbers G [1 — 1, 1] such that vectors vk (tkl)) are bounded in E uniformly in k and the bound depends on I. Taking a subsequence, if necessary, we may assume that t^ ^ t as k ^ for some tl G [1 — 1,1]. For a fixed I let us consider C := {#4° (qk) | k = 1, 2,...} and B := {nvk (tkl)) | k = 1, 2,...}. Clearly, C is precompact and B is bounded. Then vk(tkl)) lies in the precompact (in E) set <<(C, B). Thus, there exist a converging subsequence. Using Cantor's diagonal argument, we may assume that vk(tkl)) converges in E to some vl G E as k ^
for all I = —1, —2,____ Let us define vl(t) := (q),vl) for t > tl. From
the continuity of the cocycle we have that vk(t) ^ ^(t) as k ^ for t > tl. This implies that vl—i(t) = vl (t) for all I and all t > t^. For any t G R define v*(t) := vl(t), where 1 such that t > t^. Then v*(•) is a complete trajectory at q and vk(t) ^ v*(t) in E for a 111 G R. From (A.7) it follows that v*(^) is amenable. Since nvk(0) = Zk ^ nv*(0) and Zk ^ Z, we must have nv*(0) = But since v(^) and v*(•) are both amenable at q and n: A(q) ^ E— is a homeomorphism, we must have v(-) = v*(^) and, consequently, vk(0) ^ v(0) in E. This leads to a contradiction. □
Remark 13. Let us discuss the conditions of Theorem A.2. Assumption (BA) is natural in the following two cases. The first one is when (Q,$) is a minimal
almost periodic (in the sense of Bohr) flow. In this case this condition will be
q
the limit dynamics is nontrivial. The second case is when the cocycle is linear (as
q
The assumption of uniform compactness for <£> is also natural. It will be satisfied if we assume that assumptions (UCOM) and (ULIP) below hold. Thus, this condition is also natural.
(UCOM) There exists Tucom > 0 such that for any precompact set C C Q and any bounded set B C E the set 0Tucom (C, B) is precompact in E.
(ULIP) There exists ts > 0 such that for any T > 0 there is a constant LT > 0 such that for all vi, v2 E E and q E Q we have
- ^t(q, V2) ||e < LT|vi - v2|h for t E [ts,ts + T] (A.8)
and also
||^(q,vi) - ^t(q, V2) |e < C?||vi - v2||e for t E [0,T]. (A.9)
Lemma A.l. Under the hypotheses of Theorem A.l suppose in addition thatA(q)
q E Q
Then $ is uniformly compact.
Proof. Consider a precompact set C C Q and a bounded set Be E. Consider any sequence v0,k E $(C x B), wher e k = 1, 2,..., for which we shall find a converging subsequence. Then v0,k = $(qk, Zk) for some qk E C, Zk E B and there are amenable trajectories (•) at qk such that vk (0) = v0,k. We may assume that qk converges to some q as k ^ Similarly to (A.5) from (ULIP) for some
constant M > 0 we get
/0 /•—T
e2vs|vk(s) - <(s)|Hds > e2vs|vk(s) - <(s)||ds =
-TO J —T-1
= e2vsk|vk(Sk) - <(Sk)|H > e-2v(T+1)(C+i)-2|K(-rwcom) - <(-TUCOm)||E,
(A.10)
where t = max(rs, Tucom} and sk E [-t - 1, -t] are some numbers. From this and (BA) we get that vk(-Tucom) lies in some bounded set B' and, consequently, vk(0) E (C', B'), where C' = (qk) | k = 1, 2,...}. Then by (UCOM)
one can find a converging subsequence of vk(0) = v0,k. The lemma is proved. □
Lemma A.2. Under the hypotheses of Theorem A.l suppose in addition that (ULIP) is satisfied. Let A(q) be nonempty for some q E Q. Then : E- ^ E
q E Q
particular, A(q) is a Lipschitz submanifold in E.
Proof. Similarly to (A.5) for any two amenable trajectories v^-) and v|(•) at q we have
r 0
r iP|| • ||nvi(0) - nv2(0)||E > e2vs|v i(s) - v£(s)||ds >
,-ts " (A-U)
> e*>i(s) - v2(s)|H = e2vSo|viM - vj(S0)|H
-TS-1
(A.12)
$ q(Zi) - $ q(C2)||e = IK(0) - Vj(0)he < CTs+iev(Ts• HCl - C2^e.
(A.13)
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