On the Exponents of Exponential Dichotomies

An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations.


Introduction
Exponential dichotomy of a linear system of differential equations is a type of conditional stability that goes back to an idea in Perron [1]. It was revealed to be a very important tool for the study of nonlinear systems because of its roughness. Indeed, it has been used to show the existence of chaotic behaviour in non autonomous perturbations of autonomous nonlinear equations having a homoclinic solution, since transverse intersection of stable and unstable manifolds along a homoclinic solution corresponds to the fact that the linearization of the nonlinear system along it has an exponential dichotomy on R [2]. Exponential dichotomies are also related with the so called reducibillty. A linear system of differential equationsẋ = A(t)x is said to be reducible if there exists an invertible C 1 matrix S(t) such that the change of variables x = S(t)y transforms the system into a block diagonal systeṁ In [3], it is proven that a system is reducible if and only if the original system has an exponential or ordinary dichotomy. The difference between the two cases is that in ordinary dichotomy the exponents are equal to zero. Another interesting property is the following (see [3]). The linear systemẋ = A(t)x has an exponential dichotomy on R + if and only if for every locally integrable function f (t), t ∈ R + , such that sup t≥0 t+1 t f (s)ds < ∞, the inhomogeneous linear systemẋ = A(t)x + f (t) has a bounded solution. Exponential dichotomies have also relations with such notions as integral separation or spectral theory, see for example [4,5]. Recently, it has been proved in [6] that if a bounded linear Hamiltonian system is exponentially separated into two subspaces of the same dimension, then it must have an exponential dichotomy.
Let us start with the definition of exponential dichotomy. A linear systeṁ where A(t) is a piecewise continuous n × n matrix, is said to have an exponential dichotomy on an interval I ⊂ R (usually R, R + , R − ) with projection P, constant k ≥ 1 and exponents α, β > 0 if the fundamental matrix X(t) of the Equation (1) (with X(0) = I) satisfies the following conditions: Here R = (−∞, ∞), R + = [0, ∞) and R − = (−∞, 0]. It follows immediately from the definition that, if α ≤ α and β ≤ β then α and β are also exponents of the dichotomy with the same projection P and constant k and also that the linear system (1) has an exponential dichotomy on an interval J ⊆ I if it has one in the interval I. Next, from Gronwall inequality it follows that, on a compact interval, the linear system (1) has an exponential dichotomy with any projection P and exponents α and β (but the constant may change).
We give few examples of systems having an exponential dichotomy. An autonomous systeṁ x = Ax has an exponential dichotomy on R if and only if all the eigenvalues of A have nonzero real parts. A periodic systemẋ = A(t)x has an exponential dichotomy on R if and only if all the Floquet exponents have nonzero real parts. A scalar equationẋ = a(t)x has an exponential dichotomy on where the limits are taken as t − s → ±∞ in case I = R ± respectively.
Suppose the linear systemẋ = A(t)x has an exponential dichotomy on R + with exponents α and β. The result that motivates this paper is the following, see [3] (Proposition 1, p. 34).
As a matter of fact in [3] (Proposition 1, p. 34), an estimate on the size of ε is also given, showing that, if β = α and ε < α x has an exponential dichotomy on R + with exponent α − 2kε. So if β = α andα =β < α we have ε = α−α 2k . We emphasize the fact that in [7] the assumptions on B(t) have been weakened to obtain a roughness result valid also for unbounded perturbations.
However, the exponents of the dichotomy determine the rate of convergence to zero of bounded solution either at ∞ (when the dichotomy is in R + ) or at −∞ (when the dichotomy is in R − ). Sometimes it becomes important to determine this rate of convergence, and hence the exponents of the dichotomy, for example when studying chaotic behaviour of discontinuous systems [8] or developing Melnikov theory for implicit nonlinear differential equations [9]. As a matter of fact in [8] the following result has been proved. Theorem 2. Letẋ = A(t)x have an exponential dichotomy on R + with exponents α, β. Then there exists ε > 0 such that if B(t) is a piecewise continuous function such that, for some T > 0, sup t≥T |B(t)| < ε and ∞ T |B(t)|dt < 1 k then the linear systemẋ = [A(t) + B(t)]x has an exponential dichotomy on [T, ∞) (and hence also on R + ) with the same exponents α, β.
Of course Theorems 1 and 2 hold equally well when the dichotomy ofẋ = A(t)x is on R − . The proof given in [8] follows an idea in [3] where an exponential estimate is derived for bounded solutions of certain integral inequalities. In this paper we want to give another, more direct, proof of the same result. As a matter of fact we work directly in the space of continuous functions decaying to zero as t → ∞ at a certain given rate. This approach leads us to derive the first of the two exponential estimates given in (2). The second is derived passing to the adjoint system and using the fact that one has a certain freedom in choosing the projection of the dichotomy (see Proposition 2).
Our method has also the advantage that relates the projection of the dichotomy of the perturbed system with the one of the unperturbed. As a matter of fact, we will give an estimate of the norm of the difference between the two projections in term of sup t∈I |B(t)|, where I = R + , R − is the interval where the exponential dichotomy is considered. This estimate allows us to prove the same result also when the dichotomy of the unperturbed system is on R, a fact that was not noted in [8].
We now briefly resume the content of this paper. In Section 2 we recall basic properties of exponential dichotomy, stable and unstable spaces, roughness, freedom in the choice of the projection etc. Section 3 is devoted to the proof of our main result. Finally, Section 4 contains applications to asymptotically constant matrices and to the linearization of nonlinear systems.
We conclude this section by giving some notations used in the paper. For a linear map L from a Banach space into another, we denote by RL and N L its range, resp. its kernel.

Properties of Exponential Dichotomies
First we start with a remark. Let ν ∈ R be a real number. Then Y(t) = X(t)e νt is a fundamental matrix of the linear systemẋ Assuming that (1) has an exponential dichotomy on I with exponents α, β, we have, for s, t ∈ I, with s ≤ t: that is (3) has an exponential dichotomy on I with the same projection P, constant k and exponents (α − ν) and (β + ν). Viceversa, if (3) has an exponential dichotomy on I with projections P, constant k and exponentsα,β, then (1) has an exponential dichotomy on I with the same projections P and constant k, and exponents α =α So, starting from a linear system with an exponential dichotomy, shifting the coefficient matrix by νI, ν = α−β 2 , we can assume that the exponents are the same.

Proposition 1.
Suppose that (1) has an exponential dichotomy of the intervals I 1 and I 2 with the same projection and exponents. Suppose, also, that I 1 ∩ I 2 = ∅. Then (1) has an exponential dichotomy of the interval I 1 ∪ I 2 with the same projection and exponents but possibly different constant.
Proof. If I 1 ⊂ I 2 or I 2 ⊂ I 1 there is nothing to prove. So we assume that I = I 1 ∩ I 2 is different from both I 1 and I 2 . We can also assume that I 1 is on the left and I 2 is on the right that is: if t 1 ∈ I 1 \ I and It is clear that (2) holds if s, t ∈ I 1 or s, t ∈ I 2 . So, let s ∈ I 1 \ I and t ∈ I 2 \ I. Taket ∈ I. Then s <t < t and we have: the proof is complete.
Since in compact intervals I = [a, b] a linear system (1) has an exponential dichotomy with any projection and any exponents, it follows from Proposition 1 that if a linear system has an exponential dichotomy on an interval [T, ∞) (resp. (−∞, −T]) then it has an exponential dichotomy with the same exponents and projection on R + = [0, ∞) (resp. R − = (−∞, 0]). Hence, in the following we will only consider When the dichotomy is on R + (or on R − ) we have some freedom in the choice of the projection. Indeed we have the following Proposition 2. [3] (p. 16-17). Suppose (1) has an exponential dichotomy on R + with projection P. Then but the kernel of P, N P, can be any complement of RP. Moreover if Q : R n → R n is another projection such that RQ = RP then there exist a constant k Q such that (2) holds with Q and k Q instead of P and k (with the same exponents). If the dichotomy is on R − then it is N P which is uniquely defined being Moreover RP can be any complement of N P and if Q : R n → R n is another projection such that N Q = N P then there exist a constant k Q such that (2) holds with Q and k Q instead of P and k (with the same exponents).
A consequence of the roughness Theorem 1 is the following. Corollary 1. Suppose the linear system (1) has an exponential dichotomy on R + [resp. R − ] with projection P and exponents α and β. Let B(t) be a matrix such that where the limit is taken at +∞ if I = R + and at −∞ when I = R − . Then, givenα < α andβ < β, the linear systemẋ = [A(t) + B(t)]x has an exponential dichotomy on R + [resp. R − ] with exponentα andβ and projectionP such that NP = N P [resp. RP = RP].
Proof. Letα andβ be as in the statement of the theorem and let ε > 0 be as in Theorem 1. It follows from the assumption the existence of T such that for t ≥ T we have |B(t)| ≤ ε and the linear systeṁ x = A(t)x has an exponential dichotomy on [T, +∞) with projection P and exponents α and β. Then from Theorem 1 it follows thatẋ = [A(t) + B(t)]x has an exponential dichotomy on [T, ∞) with exponentα andβ and projection as in the statement of the Corollary. However, we have already observed that on [0, T],ẋ = [A(t) + B(t)]x has an exponential dichotomy with the same projection and exponents. Then the conclusion follows from Proposition 1.
Example. Consider the scalar equationẋ = −1 + 1 t+1 x. The unperturbed equationẋ = −x has an exponential dichotomy on R (and hence on both R + and R − ) with k = 1, α = 1 and projection P = I. The solution of the perturbed equation with x(0) = 1 is x(t) = (t + 1)e −t and Let α < 1. The function (t + 1)e −(1−α)t is increasing on 0, α 1−α and decreasing on α for any s ≤ t. Next, observe that for 0 ≤ s ≤ t we have So the equationẋ = −1 + 1 t+1 x has an exponential dichotomy on R + with exponent α < 1 but not with exponent = 1 since otherwise there should exists k ≥ 1 such that for any 0 ≤ s ≤ t which is absurd. However, the fundamental solution of scalar equationẋ = −1 for any 0 ≤ s ≤ t. So, the scalar equationẋ = −1 + 1 t 2 +1 x has an exponential dichotomy on R + with exponent α = −1.
The difference between the two examples is that the integral of 1 t+1 in [0, ∞) is divergent whereas the integral of 1 t 2 +1 in [0, ∞) is convergent. Thus we guess that that the statement of Theorem 1 can be improved when ∞ 0 |B(t)|dt < ∞.

The Main Result
In this section we prove the following result. A similar result holds when the dichotomies are considered on R − and on R.
Proof. First, replacing A(t) with A(t) = A(t) + νI, ν = α−β 2 , we may assume that the exponents are equal. Denote them by δ. Next, consider the perturbed systeṁ Letδ < δ and take ε > 0 as in Theorem 1. Then (4) has an exponential dichotomy on R + with projection, say,P and exponentδ. We now follow the approach in [10] to construct a suitable projection for the dichotomy of the perturbed equation.
Let X B (t) be the fundamental matrix of system (4) and X(t) be the fundamental matrix oḟ x = A(t)x. A well known standard argument shows that a bounded solution of (4) satisfies the fixed point equation for some ξ ∈ R n . It is easy to see that if x(t), x 1 (t) and x 2 (t) are bounded functions then So taking ε > 0 such that 2kε < δ, we see that the map x(t) →x(t) is a uniform contraction (with respect to ξ) on the space C 0 b (R + ) of bounded continuous functions of R + . So, for any ξ ∈ R n the map x(t) →x(t) has a unique fixed point x(t, ξ) such that Note that x(t, Pξ) is the unique fixed point of and then x(t, Pξ) = x(t, ξ), because of P 2 = P and the uniqueness of the fixed point. It is straightforward to see that such a fixed point is a solution of (4) and that it is linear with respect to ξ. So We pause for a moment to observe that From the previous considerations it follows that X B (t)ξ is a bounded solution of (4) if and only if ξ = Qξ. Moreover, we have PQ = P and QPξ = x(0, Pξ) = x(0, ξ) = Qξ that is Q is a projection. Next, if ξ ∈ N Q then Pξ = PQξ = 0 and if ξ ∈ N P then Qξ = QPξ = 0. So Finally, ξ ∈ RQ if and only if X B (t)ξ = X B (t)Qξ is a bounded solution of (4). From Proposition 2 it follows, then, that Q is a projection for the dichotomy of (4). So for some K ≥ 1, or, if we go back to the original system with A(t) instead of A(t) + νI: From the previous part we know that We want to show thatδ can be replaced by δ. To this end we consider the map x(t) →x(t): or else x(·) ≤ k (|ξ| + ∆ x(·) ) and similarly So we have proved the following Proposition 3. Suppose the linear system (1) has an exponential dichotomy on R + with projection P constant k and exponents α = β = δ. Let B(t) be a matrix and suppose there exists T ≥ 0 such that such that where ε > 0 is sufficiently small and ∆ satisfies 2k∆ ≤ 1. Then for any s ≥ T the map (7) is a contraction on the set C 0 δ ([s, ∞)) and contraction constant = 1 2 . Thus its unique fixed point x(t, s, ξ) belongs to C 0 δ ([s, ∞)) and x(t, s, ξ) ≤ 2k|ξ|.
Hence we proved that for any t ≥ s ≥ T, and we extend this inequality for any t ≥ s ≥ 0 provided we change 2k with a possibly larger constant K 1 . Next, from Proposition 2, we also know that for 0 ≤ s ≤ t and possibly another constant K 2 , since we know that Q can be taken as a projection of the dichotomy of the perturbed system. Thus: (whereδ < δ) for any t ≥ s ≥ 0 and K = max{K 1 , K 2 }.
To complete the proof we still have to prove that, for T ≤ s ≤ t, it results for possibly another constant K. The fundamental matrix Y(t) = X(t) −1 * of the adjoint systeṁ has an exponential dichotomy on R + with projection (I − P * ). Indeed: for any 0 ≤ s ≤ t. From the previous part applied to the systemẋ = −[A(t) + B(t)] * x we see that a projectionQ * exists such that RQ * = RP * and where Y B (t) = X B (t) −1 * is the fundamental matrix of the perturbed systemẋ = −[A(t) + B(t)] * x. Going back to X B (t) we see that for 0 ≤ s ≤ t and a certain constant K (possibly different from the previous one, however we do not introduce other notations for these constants since at the end we can take the larger of all). From the first inequality it follows that, if ξ ∈ RQ then |X B (t)ξ| ≤ Ke −δt and hence RQ ⊂ RQ. So RQ = RQ since rankQ = rankQ * = rankP * = rankP = rankQ. Next: But in the same way we see that [RP * ] ⊥ = N P and hence As a consequence Q =Q and we have Going back to the original system (that is before the shifting from A(t) to A(t) + νI) we see that for 0 ≤ s ≤ t. This completes the proof when the dichotomy is on R + . When the dichotomy is on R − , we reduce to the case of R + by changing t with −t, X(t) with X(−t) and A(t) with −A(−t). Whenẋ = A(t)x has an exponential dichotomy on R, we apply the previous result to see thatẋ = [A(t) + B(t)]x has an exponential dichotomy on R + with projection Q + and on R − with projection Q − . Then RQ + ∩ N Q − = {0} because both projections are close to P and RP ∩ N P = {0} sinceẋ = A(t)x has an exponential dichotomy on R. The conclusion follows from [3] [p. 19], (see also [2] (Proposition 2.1)).
Let X 0 (t) be the fundamental matrix ofẋ = Ax such that X 0 (0) = I. Since −α and β are semi-simple eigenvalues, their generalized eigenspaces, that we denote with V s and V u , consist of eigenvectors of −α and β, that is for any v ∈ V s (resp. v ∈ V u ) we have X 0 (t)v = ve −αt (resp. X 0 (t)v = ve βt ). Write where V ss is the generalized eigenspace of the eigenvalues of A with real parts less than −α − µ and V uu is the generalized eigenspace of the eigenvalues of A with real parts greater than α + µ. Let d ss , d s , d u d uu be the dimensions of V ss , V s V u , V uu respectively. Let P ss : R n → R n be the projection onto V ss with kernel V s ⊕ V u ⊕ V uu , P s : R n → R n be the projection onto V s with kernel V ss ⊕ V u ⊕ V uu , P u : R n → R n be the projection onto V u with kernel V ss ⊕ V s ⊕ V uu , and P uu : R n → R n be the projection onto V uu with kernel V ss ⊕ V s ⊕ V u .
Next, V ss , V s , V u and V uu are all invariant under X 0 (s), that is for V = V ss , V s , V u , V uu . So we have, for example X 0 (t)P ss ξ = P ss X 0 (t)P ss ξ and P ss X 0 (t)(I − P ss ) = 0 because X 0 (t)(I − P ss ) = X 0 (t)P s + X 0 (t)P u + X 0 (t)P uu ∈ V s + V u + V uu . So X 0 (t)P ss = P ss X 0 (t).
Now we observe that We conclude this Section with an application of Proposition 4 to nonlinear systems. Let f (x) be a C 1 -function such that f (x) is Lipschitz with L f as Lipschitz constant. Suppose the systemẋ = f (x) has two hyperbolic fixed points x = x − and x + (that may coincide, i.e., x − = x + ) together with a heteroclinic orbit u(t), i.e., a bounded solution such that lim t→±∞ u(t) = x ± .
The fixed points being hyperbolic means that the matrices f (x − ) and f (x + ) have no eigenvalues with zero real part. Then both systemsẋ = f (x ± )x have an exponential dichotomy on R with projections, say, P + and P − . It is known that rankP ± equals the number of eigenvalues of f (x ± ) having negative real parts counted with multiplicities. Let α ± and β ± be the exponents of the dichotomy ofẋ = f (x ± )x respectively. First we observe thatu(t) = f (u(t)) is bounded and it is also a solution ofẋ = f (u(t))x. From the roughness theorem we know that this system has an exponential dichotomy on R + with exponentsα + andβ + slightly less that α + and β + respectively. Hence we get |u(t) − x + | ≤ Ke −α + t for t ≥ 0. Similarly we get |u(t) − x − | ≤ Keβ − t , for t ≤ 0. So we see that f (u(t)) − f (x + ) ∈ L 1 (R + ) and f (u(t)) − f (x − ) ∈ L 1 (R − ). A simple application of Theorem 3 gives then the following Theorem 4. Let f (x) be a C 1 -function with Lipschitz continuous derivative. Suppose there exists x = x − and x + such that f (x − ) = f (x + ) = 0 and f (x − ), f (x + ) has no eigenvalues with zero real parts. Then both linear systemsẋ = f (x − )x andẋ = f (x + )x have an exponential dichotomy on R. Let α ± , β ± and P ± be the corresponding exponents and projections. Suppose further that the (nonlinear) equationẋ = f (x) has a solution u(t) such that lim t→±∞ u(t) = x ± Then the linear equationẋ = f (u(t))x has an exponential dichotomy on both R − and R + with exponents α − , β − and α + , β + respectively, and projections Q ± such that rank Q ± = rank P ± .

Conclusions
We have given a new proof of a roughness result for linear systems with an exponential dichotomy different than the one in [8]. This new proof has the advantage that it is is more direct, can be easily extended to system having an exponential dichotomy on the whole line and gives a precise estimate on the norm of the difference of the projections of the dichotomies of the perturbed and the unperturbed system. Moreover it extends also to more general situations. Indeed the assumptions that sup t∈I |B(t)| < ε is used just to prove that the map where Γ(t, s) = X(t)PX(s) −1 if 0 ≤ s ≤ t −X(t)(I − P)X(s) −1 if 0 ≤ t < s, is a contraction on C 0 b . According to [7] this holds also under the weaker assumption that inf T>0 sup t≥0 t+T t |B(s)|ds k 1 − e −αT + k 1 − e −βT < 1 (10) and the fixed point x(t, s) satisfies again x ≤ C|ξ|, for a suitable constant C. The remaining part of the proof showing that this fixed point indeed belongs to C 0 δ just depends on the fact that |B(t)| ∈ L 1 (R + ). Hence Theorem 3 holds also under the weaker condition (10) instead of sup t≥0 |B(t)| < ε.