CONVERSE THEOREM ON A GLOBAL CONTRACTION METRIC FOR A PERIODIC ORBIT

. Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the ﬂow are contracted, then there exists a unique periodic orbit, which is exponentially stable and we obtain an upper bound on the rate of exponential attraction. In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the upper bound on the rate of exponential attraction.

1. Introduction. The stability and the basin of attraction of periodic orbits provide important information in many applications. However, already the determination of a periodic orbit is a non-trivial task as it involves solving the differential equation. There are several methods to study the stability and the basin of attraction of periodic orbits, among them Lyapunov functions and contraction metrics. A Lyapunov function, as well as the classical definition of stability, requires the knowledge of the position of the periodic orbit which in many applications can only be approximated. Contraction analysis, on the other hand, does not require us to know the location of the periodic orbit.
Throughout the paper we will study the autonomous ODĖ where f ∈ C σ (R n , R n ) with σ ≥ 1. We denote the solution x(t) with initial condition x(0) = x 0 by S t x 0 = x(t) and assume that it exist for all t ≥ 0. We treat R n as a Riemannian manifold, equipped with a Riemannian metric, which can be expressed by a matrix-valued function M (x). In particular, M (x) defines a point-dependent scalar product through v, w M (x) = v T M (x)w for all v, w ∈ R n from the tangent space at x. The rate of expansion over time of the distance between solutions of (1) through x and x + δv for small δ > 0 with respect to the Riemannian metric is expressed by L M (x; v), see (2). The distance between the solution through x and adjacent solutions through x + v is decreasing if L M (x; v) < 0. If the distance between x and all adjacent solutions in direction The Riemannian metric M is called contraction metric in Note that L M (·) is a continuous function and, as we will show in the paper, also locally Lipschitz-continuous. Due to the maximum, however, it is not differentiable in general.
The following theorem shows that the existence of a contraction metric implies the existence, uniqueness and stability of a periodic orbit. Moreover, it provides information about its basin of attraction. Note that the conditions on M are local and can easily be checked for a given function M , while the implications are global. Theorem 1.2. Let ∅ = K ⊂ R n be a compact, connected and positively invariant set which contains no equilibrium. Let M be a contraction metric in K with exponent −ν < 0, see Definition 1.1.
Then there exists one and only one periodic orbit Ω ⊂ K. This periodic orbit is exponentially stable, and the real parts of all Floquet exponents -except the trivial one -are less than or equal to −ν. Moreover, the basin of attraction A(Ω) contains K.
This theorem goes back to Borg [2] with M (x) = I, and has been extended to a general Riemannian metric [13]. For more results on contraction analysis for a periodic orbit see [9,8,10,11].
Note that a similar result holds with an equilibrium if the contraction takes place in all directions v, i.e. if L M (x) ≤ −ν in (3) is replaced by L M (x) := max v T M (x)v=1 L M (x; v) ≤ −ν. For more references on contraction analysis see [12], and for the relation to Finsler-Lyapunov functions see [4]. The theory of normally hyperbolic invariant manifolds [14] considers more general invariant manifolds, not necessarily attracting, and studies their persistence under perturbations of the underlying system.
In this paper we are interested in global converse results, i.e. given an exponentially stable periodic orbit, does a Riemannian contraction metric as in Definition 1.1 exist in the whole basin of attraction? [12] gives a converse theorem, but here M (t, x) depends on t and will, in general, become unbounded as t → ∞. In [6] the existence of such a contraction metric was shown on a given compact subset of CONVERSE THEOREM ON CONTRACTION METRIC 5341 A(Ω), first on the periodic orbit, using Floquet theory, and then on K, using a Lyapunov function. The local construction, however, contained an error: the Floquet representation of solutions of the first variational equation along the periodic orbit is in general complex. We will show in this paper, that, by choosing the complex Floquet representation appropriately, the constructed Riemannian metric is realvalued. Moreover, we will show the existence of a Riemannian metric on the whole, possibly unbounded basin of attraction by using a new construction. The Riemannian metric will be arbitrarily close to the true rate of exponential attraction. Let us summarize the main result of the paper in the following theorem.
Let Ω be an exponentially stable periodic orbit with basin of attraction A(Ω) ofẋ = f (x), let −ν < 0 be the largest real part of all its non-trivial Floquet exponents and f ∈ C σ (R n , R n ) with σ ≥ 3.
holds for all x ∈ A(Ω).
The metric is constructed in several steps: first on the periodic orbit, then in a neighborhood, and finally in the whole basin of attraction. In the proof, we define a projection of points x in a neighborhood of the periodic orbit onto the periodic orbit, namely onto p ∈ Ω, such that to (x − p) T M (p)f (p) = 0. This is then used to synchronize the times of solutions through x and p, and to define a time-dependent distance between these solutions, which decreases exponentially.
Let us compare our result with other converse theorems for a contraction metric: In [6], a contraction metric on a given compact subset of the basin of attraction was constructed, while in this paper we construct one on the whole basin of attraction. In [5], a construction metric is characterized as the solution of a linear matrixvalued PDE. This is beneficial for its computation by solving the PDE, however, the exponential rate of attraction cannot be recovered, which is an advantage of the approach in this paper. Similar converse theorems for a contraction metric for an equilibrium were obtained in [7] in Theorems 4.1, 4.2 and 4.4.
Let us give an overview over the paper: In Section 2 we prove a special Floquet normal form to ensure that the contraction metric that we later construct on the periodic orbit is real-valued. In Section 3 we prove the main result of the paper, Theorem 1.3, showing the existence of a Riemannian metric on the whole basin of attraction. In the appendix we prove that L M is locally Lipschitz-continuous.

2.
Floquet normal form. Before we consider the Floquet normal form, we will prove a lemma which calculates L M (x) for the Riemannian metric M (x) = e 2V (x) N (x).
Lemma 2.1. Let N : R n → S n be a Riemannian metric and V : R n → R a continuous function such that the orbital derivative V exists and is continuous.
Then M (x) = e 2V (x) N (x) is a Riemannian metric and Proof. It is clear that M (x) is a positive definite for all x since e 2V (x) > 0. We have This shows the lemma.
In order to show later that our constructed Riemannian metric M is real-valued, we will construct a special Floquet normal form in Proposition 1 such that the matrix in (6) is real-valued. In Corollary 1 we will show estimates in the case that (5) is the first variational equation of a periodic orbit. The proof of the following proposition is inspired by [3]. In the following, we denote A * = A T for a matrix A ∈ C n×n . Proposition 1. Consider the periodic differential equatioṅ y = F (t)y (5) where F ∈ C s (R, R n×n ) is T -periodic, s ≥ 1 and denote by Φ ∈ C s (R, R n×n ) its principal fundamental matrix solution with Φ(0) = I. Then there exists a T -periodic function P ∈ C s (R, C n×n ) with P (0) = P (T ) = I and a matrix B ∈ C n×n such that for all t ∈ R Φ(t) = P (t)e Bt .
Proof. Since F ∈ C s (R, R n×n ), we also have Φ ∈ C s (R, R n×n ). Noting that Ψ(t) := Φ(t + T ) solves (5) with Ψ(0) = Φ(T ), we obtain from the uniqueness of solutions that Consider C := Φ(T ) ∈ R n×n which is non-singular and hence all eigenvalues of Φ(T ) are non-zero. Let := 1 2 min T 2 , 1 and S ∈ R n×n be such that S −1 CS =: J is in real Jordan normal form with the 1 replaced by |λ j | for each eigenvalue λ j , i.e. J is a block-diagonal matrix with blocks J j of the form ∈ R 2mj ×2mj for each pair of complex eigenvalues α j ± iβ j of C, where r j = α 2 j + β 2 j and m j denotes the dimension of the generalized eigenspace of one of them; note that we have pairs of complex conjugate eigenvalues since C is real. This can be achieved by letting S 1 ∈ R n×n be an invertible matrix such that S −1 1 CS 1 is the standard real Jordan Normal Form with 1 on the super diagonal. Then define S 2 to be a matrix of blocks diag(1, |λ j |, ( ) 2 |λ j | 2 , . . . , ( ) mj −1 |λ j | mj −1 ) for real λ j and diag(1, 1, |λ j |, |λ j |, . . . , ( ) mj −1 |λ j | mj −1 , ( ) mj −1 |λ j | mj −1 ) for a pair of complex conjugate eigenvalues λ j and λ j . Setting S = S 1 S 2 yields the result.
For each of the blocks, we will now construct a matrix K j ∈ C mj ×mj for real eigenvalues λ j and K j ∈ R 2mj ×2mj for each pair of complex eigenvalues α j ± iβ j such that We distinguish between three cases: λ j being real positive, real negative or complex. Using the series expansion of ln(1 + x) we obtain for a nilpotent matrix note that the sum is actually finite.
Since I and N commute, we have with (9) and Since I and N commute, and N k = 0 for k ≥ m j we have with (9) We only consider one of the two complex conjugate eigenvalues λ j and λ j of Φ(T ). Writing λ j in polar coordinates gives λ j = α j + iβ j = r j e iθj = r j cos θ j + ir j sin θ j with r j > 0 and θ j ∈ (0, 2π). Then, defining R j = r j cos θ j − sin θ j sin θ j cos θ j , R = blockdiag(R j , R j , . . . , R j ) ∈ R 2mj ×2mj and the nilpotent matrix N ∈ R 2mj ×2mj Since I, blockdiag(Θ, Θ, . . . , Θ) and N commute, we have, using N k = 0 for k ≥ 2m j and (9) We can now define P ∈ C s (R, C n×n ) by P (t) = Φ(t)e −Bt , which satisfies P (0) = I and (7) = P (t) by (8) for all t ≥ 0, so in particular P (T ) = P (0) = I. We can now write This shows the first statement of the proposition.
We now evaluate A * + A = blockdiag(K * 1 + K 1 , . . . , K * r + K r ). Let us consider K j as in the three cases above. If m j = 1, then K j below does not contain the last sum with in the following arguments, and the form of c j is immediately clear.
where Θ = 0 −θ j θ j 0 and the nilpotent matrix N has 2 × 2 blocks of cos θ j sin θ j − sin θ j cos θ j on its super diagonal. Note that all entries of N k , k ∈ N are real and have an absolute value of ≤ 1 as they are of the form cos(kθ j ) and ± sin(kθ j ) for k = 1, 2, . . .. Hence, for w ∈ C 2mj This shows the second statement of the proposition. To show that (P −1 (t)) * (S −1 ) * S −1 P −1 (t) has real entries, note that It is thus sufficient to show that (e At ) * e At is real-valued, since all other matrices are real-valued. Note that since A = blockdiag(K 1 , . . . , K r ), we have e At = blockdiag(e tK1 , . . . , e tKr ), (e tA ) * e tA = blockdiag((e tK1 ) * e tK1 , . . . , (e tKr ) * e tKr ) and the blocks where K j have only real entries are trivially real-valued (cases 1 and 3). In case 2, K j = 1 T ((iπ + ln |λ j |)I + N ), where N ∈ R mj ×mj is a nilpotent, upper triangular matrix. Then, noting that I and N commute, which has real entries.
, σ ≥ 2 and let S t q be an exponentially stable periodic solution with period T and q ∈ R n . Then the first variational equationẏ = Df (S t q)y is of the form as in the previous proposition with s = σ − 1; 1 is a single eigenvalue of Φ(T ) with eigenvector f (q) and all other eigenvalues of Φ(T ) satisfy |λ| < 1. More precisely, if −ν < 0 is the maximal real part of all non-trivial Floquet exponents, we have ln |λ| T ≤ −ν. With the notations of Proposition 1 we can assume that λ 1 = 1 and Se 1 = f (q).
Then we have for all > 0 Proof. Since f (S t q) solves (5), we have f (S t q) = P (t)e Bt f (q) and, in particular for 3. Converse theorem. We will prove Theorem 1.3, showing that a contraction metric exists for an exponentially stable periodic orbit in the whole basin of attraction. Moreover, we can achieve the bound −ν + for L M for any fixed > 0, where −ν denotes the largest real part of all non-trivial Floquet exponents. Note that we consider contraction in directions v perpendicular to f (x) with respect to the metric M , i.e. v T M (x)f (x) = 0. One could alternatively consider directions perpendicular to f (x) with respect to the Euclidean metric, i.e. v T f (x) = 0, but then the function L M needs to reflect this, see [5,1].
In the proof we will first construct M = M 0 on the periodic orbit Ω using Floquet theory. Then, we define a projection π of points in a neighborhood U of Ω onto Ω such that (x − π(x)) T M 0 (π(x))f (π(x)) = 0, which will be used to synchronize the time of solutions such that π(S τ x) = S θx(τ ) π(x). Finally, M will be defined through a scalar-valued function V by M (x) = M 1 (x)e 2V (x) , where M 1 = M 0 on the periodic orbit.
Proof. (of Theorem 1.3) Note that we assume f ∈ C σ (R n , R n ) to achieve more detailed results concerning the smoothness and assume lower bounds on σ as appropriate for each result; we always assume at least σ ≥ 2.
I. Definition and properties of M 0 on Ω We fix a point q ∈ Ω and consider the first variational equatioṅ which is a T -periodic, linear equation for y, and Df ∈ C σ−1 . By Proposition 1 and Corollary 1 the principal fundamental matrix solution Φ ∈ C σ−1 (R, R n×n ) of (10) with Φ(0) = I can be written as where B ∈ C n×n ; note that P ∈ C σ−1 (R, C n×n ) can be defined on the periodic orbit as it is T -periodic. By the assumptions on Ω, the eigenvalues of B are 0 with algebraic multiplicity one and the others have a real part ≤ −ν < 0. We define S as in Proposition 1 and define the C σ−1 -function Note that M 0 (S t q) is real by Proposition 1, symmetric, since it is Hermitian and real, and positive definite by and since S −1 P −1 (S t q) is non-singular.
We will now calculate L M0 (S t q; v). First, we have for the orbital derivative Furthermore, by using (P −1 (S t q)P (S t q)) = 0, we obtain (P −1 (S t q)) = −P −1 (S t q)(P (S t q)) P −1 (S t q).
In addition, since t → P (S t q)e Bt is a solution of (10), we have (P (S t q)) = Df (S t q)P (S t q) − P (S t q)B. Altogether, we get Hence, Thus, we obtain where w : and, using e 1 = S −1 P −1 (S t q)f (S t q) from Corollary 1 This shows with Corollary 1 and (14) II. Projection Fix a point q ∈ Ω on the periodic orbit. For x near the periodic orbit we define the projection π(x) = S θ q on the periodic orbit orthogonal to f (S θ q) with respect to the scalar product v, w M0(S θ q) = v T M 0 (S θ q)w implicitly by (16) below. The following lemma is based on the implicit function theorem and shows that the projection can be defined in a neighborhood of the periodic orbit, not just locally.
Then there is a compact, positively invariant neighborhood U of Ω with U ⊂ A(Ω) and a function π ∈ C σ−1 (U, Ω) such that π(x) = x if and only if x ∈ Ω. Moreover, for all x ∈ U we have (x − π(x)) T M 0 (π(x))f (π(x)) = 0. (16) Proof. Fix a point q ∈ Ω and define M 0 by (11). Define the C σ−1 function Define the following constants: with the matrix norm · = · 2 , which is induced by the vector norm · = · 2 and is sub-multiplicative. We will first prove the following quantitative version of the local implicit function theorem, using that θ is one-dimensional.
Now we want to define π by using the p x for finitely many points x ∈ Ω. Denote the (minimal) period of the periodic orbit by T ; we can assume that < T . Define We can conclude that if S θ p − p ≤ c/2 with p ∈ Ω and |θ| ≤ T /2, then |θ| < /2.
Let δ = min(δ/2, c/4). Since Ω is compact and Ω ⊂ x0∈Ω B δ (x 0 ), there is a finite number of x i = S θi q ∈ Ω, i = 1 . . . , N , with such that U is an open neighborhood of Ω. We want to show that the p xi = p i define a unique function p : U → S 1 T , where S 1 T are the reals modulo T such that p = p i on B δ (x i ). We need to show that if x ∈ B δ (x i ) ∩ B δ (x j ), then p i (x) = p j (x).

PETER GIESL
Let x ∈ B δ (x i ) ∩ B δ (x j ) and, without loss of generality, |θ j − θ i | ≤ T /2 since the θ i and θ j are uniquely defined only modulo T . Then Since Ω is stable, we can choose Ω ⊂ U • ⊂ U ⊂ U such that U is compact and positively invariant. For x ∈ U define π(x) = S p(x) q. Since p is defined by p xi , we have by Lemma 3.2 that 0 = G(x, p(x)) = (x − π(x)) T M 0 (π(x))f (π(x)).

III. Synchronization
In this step we synchronize the time between the solution S t x and the solution on the periodic orbit S θ π(x) such that (19) holds. This will enable us later to define a distance between S t x and Ω in Step IV.
for all t ≥ 0. We havė The denominator of (20) is strictly positive for all t ≥ 0 and x ∈ U .
Proof. Denote π(x) =: p ∈ Ω. Observe, that both sides of (19) equal for t = 0. For any t ≥ 0, we have S t x ∈ U , and since π(S t x) denotes a point on the periodic orbit, we can write it as π(S t x) = S θx(t) p. Note that θ x (t) is only uniquely defined modulo T , however, it is uniquely defined by the requirement that θ x is a continuous function. By (16), we have Hence, θ x (t) is implicitly defined by Note that θ x ∈ C σ−1 (R + 0 , R) by the Implicit Function Theorem which implies With the notations of the proof of Lemma 3.1, for S t x ∈ U there is a point by (17).
Proof. We apply (19) to the point S τ x and the time t, obtaining S θ Sτ x (t) π(S τ x) = π(S t S τ x).
As both right-hand sides are equal by the semi-flow property, this proves the statement.

IV. Distance to the periodic orbit
In the following lemma we define a distance of points in U to the periodic orbit, and we show that it decreases exponentially. Note that the in the following lemma is not related to the in the previous steps.
Lemma 3.5. Let 0 < < min(1, ν/2) and σ ≥ 2. Then there is a positively invariant, compact neighborhood U of the periodic orbit Ω such that the function d ∈ C σ−1 (U, R + 0 ), defined by satisfies d(x) = 0 if and only if x ∈ Ω. Moreover, d (x) < 0 for all x ∈ U \ Ω and d(S t x) ≤ e 2(−ν+2 )t d(x) for all x ∈ U and all t ≥ 0, Proof. Note that d is C σ−1 as all of its terms are. As M 0 (x) is positive definite, d(x) = 0 if and only if x = π(x), i.e. x ∈ Ω by Lemma 3.1. Define where the constants where defined in Step II, proof of Lemma 3.1.
For y ∈ U we use the Taylor expansion around π(y) ∈ Ω. Hence, there is a function ψ(y) satisfying f (y) = f (π(y)) + Df (π(y))(y − π(y)) + ψ(y) with ψ(y) ≤ c 4 y − π(y) for all y ∈ U , noting that Ω is compact, where we choose U still to be a positively invariant, compact neighborhood of Ω, possibly smaller than before and such that also have Recall that, due to the definition of M 0 and (19) we have Now let us calculate the orbital derivative, denoting θ(t) := θ x (t).
Hence, we have from (28) and (29) d (S t x) , we obtain, using (30) and (27) by (24) and (22). Noting that Let us summarize the results obtained so far in the following corollary.

Corollary 2.
Let Ω be an exponentially stable periodic orbit ofẋ = f (x) with f ∈ C σ (R n , R n ) and σ ≥ 2, such that −ν < 0 is the maximal real part of all non-trivial Floquet exponents.
Furthermore, for a fixed x ∈ U , there is a bijective Finally, there is a constant C > 0 such that for all θ ≥ 0 and all x ∈ U .

V. Definition of M 1 and M in A(Ω)
For all x ∈ U we have defined the distance in Lemma 3.5 which is C σ−1 . Let ι > 0 be so small that the set Ω 2ι : It is clear that M 1 (x) is positive definite for all x ∈ R n , M 1 is C σ−1 and M 1 (π(x)) = M 0 (π(x)) for all x ∈ Ω 4 3 ι . We will define the Riemannian metric M through M 1 and a scalar-valued function V : A(Ω) → R, which will be defined later. Let us denote µ := ν − > 0. The function V will be continuous and its orbital derivative V exists and is continuous. It satisfies Note that r(x) ≤ −µ for all x ∈ R n . Indeed, for x ∈ Ω ι we have L M1 (π(x)) = L M0 (π(x)) ≤ −µ as π(x) ∈ Ω, see (15), and thus Then we define We obtain by Lemma 2.1 This shows the theorem. In the last steps we will define the function V and prove the properties stated above.

VI. Definition of V loc
We define V loc (x) for x ∈ Ω ι . Note that Ω ι is positively invariant by Lemma 3.5, so S t x ∈ Ω ι for all t ≥ 0. We define We have V loc (x) = 0 for all x ∈ Ω. We will show that V loc is well-defined, continuous, its orbital derivative V exists and is continuous for all x ∈ Ω ι and that (33) holds for all x ∈ Ω ι/3 . For x ∈ U , define By Lemma 3.5 there is a constant C > 0 such that, defining p := π(x) ∈ Ω, for all t ≥ 0 and all x ∈ U with µ 0 := ν − 2 > 0; note that S θx(t) p = π(S t x) by (19). Now, we use Lemma A.1 and σ ≥ 3, showing that L M1 is Lipschitz-continuous on the compact set U ; note that σ − 1 ≥ 2. Hence, by (36), which is integrable over [0, ∞). Hence, by Lebesgue's dominated convergence theorem, the function g T (τ, x) converges point-wise for T → ∞ for all τ ≥ 0 and x ∈ U .

VII. Definition of V glob in A(Ω)
For the global part, note that V loc is defined and smooth in Ω ι and we have V loc (x) = −L M1 (x)+r(x) for all x ∈ Ω ι/3 . The global function V glob : A(Ω)\Ω → R is defined as the solution of the non-characteristic Cauchy problem where Γ = {x ∈ U | d(x) = ι/3}. It is clear that V glob is continuous, and V glob exists and is continuous on A(Ω) \ Ω.
Note that we have V glob (x) = V loc (x) for x ∈ Ω ι/3 \ Ω, and hence V glob can be extended to a continuous function V on A(Ω) by setting V glob (x) := V loc (x) = 0 for all x ∈ Ω. Then also its orbital derivative V glob exists and is continuous and V glob satisfies (33). This proves the theorem.
Conclusions. In this paper we have studied contraction metrics, which are Riemannian metrics on the phase space R n . Moreover, the distance, defined by the induced norm, of adjacent solution trajectories is decreasing over time. Here, only adjacent solutions in direction perpendicular to the flow with respect to the induced scalar product are considered. The existence of such a contraction metric in a compact, connected and positively invariant set, which contains no equilibrium, implies the existence of a unique periodic orbit as well as its exponential stability. Moreover, it provides an upper bound on the non-trivial Floquet exponents and determines a subset of the basin of attraction of the periodic orbit. This paper has considered the converse question, namely the existence of such a contraction metric. We have proved the existence of a contraction metric for an exponentially stable periodic orbit in its basin of attraction, and the upper bound on the function L M is arbitrarily close to the true exponential rate of attraction. The construction is achieved by first defining the contraction metric on the periodic orbit, then in a neighborhood and finally in the whole basin of attraction as the solution of a non-characteristic Cauchy problem.