Computation of a contraction metric for a periodic orbit using meshfree collocation

Article (Accepted Version)

http://sro.sussex.ac.uk

Giesl, Peter (2019) Computation of a contraction metric for a periodic orbit using meshfree collocation. SIAM Journal on Applied Dynamical Systems, 18 (3). pp. 1536-1564. ISSN 1536-0040

This version is available from Sussex Research Online: http://sro.sussex.ac.uk/id/eprint/84768/

This document is made available in accordance with publisher policies and may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher’s version. Please see the URL above for details on accessing the published version.

Copyright and reuse: Sussex Research Online is a digital repository of the research output of the University.

Copyright and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable, the material made available in SRO has been checked for eligibility before being made available.

Copies of full text items generally can be reproduced, displayed or performed and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes without prior permission or charge, provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way.

COMPUTATION OF A CONTRACTION METRIC FOR A PERIODICORBIT USING MESHFREE COLLOCATION

PETER GIESL∗

Abstract. Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamicalsystem. We consider a contraction metric, i.e. a Riemannian metric with respect to which the distancebetween adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow arecontracted, then there exists a unique periodic orbit, which is exponentially stable.

In this paper we propose a construction method using meshfree collocation to approximately solvea matrix-valued PDE problem. We derive error estimates and show that the approximation is itself acontraction metric if the collocation points are sufficiently dense. We apply the method to several examples.

Keywords. Periodic orbit; contraction metric; matrix-valued Partial Differential Equation; meshfreecollocation; error estimates.

AMS subject classifications. 34C25, 65N15, 37C27, 65N35.

1. Introduction. Ordinary differential equations arise in many applications in biology,physics, and various other areas. The determination of periodic orbits, their stability andbasins of attraction are important to analyze systems and to develop models.

We consider a general autonomous ODE of the form

x = f(x), (1.1)

where f ∈ C1(Rn,Rn). We denote the solution of the initial value problem (1.1) withx(0) = ξ by Stξ = x(t) and assume that it exists for all t ≥ 0.

The basin of attraction of a periodic orbit can be determined using a Lyapunov func-tion, however, its definition requires the exact position of the periodic orbit. A contractionmetric, on the other hand, can prove the existence, uniqueness and stability of a periodicorbit without knowledge of its position. Moreover, a contraction metric is robust to smallperturbations of the system or the metric. This means that a sufficiently good approxima-tion to a certain contraction metric, e.g. using numerical methods, is itself a contractionmetric.

A contraction metric is a Riemannian metric such that the distance between adjacenttrajectories decreases over time with respect to the Riemannian metric. This type of stability,comparing adjacent solution with each other, is called incremental stability and a contractionmetric is a special type of a Finsler-Lyapunov function [4].

A contraction metric for a periodic orbit can be expressed as a matrix-valued functionM ∈ C1(Rn, Sn×n), where Sn×n denotes the symmetric Rn×n matrices, such that M(x) ispositive definite and thus ⟨v,w⟩x = vTM(x)w defines a point-dependent scalar productfor v,w ∈ Rn. The contraction condition is expressed by LM (x) ≤ −ν < 0, where LM isdefined in (1.3) below.

We first define for all x ∈ Rn with f(x) = 0

V (x) = Df(x)− f(x)f(x)T (Df(x) +Df(x)T )

∥f(x)∥2. (1.2)

∗Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom,p.a.giesl@sussex.ac.uk

1

Then we define

LM (x) = maxv∈Rn,vTM(x)v=1,vT f(x)=0

LM (x;v) (1.3)

LM (x;v) =1

2vT(M ′(x) + V (x)TM(x) +M(x)V (x)

)v .

Here, (M ′(x))i,j=1,...,n = (∇Mij(x))T f(x) is the matrix of the orbital derivatives of Mij

along solutions of (1.1) and ∥ · ∥ = ∥ · ∥2 denotes the Euclidean norm in Rn.The function LM (x;v) for v with vT f(x) = 0 is negative, if the distance between

solutions through x and x+ δv for small δ > 0 with respect to the metric M(x) decreases.Let us give a heuristic explanation. To measure the distance, we synchronize the times suchthat the difference vector between the solutions is perpendicular to the flow. In particular,we define θ(t) such that θ(0) = 0 and(

Sθ(t)(x+ δv)− Stx)T

f(Stx) = 0 for all t ≥ 0.

The implicit function theorem shows that

θ(0) =∥f(x)∥2 − δvTDf(x)f(x)

f(x+ δv)T f(x)≈ 1− δ

vT (Df(x)T +Df(x))f(x)

∥f(x)∥2 + δvTDf(x)T f(x)(1.4)

for small δ > 0. Now we consider the squared distance between the trajectories with respectto the Riemannian metric

d(t) =(Sθ(t)(x+ δv)− Stx

)TM(Stx)

(Sθ(t)(x+ δv)− Stx

)and take the derivative. We obtain, using Taylor expansion,

d

dtd(t)

∣∣∣∣t=0

=(θ(0)f(x+ δv)− f(x)

)TM(x)δv + δ2vTM ′(x)v

+δvTM(x)(θ(0)f(x+ δv)− f(x)

)≈ δ(θ(0)− 1)[f(x)TM(x)v + vTM(x)f(x)]

+δ2θ(0)[(Df(x)v)TM(x)v + vTM(x)Df(x)v] + δ2vTM ′(x)v

≈ δ2[− vT (Df(x)T +Df(x))f(x)

∥f(x)∥2[f(x)TM(x)v + vTM(x)f(x)]

+(Df(x)v)TM(x)v + vTM(x)Df(x)v + vTM ′(x)v

]by (1.4)

= δ2vT[V (x)TM(x) +M(x)V (x) +M ′(x)

]v = 2δ2LM (x;v) .

If LM (x;v) is bounded by a negative constant −ν, then d(t) is exponentially decreasing.We now cite the following implication for the existence, uniqueness and stability of a

periodic orbit and its basin of attraction, see [7, Theorem 2.1].Theorem 1.1. Let K ⊂ Rn be compact, connected and positively invariant set that

does not contain an equilibrium of (1.1), i.e. for all x ∈ K we have f(x) = 0.Let M ∈ C1(Rn, Sn×n) such that M(x) is positive definite for all x ∈ Rn. Moreover,

assume that LM (x) ≤ −ν < 0 holds for all x ∈ K, see (1.3).Then there exists a unique periodic orbit Ω ⊂ K, Ω is exponentially stable and the largest

real part of all non-trivial Floquet exponents is at most −ν. Moreover, K is a subset of thebasin of attraction A(Ω).

2

A matrix-valued function M satisfying the assumptions of Theorem 1.1 is called con-traction metric. It provides information about the basin of attraction of a periodic orbit,without requiring any information about its existence or location.

Contraction metrics for periodic orbits have been studied by Borg [1] with the Eu-clidean metric and Stenstrom [21] with a general Riemannian metric. For further results oncontraction metrics see [13, 14, 15, 16].

The converse question, namely the existence of a contraction metric defined in the basinof attraction of an exponentially stable periodic orbit, has been studied in [5]. In [7], acontraction metric was characterized as the unique solution of a matrix-valued PDE.

Computational methods for contraction methods have been proposed in [8] for periodicorbits in time-periodic systems, where the contraction metric was a continuous piecewiseaffine (CPA) function and the contraction conditions were transformed into constraints of asemidefinite optimization problem. This is similar to the construction of a Lyapunov func-tion, which, however, can be solved using linear optimization, where much larger optimiza-tion problems can be tackled. The phase space is triangulated and the number of constraintsbecomes very large. While the method includes error estimates, which guarantee that thefeasibility of the semidefinite optimization problem implies the rigorous determination of acontraction metric, the method is computationally demanding.

In [17, Theorem 3] a contraction metric for periodic orbits was constructed using LinearMatrix Inequalities and SOS (sum of squares). This method is applicable to polynomialsystems, and can be generalized. The computational demand depends on the requireddegree of the polynomial for the construction of the contraction metric. Due to the fact thatnot all positive polynomials can be written as a sum of squared polynomials, the method isnot always guaranteed to succeed.

In this paper we propose a method to construct a contraction metric by approximatelysolving the matrix-valued PDE from [7] using meshfree collocation. Meshfree collocation isa powerful method to solve interpolation and PDE problems, it works in any dimension anddoes not require a triangulation of the phase space. Instead, the PDE is required to holdat a set of scattered collocation points, the approximation is computed as the solution of asystem of linear equations and it is the norm-minimal interpolant in a reproducing kernelHilbert space, in our case a Sobolev space. The size of the linear system corresponds to thenumber of collocation points. The method is particularly suitable for refinement due to thescattered collocation points.

It has been shown in [7] that the unique solution of the PDE problem is a contraction

metric. In this paper, we approximate the solution of the PDE problem by M via mesh-free collocation and prove that the approximation is also a contraction metric, using errorestimates.

Let us give an overview over the paper: in Section 2 we introduce the PDE problem andcite an existence and uniqueness result. In Section 3 we show that if M is an approximatesolution of the PDE problem, then M is a contraction metric, i.e. M is positive definiteand L

Mis negative definite. In Section 4 we discuss meshfree collocation and prove error

estimates. Section 5 applies the method to several examples. The appendix provides explicitformulas for the calculations and explains how to check the conditions that M is positivedefinite and L

Mis negative.

2. PDE characterizing a contraction metric. In [7] the existence and uniquenessof a solution of a linear matrix-valued PDE has been shown, see Theorem 2.1. Moreover,the solution of this problem is a contraction metric.

To introduce the PDE problem, we define for all x ∈ Rn with f(x) = 0 the linear

3

differential operator L, acting on M ∈ C1(Rn, Sn×n) by

LM(x) :=M ′(x) + V (x)TM(x) +M(x)V (x), (2.1)

where V was defined in (1.2). Moreover, we define the projection Px for all x ∈ Rn withf(x) = 0 onto the (n−1)-dimensional space perpendicular to f(x), i.e. P 2

x = Px, Pxf(x) = 0and Pxv = v if vT f(x) = 0, by

Px := I − f(x)f(x)T

∥f(x)∥2. (2.2)

The following theorem is from [7, Theorem 3.1, Theorem 4.2].Theorem 2.1. Let Ω be an exponentially stable periodic orbit of x = f(x), f ∈

Cs(Rn,Rn), where s ≥ 2, with basin of attraction A(Ω). Fix x0 ∈ A(Ω) and c0 ∈ R+.Let B ∈ Cs−1(A(Ω), Sn×n) be such that B(x) is positive definite for all x ∈ A(Ω) and defineC ∈ Cs−1(A(Ω), Sn×n) by (see (2.2))

C(x) = PTx B(x)Px.

Then there exists a unique solution M ∈ Cs−1(A(Ω), Sn×n) of the linear matrix-valuedPDE (see (2.1))

LM(x) = −C(x) for all x ∈ A(Ω) (2.3)

satisfying f(x0)TM(x0)f(x0) = c0∥f(x0)∥4. (2.4)

The solution M(x) is positive definite for all x ∈ A(Ω) and it is of the form

M(x) =

∫ ∞

0

Φ(t, 0;x)TC(Stx)Φ(t, 0;x) dt+ c0f(x)f(x)T ,

where Φ(t, 0;x) denotes the principal fundamental matrix solution of ϕ(t) = D(Stx)ϕ(t)with Φ(0, 0;x) = I.

Note that since LM (x;v) = 12v

TLM(x)v, see (1.3), a function Msatisfying (2.3) gives LM (x;v) = −1

2vTPTx B(x)Pxv, and thus LM (x) =

−12 minv∈Rn,vTM(x)v=1,vT f(x)=0 v

TB(x)v, which can be bounded by a negative con-stant −ν for all x within a compact set K ⊂ A(Ω). Moreover, M , satisfying (2.3) and (2.4)is positive definite and, hence, is a contraction metric.

In this paper we seek to approximate the solution of (2.3) and (2.4) by M , satisfying

LM(x) = −C(x) by meshfree collocation. Note that we cannot assume that C(x) is of the

form PTx B(x)Px. We will, however, assume that C and C are close to each other, as well as

their first derivatives. We will show that then the approximation M is a contraction metric.This proves a constructive converse theorem.

3. Approximation of PDE is a contraction metric. In this section we will showthat an approximate solution of (2.3) and (2.4), which is sufficiently close, is a contractionmetric, in particular, that it is positively invariant. Sufficiently close is expressed by thefact that the difference between LM and LM , see (3.1), as well as their first derivatives, issufficiently small, see (3.3) and (3.4). Later, we will show error estimates which ensure that

(3.3) and (3.4) are satisfied for M being an approximation using meshfree collocation. Forthe following theorem we denote γ+(K) =

∪t≥0 StK.

Theorem 3.1. Let the assumptions of Theorem 2.1 hold. Let K ⊂ A(Ω) be a compact

set with Ω ⊂K and let x0 ∈ K as well as c0 ∈ R+.

4

Then there is an ϵ > 0 such that for all M, C ∈ C1(γ+(K),Sn×n) satisfying

LM(x) = −C(x) for all x ∈ γ+(K) (3.1)

f(x0)T M(x0)f(x0) = c0∥f(x0)∥4 (3.2)

∥C(x)− C(x)∥ ≤ ϵ for all x ∈ γ+(K) (3.3)∥∥∥∥ d

dxi

(C(x)− C(x)

)∥∥∥∥ ≤ ϵ for all x ∈ γ+(K) and i = 1, . . . , n (3.4)

we have that M(x) is positive definite for all x ∈ K. Moreover, there is a constant ν > 0such that

LM(x) ≤ −ν

holds for all x ∈ γ+(K), where LM was defined in (1.3).Remark 3.2. Note that for a compact and positively invariant set we have γ+(K) = K.Proof. Step I

Denote by T > 0 the (minimal) period of the periodic orbit Ω and by −ν < 0 the maximal

real part of all its non-trivial Floquet exponents. Define f0(x) =f(x)

∥f(x)∥2 for all x ∈ Rn with

f(x) = 0. Set

F0 := minx∈γ+(K)

∥f(x)∥ > 0, then

maxx∈γ+(K)

∥f0(x)∥ = maxx∈γ+(K)

1

∥f(x)∥=

1

minx∈γ+(K)

∥f(x)∥=

1

F0.

Define F := maxx∈γ+(K)

∥f(x)∥,

F1 := maxi=1,...,n

maxx∈γ+(K)

∥∥∥∥ ∂

∂xif0(x)

∥∥∥∥ .Choose ϵ0 = 1

2 min(ν, 1). We use [7, Lemma 3.3], which proves the existence of acompact, positively invariant neighborhood U of Ω with Ω ⊂ U ⊂ U ⊂ A(Ω) and a mapπ ∈ Cs−1(U,Ω) with π(x) = x if and only if x ∈ Ω. For a fixed x ∈ U there is a bijectiveCs−1 map θx : [0,∞) → [0,∞) with inverse tx = θ−1

x ∈ Cs−1([0,∞), [0,∞)) such thatθx(0) = 0 and

π(Stx) = Sθx(t)π(x)

for all t ∈ [0,∞). Moreover, θx(t) ∈ [1− ϵ0, 1 + ϵ0] for all t ≥ 0 and tx(θ) ∈ [1− ϵ0, 1 + ϵ0]for all θ ≥ 0.

Finally, there is a constant C > 0 such that

|θx(t)− 1| ≤ Ce−µ0t for all t ≥ 0 (3.5)

∥Stx(θ)x− Sθπ(x)∥ ≤ Ce−µ0θ∥x− π(x)∥ for all θ ≥ 0 (3.6)

and all x ∈ U , where µ0 = ν − ϵ0 > 0; for (3.5) see the proof of [6, Corollary 3.6]. Wedefine cU := maxx∈U ∥x − π(x)∥. Since K ⊂ A(Ω) is compact, there is a T0 > 0 such thatStK ⊂ U for all t ≥ T0.

By [7, Lemma 3.4], with similar arguments as in [7, Lemma 3.5] to extend it to thecompact set K, there exists κ > 0 such that for all x ∈ K we have

∥PStxΦ(t, 0;x)∥ ≤ C0e−κt (3.7)

5

for all t ≥ 0, where Φ(t, 0;x) denotes the principal fundamental matrix solution of the firstvariational equation ϕ(t) = Df(Stx)ϕ(t) with Φ(0, 0;x) = I.

There are Λ, λ > 0 such that vTB(x)v ≥ λ∥v∥2 and vT M(x)v ≤ Λ∥v∥2 hold for all

x ∈ γ+(K) and all v ∈ Rn, since B is positive definite and continuous, and M is continuouson the compact set γ+(K).

For x with f(x) = 0 define

A(x) = Df(x)− f(x)f(x)T [Df(x) +Df(x)T ]

∥f(x)∥2. (3.8)

We denote by µ(A) = limh→0+∥I+hA∥−1

h the logarithmic norm of a matrix A ∈ Rn×n, whichcan also be negative. For the matrix norm ∥ · ∥ induced by the vector norm ∥ · ∥ = ∥ · ∥2,i.e. ∥A∥ = supx=0

∥Ax∥∥x∥ , we have that µ(A) is the largest eigenvalue of 1

2 (AT + A). Since

the eigenvalues vary continuously with the matrix elements, we can define

α0 := maxy∈γ+(K)

µ(−A(y)) . (3.9)

We will see in Step III that α0 > 0.Define the positive constants

c1 := min

(λ

2α0, c0F

2

)(3.10)

c2 :=T0F 20

+√n2F1F0 + 1

F 20

C

µ0cU (1 + ϵ0) +

T

F 20

+C

F 20 µ0(1− ϵ0)

(3.11)

c3 :=C2

0

2κ+ 2

C0F

F0κ+ 2c2F

2 (3.12)

ϵ := min

(λ

2,c12c3

). (3.13)

Step II

We first show LM(x) ≤ −ν := − λ

4Λ for all x ∈ γ+(K). We have for all x ∈ γ+(K), sincevT f(x) = 0 implies Pxv = v

2LM(x) ≤ max

v∈Rn,vT M(x)v=1,vT f(x)=0vTLM(x)v

+ maxv∈Rn,vT M(x)v=1,vT f(x)=0

vT [LM(x)− LM(x)]v

≤ − minv∈Rn,vT M(x)v=1,vT f(x)=0

vTPTx B(x)Pxv

+ maxv∈Rn,vT M(x)v=1,vT f(x)=0

vT [C(x)− C(x)]v

= − minv∈Rn,vT M(x)v=1,vT f(x)=0

vTB(x)v + maxv∈Rn,vT M(x)v=1,vT f(x)=0

ϵ∥v∥2

≤ (−λ+ ϵ) maxv∈Rn,vT M(x)v=1,vT f(x)=0

∥v∥2

≤ − λ

2Λ,

using (3.13). This shows the statement.

6

Step III

To show that M is positive definite, fix x ∈ K and w ∈ Rn. We write with c = f(x)Tw ∈ R

w = Pxw︸︷︷︸=:v

+cf(x)

∥f(x)∥2︸ ︷︷ ︸=f0(x)

(3.14)

such that

∥w∥2 = ∥v∥2 + c2

∥f(x)∥2. (3.15)

Note that c2

∥f(x)∥2 ≤ ∥w∥2, i.e.

|c| ≤ ∥w∥F for all x ∈ γ+(K). (3.16)

We have

wT M(x)w ≥ wTM(x)w −∣∣∣wT [M(x)−M(x)]w

∣∣∣ . (3.17)

Let us estimate the first term in (3.17). Using Theorem 2.1 for the form of M(x) we have

wTM(x)w =

∫ ∞

0

(PStxΦ(t, 0;x)w)TB(Stx)PStxΦ(t, 0;x)w dt+ c0wT f(x)f(x)Tw

=

∫ ∞

0

(PStxΦ(t, 0;x)w)TB(Stx)PStxΦ(t, 0;x)w dt+ c0c2. (3.18)

Note that ϕ(t) = Φ(t, 0;x)w satisfies ϕ(t) = Df(Stx)ϕ(t). Similar to [7, (3.25)] we havefor any solution ϕ(t) of ϕ(t) = Df(Stx)ϕ(t)

d

dt(PStxϕ(t)) =

d

dt

((I − f(Stx)f(Stx)

T

∥f(Stx)∥2

)ϕ(t)

)= −Df(Stx)f(Stx)f(Stx)

T + f(Stx)f(Stx)TDf(Stx)

T

∥f(Stx)∥2ϕ(t)

+f(Stx)f(Stx)

T f(Stx)T [Df(Stx) +Df(Stx)

T ]f(Stx)

∥f(Stx)∥4ϕ(t)

+

(I − f(Stx)f(Stx)

T

∥f(Stx)∥2

)Df(Stx)ϕ(t)

=

(Df(Stx)−

f(Stx)f(Stx)T [Df(Stx) +Df(Stx)

T ]

∥f(Stx)∥2

)PStxϕ(t)

= A(Stx)PStxϕ(t),

where A was defined in (3.8). By [3, Theorem 3, p. 58] we have

∥PStxϕ(t)∥ ≥ ∥Pxϕ(0)∥ exp(−∫ t

0

µ(−A(Ssx)) ds)

≥ ∥Pxϕ(0)∥ exp (−α0t)

where α0 was defined in (3.9).

7

Denoting ϕ(t) = Φ(t, 0;x)w we have with (3.7)

exp(−α0t)∥Pxw∥ ≤ ∥PStxΦ(t, 0;x)w∥ ≤ C0e−κt∥w∥

for any w ∈ Rn. Hence, we can conclude that α0 > 0.Moreover, we have ∫ ∞

0

(PStxΦ(t, 0;x)w)TB(Stx)PStxΦ(t, 0;x)w dt

≥ λ

∫ ∞

0

∥PStxϕ(t)∥2 dt

≥ λ

∫ ∞

0

∥Pxϕ(0)∥2 exp(−2α0t) dt

=λ

2α0∥Pxw∥2,

where λ was defined in Step I.Altogether, we have with (3.18), (3.10) and (3.15)

wTM(x)w ≥ λ

2α0∥v∥2 + c0c

2

≥ c1∥v∥2 +c1F 20

c2

= c1

(∥v∥2 + c2

F 20

)≥ c1∥w∥2 . (3.19)

Step IVNow we focus on the second term in (3.17). We have with (3.14)∣∣∣wT [M(x)− M(x)]w

∣∣∣≤∣∣∣wTPTx [M(x)− M(x)]Pxw

∣∣∣+ ∣∣∣wTPTx [M(x)− M(x)]c f0(x)∣∣∣

+∣∣∣c f0(x)T [M(x)− M(x)]Pxw

∣∣∣+ ∣∣∣c2 f0(x)T [M(x)− M(x)]f0(x)∣∣∣ . (3.20)

We will now derive bounds for each of the terms in (3.20). In the following we write Φ(t)for Φ(t, 0;x).Step V: first term in (3.20)Using Theorem 2.1 for the form of M(x) as well as ∥Pxw∥ ≤ ∥w∥ and

PTx C(x)Px = PTx PTx B(x)PxPx = PTx B(x)Px = C(x)

we have∣∣∣wTPTx [M(x)− M(x)]Pxw∣∣∣ ≤ ∥∥∥∥PTx M(x)Px −

∫ ∞

0

Φ(s)TPTSsxC(Ssx)PSsxΦ(s) ds

∥∥∥∥ ∥w∥2.

We have by [7, Lemma 4.1, (4.1)]

d

dt

[Φ(t)TPTStxM(Stx)PStxΦ(t)

]= Φ(t)TPTStxLM(Stx)PStxΦ(t)

= −Φ(t)TPTStxC(Stx)PStxΦ(t) .

8

Hence,

PTx M(x)Px = Φ(t)TPTStxM(Stx)PStxΦ(t) +

∫ t

0

Φ(s)TPTSsxC(Ssx)PSsxΦ(s) ds. (3.21)

As t → ∞, the first term on the right-hand side vanishes by (3.7). For the second term of(3.21) we have by (3.7)∥∥∥∥∫ ∞

0

Φ(s)TPTSsxC(Ssx)PSsxΦ(s) ds−∫ ∞

0

Φ(s)TPTSsxC(Ssx)PSsxΦ(s) ds

∥∥∥∥≤ C2

0 ϵ

∫ ∞

0

e−2κs ds

=C2

0

2κϵ .

Hence, altogether ∣∣∣wTPTx [M(x)− M(x)]Pxw∣∣∣ ≤ C2

0

2κϵ ∥w∥2 . (3.22)

Step VI: second and third terms in (3.20)Due to the form of M(x), see Theorem 2.1, we have, using Φ(t)f(x) = f(Stx) andPStxf(Stx) = 0

M(x)f0(x) =1

∥f(x)∥2

∫ ∞

0

Φ(t)TPTStxB(Stx)PStxΦ(t)f(x) dt+ c0f(x)

=1

∥f(x)∥2

∫ ∞

0

Φ(t)TPTStxB(Stx)PStxf(Stx) dt+ c0f(x)

= c0f(x) .

Hence, f0(x)TM(x)Pxw = 0. This shows∣∣∣c f0(x)T [M(x)− M(x)]Pxw

∣∣∣ = ∣∣∣c f0(x)T M(x)Pxw∣∣∣

≤ |c| ∥w∥∥∥∥f0(x)T M(x)Px

∥∥∥ .We have by [7, Lemma 4.1, (4.4)]

d

dt

[f0(Stx)

T M(Stx)PStxΦ(t)]= f0(Stx)

TLM(Stx)PStxΦ(t)

= −f0(Stx)T C(Stx)PStxΦ(t)

f0(x)T M(x0)Px = f0(Stx)

T M(Stx0)PStxΦ(t)

+

∫ t

0

f0(Ssx)T C(Ssx)PSsxΦ(s) ds.

As t→ ∞, the first term on the right-hand side vanishes by (3.7). For the second term, wehave with (3.7) and since C(Ssx)f0(Ssx) = PTSsx

B(Ssx)PSsxf0(Ssx) = 0∥∥∥∥∫ ∞

0

f0(Ssx)T C(Ssx)PSsxΦ(s) ds

∥∥∥∥ =

∥∥∥∥∫ ∞

0

f0(Ssx)T [C(Ssx)− C(Ssx)]PSsxΦ(s) ds

∥∥∥∥≤ C0

F0ϵ

∫ ∞

0

e−κs ds

=C0

F0κϵ.

9

Hence, altogether by (3.16)∣∣∣c f0(x)T [M(x)− M(x)]Pxw∣∣∣ ≤ |c| ∥w∥ C0

F0κϵ

≤ ∥w∥2 C0F

F0κϵ . (3.23)

A similar estimate holds true for the second term in (3.20).Step VII: last term in (3.20)We have by [7, Lemma 4.1, (4.2)]

d

dt

[f0(Stx)

T M(Stx)f0(Stx)]= f0(Stx)

TLM(Stx)f0(Stx)

= −f0(Stx)T C(Stx)f0(Stx)

and hence

f0(Stx)T M(Stx)f0(Stx)

= f0(x)T M(x)f0(x)−

∫ t

0

f0(Ssx)T C(Ssx)f0(Ssx) ds. (3.24)

We have for x ∈ K and t ≥ T0, noting that f0(Ssx)TC(Ssx)f0(Ssx) = 0,∥∥∥∥∫ t

0

f0(Ssx)T C(Ssx)f0(Ssx) ds

∥∥∥∥ ≤∫ T0

0

∥∥∥f0(Ssx)T [C(Ssx)− C(Ssx)]f0(Ssx)∥∥∥ ds

+

∥∥∥∥∫ t

T0

f0(Ssx)T [C(Ssx)− C(Ssx)]f0(Ssx) ds

∥∥∥∥≤ T0F 20

ϵ+

∥∥∥∥∫ t

T0

[g(Ssx)− g(π(Ssx))] ds

∥∥∥∥+

∥∥∥∥∫ t

T0

g(π(Ssx)) ds

∥∥∥∥ ,where we define g(x) = f0(x)

T [C(x)− C(x)]f0(x).For any s ∈ [T0, t] we have Ssx ∈ U . Moreover, the straight line between Ssx and

π(Ssx) is in U . Thus, since g is C1, there is a θ ∈ [0, 1] with

g(Ssx)− g(π(Ssx)) = ∇g(θSsx+ (1− θ)π(Ssx)) · (Ssx− π(Ssx)).

In particular,

∥g(Ssx)− g(π(Ssx))∥ ≤ maxy∈U

∥∇g(y)∥ ∥Ssx− π(Ssx)∥.

For ∇g we have

∂

∂xi

(f0(x)

T [C(x)− C(x)]f0(x))=

(∂

∂xif0(x)

)T[C(x)− C(x)]f0(x)

+f0(x)T ∂

∂xi[C(x)− C(x)]f0(x)

+f0(x)T [C(x)− C(x)]

(∂

∂xif0(x)

)10

and thus

maxy∈U

∥∇g(y)∥ ≤√nϵ

(2F1

F0+

1

F 20

).

Denote p := ST0x ∈ U and q := π(p) ∈ Ω. We have π(ST0+sx) = π(Ssp) =Sθp(s)π(p) ∈ Ω. Since q ∈ Ω is a point on the periodic orbit with period T we have

with f0(Ssq)TC(Ssq)f0(Ssq) = 0 and by (3.24)∫ T

0

g(Ssq) ds =

∫ T

0

f0(Ssq)T C(Ssq)f0(Ssq) ds = 0. (3.25)

Altogether, we obtain∥∥∥∥∫ t

0

f0(Ssx)T C(Ssx)f0(Ssx) ds

∥∥∥∥≤ T0F 20

ϵ+√nϵ

2F1F0 + 1

F 20

∫ t

T0

∥Ssx− π(Ssx)∥ ds+∥∥∥∥∫ t

T0

g(π(Ssx)) ds

∥∥∥∥≤ T0F 20

ϵ+√nϵ

2F1F0 + 1

F 20

∫ t−T0

0

∥Ssp− Sθp(s)π(p)∥ ds+

∥∥∥∥∥∫ t−T0

0

g(π(Ssp)) ds

∥∥∥∥∥≤ T0F 20

ϵ+√nϵ

2F1F0 + 1

F 20

∫ θp(t−T0)

0

∥Stp(θ)p− Sθπ(p)∥ dθ(1 + ϵ0) +

∥∥∥∥∥∫ t−T0

0

g(Sθp(s)q) ds

∥∥∥∥∥≤ T0F 20

ϵ+√nϵ

2F1F0 + 1

F 20

C

∫ θp(t−T0)

0

e−µ0θ dθ cU (1 + ϵ0) +

∥∥∥∥∥∫ θp(t−T0)

0

g(Sτq)

θp(τ)dτ

∥∥∥∥∥ by (3.6)

≤ T0F 20

ϵ+√nϵ

2F1F0 + 1

F 20

C

µ0cU (1 + ϵ0) +

∥∥∥∥∥∫ θp(t−T0)

0

g(Sτq) dτ

∥∥∥∥∥+

∥∥∥∥∥∫ θp(t−T0)

0

g(Sτq)

(1

θp(τ)− 1

)dτ

∥∥∥∥∥ .Let us now focus on the last two terms. We have, using (3.25)∥∥∥∥∥

∫ θp(t−T0)

0

g(Sτq) dτ

∥∥∥∥∥ ≤ maxt∈[0,T ]

∥∥∥∥∫ t

0

g(Sτq) dτ

∥∥∥∥ ≤ Tϵ

F 20

.

Furthermore, using (3.5), we have∥∥∥∥∥∫ θp(t−T0)

0

g(Sτq)1− θp(τ)

θp(τ)dτ

∥∥∥∥∥ ≤ ϵ

F 20 (1− ϵ0)

C

∫ ∞

0

e−µ0τ dτ

=ϵC

F 20 µ0(1− ϵ0)

.

Altogether, using (3.24), we obtain∣∣∣f0(Stx)T M(Stx)f0(Stx)− f0(x)T M(x)f0(x)

∣∣∣ ≤ c2ϵ (3.26)

for all t ≥ 0 and all x ∈ K, see (3.11).

11

By (3.2) we have for x = x0 and all t ≥ 0∣∣∣f0(Stx0)T M(Stx0)f0(Stx0)− f0(x0)

T M(x0)f0(x0)∣∣∣ = ∣∣∣f0(Stx0)

T M(Stx0)f0(Stx0)− c0

∣∣∣≤ c2ϵ .

For a fixed point q ∈ Ω there is a sequence tn → ∞ with Stnx0 → q as n → ∞; thisshows that ∣∣∣f0(q)T M(q)f0(q)− c0

∣∣∣ ≤ c2ϵ (3.27)

holds for all q ∈ Ω.For a fixed point x ∈ K ⊂ A(Ω) and a fixed q ∈ Ω there is a sequence tn → ∞ with

Stnx → q as n→ ∞ and thus by the form of M as well as (3.26) and (3.27)∣∣∣f0(x)T M(x)f0(x)− f0(x)TM(x)f0(x)

∣∣∣ = ∣∣∣f0(x)T M(x)f0(x)− c0

∣∣∣≤∣∣∣f0(x)T M(x)f0(x)− f0(Stnx)

T M(Stnx)f0(Stnx)∣∣∣

+∣∣∣f0(Stnx)T M(Stnx)f0(Stnx)− f0(q)

T M(q)f0(q)∣∣∣

+∣∣∣f0(q)T M(q)f0(q)− c0

∣∣∣≤ 2c2ϵ, (3.28)

letting tn → ∞.Step VIIIWe have now derived the bounds for (3.20) and obtain with (3.22), (3.23) and (3.28)∣∣∣wT [M(x)− M(x)]w

∣∣∣ ≤ C20

2κϵ ∥w∥2 + 2∥w∥2 C0F

F0κϵ+ 2c2F

2ϵ∥w∥2 using (3.16)

= ϵ∥w∥2c3, (3.29)

see (3.12).Hence, we have by (3.17) and (3.19)

wT M(x)w ≥ c1∥w∥2 − c3ϵ∥w∥2 ≥ c12∥w∥2

by (3.13), which shows that M(x) is positive definite.

4. Meshfree collocation and error estimates. Meshfree collocation, in particularby radial basis functions, is used to approximate multivariate functions and approximatelysolve partial differential equations [19, 2, 20]. For a general introduction to meshfree collo-cation and reproducing kernel Hilbert spaces see [23].

We use meshfree collocation to find an approximation M to the solution of the matrix-valued PDE (2.3) with condition (2.4). Such a framework has been developed in [12] and we

will adapt it to our situation. Moreover, we show error estimates which will ensure that Msatisfies the assumptions of Theorem 3.1, and thus proving a constructive converse theorem.

Denote by W = Sn×n the space of real-valued symmetric n × n matrices, which is aseparable Hilbert space with inner product

⟨α, β⟩W =

n∑i,j=1

αijβij , α = (αij), β = (βij)

12

and orthonormal basis Esµν ∈ Sn×n : 1 ≤ µ ≤ ν ≤ n. Here, Esµµ denotes the matrix withvalue 1 at position (µ, µ) and value zero everywhere else. For µ < ν, Esµν denotes the matrix

with value 1/√2 at positions (µ, ν) and (ν, µ) and value zero everywhere else. We also define

Eµν ∈ Rn×n to be the matrix with value 1 at position (µ, ν) and value zero everywhere else.Let O ⊂ Rn be a domain with Lipschitz-continuous boundary and consider a mapping

Φ : O×O → L(W ), where L(W ) denotes the linear space of all linear and bounded operatorsL : W →W . Φ can be represented by a tensor of order 4, i.e. we let Φ = (Φijkℓ) and defineits action on α ∈ Rn×n by

(Φ(x,y)α)ij =

n∑k,ℓ=1

Φ(x,y)ijkℓαkℓ. (4.1)

Let us introduce reproducing kernel Hilbert spaces with values in the Hilbert space W .Definition 4.1. The Hilbert space H(O;W ) of functions g : O →W with inner product

⟨·, ·⟩H(O;W ) is called a reproducing kernel Hilbert space if there is a function Φ : O × O →L(W ) with

1. Φ(·,x)α ∈ H(O;W ) for all x ∈ O and all α ∈W .2. ⟨g(x), α⟩W = ⟨g(·),Φ(·,x)α⟩H(O;W ) for all g ∈ H(O;W ), all x ∈ O and all α ∈W .

The function Φ is called the reproducing kernel of H(O;W ).Let σ > n/2 and let ϕ : O × O → R be a positive definite, reproducing kernel of

the Sobolev space Hσ(O;R). For example, Wendland’s compactly supported radial basisfunction ψℓ,k : R+

0 → R, see [22], with ℓ = ⌊n2 ⌋ + k + 1, k ∈ N is a reproducing kernel ofHσ(O;R) with σ = k + n+1

2 with equivalent norm.By [12, Lemma 3.2, Corollary 3.3], Hσ(O; Sn×n) is a reproducing kernel Hilbert space

with positive definite reproducing kernel Φ defined by

Φ(x,y)ijkℓ := ϕ(x,y)δikδjℓ (4.2)

for x,y ∈ O and 1 ≤ i, j, k, ℓ ≤ n. Note that Φ(x,y) maps Sn×n to Sn×n.After having introduced the relevant reproducing kernel Hilbert spaces, let us now focus

on the approximation of the PDE problem under consideration. In particular, given Nlinearly independent functionals λ1, . . . , λN ∈ Hσ(O; Sn×n)∗ and the N values rk ∈ R,k = 1, . . . , N , given by an element M ∈ Hσ(O;Sn×n) through rk = λk(M), we seek to

determine the optimal recovery of M , defined to be M ∈ Hσ(O; Sn×n) solving

min∥M∥Hσ(O;Sn×n) : M ∈ Hσ(O; Sn×n) with λi(M) = ri, i = 1, . . . , N.

The solution is given by, see [12, Corollary 2.7]

M(x) =N∑k=1

βk∑

1≤µ≤ν≤n

λyk(Φ(y,x)Esµν)E

sµν , (4.3)

where the superscript y denotes the application of the functional with respect to y and thecoefficients βk ∈ R are determined by

λxi

N∑k=1

βk∑

1≤µ≤ν≤n

λyk(Φ(y,x)Esµν)E

sµν

= ri

for i = 1, . . . , N .

13

For the approximation of (2.3) and (2.4) we define for all x ∈ Rn with f(x) = 0

V (x) = Df(x)− f(x)f(x)T (Df(x) +Df(x)T )

∥f(x)∥2(4.4)

so that, see (2.1)

LM(x) =M ′(x) + V (x)TM(x) +M(x)V (x).

We set σ = s − 1 and require σ > n/2 + 1. This is to ensure that for M ∈ Hσ(O; Sn×n)we have LM ∈ Hσ−1(O; Sn×n) with σ − 1 > n/2. Hence, LM is continuous and the pointevaluations in (4.5) are well defined.

We fix the pairwise distinct collocation pointsX = x1, . . . ,xN ⊂ O as well as the point

x0 ∈ O and assume that O ⊂ A(Ω). Define the linear functionals λ(i,j)k , λ0 : H

σ(O; Sn×n) →R for 1 ≤ i ≤ j ≤ n, 1 ≤ k ≤ N by

λ(i,j)k (M) := eTi LM(xk)ej (4.5)

=: eTi LkMej , (4.6)

λ0(M) = f(x0)TM(x0)f(x0). (4.7)

In the next theorem we will show the linear independence of these functionals.Theorem 4.2. Let O ⊂ A(Ω) be a domain with Lipschitz boundary. Let σ > n/2 + 1

and let Φ : O×O → L(Sn×n) be a reproducing kernel of Hσ(O; Sn×n) and let s = σ+1. LetX = x1, . . . ,xN ⊆ O be pairwise distinct points and x0 ∈ O such that f(xi) = 0 for all

i = 0, . . . , N . Let c0 ∈ R+, and let λ(i,j)k , λ0 ∈ Hσ(O; Sn×n)∗, 1 ≤ k ≤ N and 1 ≤ i ≤ j ≤ n

be defined by (4.5) and (4.7).

Then these functionals are linearly independent and there is a unique function M ∈Hσ(O;Sn×n) solving

min

∥M∥Hσ(O;Sn×n) : λ

(i,j)k (M) = −Cij(xk), 1 ≤ i ≤ j ≤ n, 1 ≤ k ≤ N

and λ0(M) = c0∥f(x0)∥4,

where C(x) = PTx B(x)Px and B(x) = (Bij(x))i,j=1,...,n is a symmetric, positive definitematrix for each x ∈ O.

It has the form

M(x) =N∑k=1

∑1≤i≤j≤n

γ(i,j)k

[ n∑µ=1

Lk(Φ(·,x)·,·,µ,µ)ijEµµ

+1

2

n∑µ,ν=1µ=ν

[Lk(Φ(·,x)·,·,µ,ν)ij + Lk(Φ(·,x)·,·,ν,µ)ij ]Eµν]

+γ0

n∑i,j=1

fi(x0)fj(x0)

[ n∑µ=1

Φ(x0,x)i,j,µ,µEµµ

+1

2

n∑µ,ν=1µ=ν

[Φ(x0,x)i,j,µ,ν +Φ(x0,x)i,j,ν,µ]Eµν

]. (4.8)

14

where the coefficients γk = (γ(i,j)k )1≤i≤j≤n and γ0 ∈ R are determined by λ

(i,j)ℓ (M) =

−Cij(xℓ) for 1 ≤ i ≤ j ≤ n, 1 ≤ ℓ ≤ N and λ0(M) = c0∥f(x0)∥4.If the kernel Φ is given by (4.2), then M is given by

M(x) =N∑k=1

n∑i,j=1

β(i,j)k

n∑µ,ν=1

Lk(Φ(·, x)·,·,µ,ν)ijEµν

+β0ϕ(x0,x)f(x0)f(x0)T (4.9)

where the coefficients βk = (β(i,j)k )1≤i,j≤n ∈ Sn×n and β0 ∈ R are given by β0 = γ0,

β(i,i)k = γ

(i,i)k and β

(i,j)k = β

(j,i)k = 1

2γ(i,j)k for i < j.

Remark 4.3. We will later solve the system of linear equations for γ of size N n(n+1)2 +1,

and then determine the β. Note that the system for γ is of a smaller size, but the form ofM in (4.9) is beneficial for computations; see also Appendix A.

Proof. To show the linear independence we assume that

N∑k=1

∑1≤i≤j≤n

d(i,j)k λ

(i,j)k + d0λ0 = 0 (4.10)

on Hσ(O; Sn×n) with certain coefficients d(i,j)k , d0 ∈ R. We need to show that all d

(i,j)k = 0

and d0 = 0.

Let g ∈ C∞0 (Rn;R) be a nonnegative, compactly supported function with support

B(0, 1), satisfying g(x) = 1 on B(0, 1/2). Fix 1 ≤ ℓ ≤ N , as well as i∗, j∗ ∈ 1, . . . , nwith i∗ ≤ j∗. Since f(xℓ) = 0, there is ι ∈ 1, . . . , n such that fι(xℓ) = 0.

The function

gℓ(x) = (x− xℓ)ιg

(x− xℓq

),

where q := minx,y∈X∪x0,x =y ∥x−y∥ denotes the separation distance of X ∪x0, satisfiesgℓ(xk) = 0 for all k = 0, . . . , N . Moreover, ∂igℓ(xk) = 0 for all i = 1, . . . , n and xk = xℓ.Finally, we have ∂igℓ(xℓ) = 0 for i = ι and ∂ιgℓ(xℓ) = 1.

Hence, defining the matrix-valued function G ∈ Hσ(O;Sn×n) by G(x) = gℓ(x)Esi∗j∗ , we

have

0 =N∑k=1

∑1≤i≤j≤n

d(i,j)k λ

(i,j)k (G) + d0λ0(G) by (4.10)

=N∑k=1

∑1≤i≤j≤n

d(i,j)k eTi [G

′(xk) + V (xk)TG(xk) +G(xk)V (xk)]ej

+d0f(x0)TG(x0)f(x0) by (4.5) and (4.7)

We have

eTi∗Esi∗j∗ej∗ =

1√2+

(1− 1√

2

)δi∗j∗ =

1 for i∗ = j∗

1√2

for i∗ = j∗.

15

Hence, by definition of G we have

0 =N∑k=1

d(i∗,j∗)k ∇gℓ(xk) · f(xk)

[1√2+

(1− 1√

2

)δi∗j∗

]= d

(i∗,j∗)ℓ fι(xℓ)

[1√2+

(1− 1√

2

)δi∗j∗

],

which shows d(i∗,j∗)ℓ = 0. Hence, we can conclude d

(i∗,j∗)ℓ = 0 for all ℓ = 1, . . . , N and

i∗, j∗ ∈ 1, . . . , n with i∗ ≤ j∗.To show that d0 = 0, choose now G(x) = I. Since all other coefficients vanish, we are

left with

0 = d0λ0(G)

= d0f(x0)TG(x0)f(x0)

= d0∥f(x0)∥22which shows d0 = 0. This shows the linear independence.

By (4.3), the minimiser has the form

M(x) =N∑k=1

∑1≤i≤j≤n

γ(i,j)k

∑1≤µ≤ν≤n

λ(i,j)k (Φ(·,x)Esµν)Esµν

+γ0∑

1≤µ≤ν≤n

λ0(Φ(·,x)Esµν)Esµν ,

where the coefficients γk = (γ(i,j)k )1≤i≤j≤n ∈ Sn×n and γ0 ∈ R are determined by λ

(i,j)ℓ (M) =

−Cij(xℓ) for 1 ≤ i ≤ j ≤ n, 1 ≤ ℓ ≤ N and λ0(M) = c0∥f(x0)∥4.We will now show (4.8). Indeed, by (4.1) we have(

Φ(·,x)Esµν)ij=

n∑k,ℓ=1

Φ(·,x)ijkℓ(Esµν)kℓ.

For µ = ν we have

λ(i,j)k (Φ(·,x)Esµµ)Esµµ = Lk(Φ(·,x)·,·,µ,µ)ijEµµ

λ0(Φ(·,x)Esµµ)Esµµ =

n∑i,j=1

fi(x0)fj(x0)Φ(x0,x)i,j,µ,µEµµ .

For µ < ν we have

λ(i,j)k (Φ(·,x)Esµν)Esµν

=1√2(Lk(Φ(·,x)·,·,µ,ν)ij + Lk(Φ(·,x)·,·,ν,µ)ij)

1√2(Eµν + Eνµ)

=1

2(Lk(Φ(·,x)·,·,µ,ν)ij + Lk(Φ(·,x)·,·,ν,µ)ij) (Eµν + Eνµ)

and

λ0(Φ(·,x)Esµν)Esµν

=1

2

n∑i,j=1

fi(x0)fj(x0) (Φ(x0,x)i,j,µ,ν +Φ(x0,x)i,j,ν,µ) (Eµν + Eνµ).

16

This shows (4.8).

We define the symmetric matrices βk ∈ Sn×n by β(j,i)k = β

(i,j)k = 1

2γ(i,j)k if i < j and

β(i,i)k = γ

(i,i)k . Moreover, let β0 = γ0.

To show (4.9), we follow the arguments in the proof of [12, Theorem 5.2] for the firstterms. For λ0 we use Φ(x0,x)i,j,µ,ν = ϕ(x0,x)δiµδjν . Hence,

n∑i,j=1

fi(x0)fj(x0)

[ n∑µ=1

Φ(x0,x)i,j,µ,µEµµ +1

2

n∑µ,ν=1µ=ν

[Φ(x0,x)i,j,µ,ν +Φ(x0,x)i,j,ν,µ]Eµν

]

=n∑µ=1

fµ(x0)fµ(x0)ϕ(x0,x)Eµµ

+1

2ϕ(x0,x)

n∑µ,ν=1µ=ν

[fµ(x0)fν(x0) + fν(x0)fµ(x0)]Eµν

= ϕ(x0,x)

n∑µ,ν=1

fµ(x0)fν(x0)Eµν

= ϕ(x0,x)f(x0)f(x0)T ,

which shows the theorem.We will now establish an error estimate, which depends on the fill distance of the points

X = x1, . . . ,xN in O defined by hX,O = supx∈Ominxj∈X ∥x − xj∥2. We thus showa constructive converse theorem, showing that we can construct a contraction metric viameshless collocation if the fill distance is sufficiently small.

Theorem 4.4. Let f ∈ Cs(Rn;Rn), N ∋ s > n/2 + 3 and set σ = s − 1. Let Ω be anexponentially stable periodic orbit of x = f(x) with basin of attraction A(Ω).

Let B ∈ Cσ(Rn, Sn×n) such that B(x) is a positive definite matrix for all x ∈ Rn andlet C(x) = PTx B(x)Px.

Let M ∈ Cσ(A(Ω), Sn×n) be the solution of (2.3) and (2.4). Let O ⊆ A(Ω) be a bounded

domain with Lipschitz boundary. Finally, let M be the optimal recovery from Theorem 4.2.Then, we have the error estimates

∥LM − LM∥L∞(O;Sn×n) ≤ Chσ−1−n/2X,O ∥M∥Hσ(O;Sn×n) (4.11)

∥∂iLM − ∂iLM∥L∞(O;Sn×n) ≤ Chσ−2−n/2X,O ∥M∥Hσ(O;Sn×n) (4.12)

for all X ⊆ O and all i = 1, . . . , n with sufficiently small hX,O. By construction we have

f(x0)T M(x0)f(x0) = c0∥f(x0)∥4 (4.13)

Let K ∋ x0 be a compact set such that γ+(K) ⊂ O. Then M is a contraction metric in

γ+(K), i.e. M(x) is positive definite for all x ∈ K and LM(x) ≤ −ν < 0 for all x ∈ γ+(K),

provided hX,O is sufficiently small.

Proof. Note that LM − LM ∈ Hσ−1(O;Sn×n) vanishes on the set X. We can nowapply a result from [18], see [11, Theorem 2.5] for references, to each entry of the matrix

LM − LM to obtain

∥LM − LM∥L∞(O;Sn×n) ≤ Chσ−1−n/2X,O ∥L(M − M)∥Hσ−1(O;Sn×n)

≤ Chσ−1−n/2X,O ∥M − M∥Hσ(O;Sn×n)

≤ Chσ−1−n/2X,O ∥M∥Hσ(O;Sn×n),

17

since L is a differential operator of order 1. Similarly, we have

∥∂i(LM − LM)∥L∞(O;Sn×n) ≤ ∥LM − LM∥W 1∞(O;Sn×n)

≤ Chσ−2−n/2X,O ∥L(M − M)∥Hσ−1(O;Sn×n)

≤ Chσ−2−n/2X,O ∥M − M∥Hσ(O;Sn×n)

≤ Chσ−2−n/2X,O ∥M∥Hσ(O;Sn×n).

We can now use Theorem 3.1 to conclude that M is a contraction metric in K if hX,Ois sufficiently small.

5. Examples. Meshfree collocation allows for the use of scattered collocation points.The smaller the fill distance, the smaller is the error. However, the smaller the separationdistance, the larger is the condition number of the collocation matrix. To balance the two,we have chosen the collocation points on a grid. The optimal choice would be a hexagonalgrid, but here we have used a cartesian one.

5.1. Unit circle. As a first example, we consider the following systemx = x(1− x2 − y2)− yy = y(1− x2 − y2) + x

(5.1)

where the unit circle is an exponentially stable periodic orbit and the origin is an unstableequilibrium.

-1

2

-0.5

1

y

0

0

LM

neg

-1

2

0.5

x

1.510.50-2 -0.5-1-1.5-2

1

0

2

0.5

1

M p

os

1.5

y

0

2

2

x

1.510.50-0.5-2 -1-1.5-2

Fig. 5.1. Left: the area where LM

(x) is negative is the area where the function attains the value 1.

Right: the area where M(x) is positive definite is the area where the function attains the value 2.

We choose B(x) = I and the collocation points X = 1.615 Z

2 ∩ (x, y) ∈ R2 | 0.25 <√x2 + y2 < 1.5 as well as the point x0 = (1, 0) with c0 = 1. We use the kernel given

by (4.2), where ϕ(x,y) = ψ6,4(∥x − y∥2) is given by the Wendland function ψ6,4(r) =(1 − r)10+ [25 + 250r + 1, 050r2 + 2, 250r3 + 2, 145r4] and x+ = x for x ≥ 0 and x+ = 0for x < 0. The corresponding Sobolev space is H5.5(O; S2×2). This results in N = 600collocation points and thus a collocation matrix of size 3N + 1 = 1, 801.

The points x where LM(x) is negative are those, where the function described in Section

B attains 1, see Figure 5.1, left. The points x where M(x) is positive definite are those,where the function described in Section B attains 2, see Figure 5.1, right.

Figure 5.2 shows the collocation points, the boundary of the area, where LM

is negative

(red), the boundary of the area, where M is positive definite (blue) and the periodic orbit

18

-1.5 -1 -0.5 0 0.5 1 1.5

x

-1.5

-1

-0.5

0

0.5

1

1.5

y

Fig. 5.2. The collocation points (black), the boundary of the area, where LM

is negative (red), the

boundary of the area, where M is positive definite (blue) and the periodic orbit (green).

(green). The area, where LM

is negative is the area, where the collocation points are placed,

while M is positive definite in an even larger area.

5.2. Van der Pol. We consider the van der Pol system, given byx = yy = −x+ (1− x2)y

(5.2)

which has an exponentially stable periodic orbit; the origin is an unstable equilibrium.

-1

4

-0.5

2

y

0

0

LM

neg

32

1-2

x

0.5

0-1

-2-4 -3

1

-24

-1

2

0

M p

os

3

1

y

0 21

2

x

0-2 -1-2-4 -3

Fig. 5.3. Left: the area where LM

(x) is negative is the area where the function attains the value 1.

Right: the area where M(x) is positive definite is the area where the function attains the value 2.

We choose B(x) = I and the collocation points X =(2.335 Z× 3.1

45 Z)

∩([−2.3, 2.3]× [−3.1, 3.1]) ∩ (x, y) ∈ R2 | 0.8 <

√x2 + y2 as well as the point x0 = (2, 0)

with c0 = 1. We use again the kernel given by the Wendland function ψ6,4. This results inN = 6, 022 collocation points and thus a collocation matrix of size 3N + 1 = 18, 067.

The points x where LM(x) is negative are those, where the function described in Section

B attains 1, see Figure 5.3, left. The points x where M(x) is positive definite are those,where the function described in Section B attains 2, see Figure 5.3, right.

19

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x

-3

-2

-1

0

1

2

3

y

Fig. 5.4. The collocation points (black), the boundary of the area where LM

is negative (red) and

where M is positive definite (blue), together with the periodic orbit (green).

Figure 5.4 shows the collocation points (black), the boundary of the area where LM

is negative (red) and where M is positive definite (blue), together with the periodic orbit

(green). The limiting factor is clearly the positive definiteness of M , which does not coverthe whole area, where the collocation points are placed. Note that this area is not positivelyinvariant, hence, since we require estimates on γ+(K), Theorem 3.1 is only applicable to asmaller set K.

-0.5

1

0.5

0

z

y

0

-0.5 10.5

x

0-1

0.5

-0.5-1

-0.5

1.5

0

z

1

0.5

0.5

y

0

-0.51.5

1

x

0.5-10

-0.5-1-1.5

-1.5

Fig. 5.5. The collocation points (black) and the periodic orbit (green) together with left: the boundary

of the area where LM

is negative (red), right: the boundary of the area where M is positive definite (blue).

5.3. Three-dimensional example. We consider the following three-dimensional sys-tem x = x(1− x2 − y2)− y + 0.1yz

y = y(1− x2 − y2) + xz = −z + xy

(5.3)

which has an exponentially stable periodic orbit.We choose B(x) = I and the collocation points X =

(1.39 Z2 × 0.1Z

)∩ (x, y, z) ∈ R3 |

0.75 <√x2 + y2 < 1.25, |z| < 0.45 as well as the point x0 = (1, 0, 0) with c0 = 1. We use

20

-0.5

1.5

1

0z0.5

y

0

0.5

1.5-0.51

0.5

x

-1 0-0.5

-1-1.5 -1.5

Fig. 5.6. The collocation points (black), the boundary of the area where LM

is negative (red) and

where M is positive definite (blue), together with the periodic orbit (green).

again the kernel given by the Wendland function ψ6,4, the corresponding Sobolev space isH6(O; S3×3). This results in N = 1, 368 collocation points and thus a collocation matrix ofsize 6N + 1 = 8, 209.

Figures 5.5 and 5.6 show the collocation points (black), the boundary of the area where

LM

is negative (red) and where M is positive definite (blue), together with the periodicorbit (green).

6. Conclusions and further work. We have proposed a computationally efficientnumerical method to construct a contraction metric for a periodic orbit. A contractionmetric is a matrix-valued function, which satisfies a contraction criterion. We consider acertain contraction metric, satisfying a linear, first-order PDE and approximate it usingmeshfree collocation. We have proved that the approximation, if sufficiently close, is itselfa contraction metric. We have obtained error estimates for the meshfree collocation, whichshow that the approximation is sufficiently close if the collocation points are dense enough.

The method can be further improved by fully exploiting the advantages of meshfreecollocation. In particular, one could start with a coarse set of collocation points and refine,where the conditions of a contraction metric are not fulfilled. A further improvement couldinclude a posteriori estimates, that can be obtained by using Taylor-type estimates or byinterpolating with a CPA function, similar to [10] and [9] for Lyapunov functions. Todetermine a positively invariant set, one could first seek to compute a Lyapunov function.In areas where the Lyapunov function does not have negative orbital derivative, such as ina neighborhood of a periodic orbit, we can then employ the method described in this paper.The advantage of this combined method is twofold: on the one hand the computation of aLyapunov function requires less computational effort, as it is a scalar-valued function. Onthe other hand, the sublevel set of a Lyapunov function provides us with a compact andpositively invariant set.

Appendix A. Explicit formulas for the calculations.

To derive explicit formulas, let us choose a radially symmetric kernel of the form

ϕ(x,y) = ψ0(∥x − y∥2) and denote ψi+1(r) = dψi(r)/drr for i = 0, 1 and r > 0. We as-

sume that ψ1 and ψ2 can be continuously extended up to r = 0; this is, e.g. the case for

21

sufficiently smooth Wendland functions. We use the kernel Φ of the form (4.2), hence

Φ(·,x)ijµν = ψ0(∥ · −x∥2)δiµδjν . (A.1)

Thus, for the linear operators Lk, see (4.5), we have

(Lk(M))ij =n∑p=1

Vpi(xk)Mpj(xk) +n∑p=1

Mip(xk)Vpj(xk)

+n∑p=1

∂pMij(xk)fp(xk)

(Lk(Φ(·,x))·,·,µ,ν)ij =n∑p=1

ψ0(∥xk − x∥2)Vpi(xk)δpµδjν

+

n∑p=1

ψ0(∥xk − x∥2)δiµδpνVpj(xk)

+n∑p=1

ψ1(∥xk − x∥2)(xk − x)pfp(xk)δiµδjν

= ψ0(∥xk − x∥2)Vµi(xk)δjν + ψ0(∥xk − x∥2)δiµVνj(xk)+ψ1(∥xk − x∥2)⟨xk − x, f(xk)⟩δiµδjν ,

where ⟨·, ·⟩ denotes the standard scalar product in Rn.

Now we can compute M(x), using (4.9) of Theorem 4.2. We have

M(x) =N∑k=1

n∑i,j=1

β(i,j)k

n∑µ,ν=1

(Lk(Φ(·,x))·,·,µ,ν)ijEµν

+β0ψ0(∥x0 − x∥2)f(x0)f(x0)T

=N∑k=1

[ n∑i,µ,ν=1

β(i,ν)k ψ0(∥xk − x∥2)Vµi(xk)Eµν

+

n∑j,µ,ν=1

β(µ,j)k ψ0(∥xk − x∥2)Vνj(xk)Eµν

+n∑

µ,ν=1

β(µ,ν)k ψ1(∥xk − x∥2)⟨xk − x, f(xk)⟩Eµν

]+β0ψ0(∥x0 − x∥2)f(x0)f(x0)

T

=N∑k=1

[ψ0(∥xk − x∥2)

[V (xk)βk + βkV (xk)

T]

+ψ1(∥xk − x∥2)⟨xk − x, f(xk)⟩βk]

+β0ψ0(∥x0 − x∥2)f(x0)f(x0)T . (A.2)

22

Hence,

LM(x) =N∑k=1

ψ0(∥xk − x∥2)[V (x)TV (xk)βk + V (x)TβkV (xk)

T

+V (xk)βkV (x) + βkV (xk)TV (x)

]+

N∑k=1

ψ1(∥xk − x∥2)⟨xk − x, f(xk)⟩[V (x)Tβk + βkV (x)

]+

N∑k=1

ψ1(∥xk − x∥2)⟨x− xk, f(x)⟩[V (xk)βk + βkV (xk)

T]

−N∑k=1

ψ1(∥xk − x∥2)⟨f(x), f(xk)⟩βk

+N∑k=1

ψ2(∥xk − x∥2)⟨xk − x, f(xk)⟩⟨x− xk, f(x)⟩βk

+β0ψ0(∥x0 − x∥2)[V (x)T f(x0)f(x0)T + f(x0)f(x0)

TV (x)]

+β0ψ1(∥x0 − x∥2)⟨x− x0, f(x)⟩f(x0)f(x0)T . (A.3)

Observe that LM(x) is a symmetric matrix if all βk, k = 1, . . . , N are symmetric.

After establishing the formulas for M and LM , let us now consider the linear systemfor the coefficients γ and β, respectively.

Let us first calculate the coefficients b(ℓ,i,j),(k,µ,ν), b0,(k,µ,ν), b(ℓ,i,j),0 and b0,0 for 1 ≤k, ℓ ≤ N , 1 ≤ i, j, µ, ν ≤ n such that

f(x0)T M(x0)f(x0) =

N∑k=1

n∑µ,ν=1

b0,(k,µ,ν)β(µ,ν)k + b0,0β0. (A.4)

LM(xℓ)i,j =N∑k=1

n∑µ,ν=1

b(ℓ,i,j),(k,µ,ν)β(µ,ν)k + b(ℓ,i,j),0β0. (A.5)

We have by (A.2)

f(x0)T M(x0)f(x0) =

N∑k=1

[ψ0(∥xk − x0∥2)f(x0)

T[V (xk)βk + βkV (xk)

T]f(x0)

+ψ1(∥xk − x0∥2)⟨xk − x0, f(xk)⟩f(x0)Tβkf(x0)

]+β0ψ0(0)∥f(x0)∥4

and thus

b0,(k,µ,ν) = ψ0(∥xk − x0∥2)[ n∑p=1

Vpµ(xk)fp(x0)fν(x0) +n∑p=1

Vpν(xk)fp(x0)fµ(x0)

]+ψ1(∥xk − x0∥2)⟨xk − x0, f(xk)⟩fµ(x0)fν(x0) (A.6)

b0,0 = ψ0(0)∥f(x0)∥4. (A.7)

23

By (A.3) we have

b(ℓ,i,j),(k,µ,ν) = ψ0(∥xk − xℓ∥2)[ n∑p=1

Vpi(xℓ)Vpµ(xk)δνj + Vµi(xℓ)Vjν(xk)

+Viµ(xk)Vνj(xℓ) + δiµ

n∑p=1

Vpν(xk)Vpj(xℓ)

]+ψ1(∥xk − xℓ∥2)⟨xk − xℓ, f(xk)⟩ [Vµi(xℓ)δνj + δiµVνj(xℓ)]

+ψ1(∥xk − xℓ∥2)⟨xℓ − xk, f(xℓ)⟩ [Viµ(xk)δνj + δiµVjν(xk)]

−ψ1(∥xk − xℓ∥2)⟨f(xℓ), f(xk)⟩δiµδjν+ψ2(∥xk − xℓ∥2)⟨xk − xℓ, f(xk)⟩⟨xℓ − xk, f(xℓ)⟩δiµδjν (A.8)

and b(ℓ,i,j),0 = ψ0(∥x0 − xℓ∥2)[ n∑p=1

Vpi(xℓ)fp(x0)fj(x0) +n∑p=1

Vpj(xℓ)fp(x0)fi(x0)]

+ψ1(∥x0 − xℓ∥2)⟨xℓ − x0, f(xℓ)⟩fi(x0)fj(x0). (A.9)

It is now easy to see that

b(ℓ,i,j),(k,µ,ν) = b(ℓ,j,i),(k,ν,µ), (A.10)

b(ℓ,i,j),(k,µ,ν) = b(k,µ,ν),(ℓ,i,j), (A.11)

b0,(ℓ,i,j) = b0,(ℓ,j,i), (A.12)

b(ℓ,i,j),0 = b0,(ℓ,i,j). (A.13)

We now compute the γ(µ,ν)k , which are defined by γ0 = β0, γ

(µ,µ)k = β

(µ,µ)k and 1

2γ(µ,ν)k =

β(µ,ν)k = β

(ν,µ)k for µ < ν. They solve the (smaller) linear system

N∑k=1

∑1≤µ≤ν≤n

c(ℓ,i,j),(k,µ,ν)γ(µ,ν)k + c(ℓ,i,j),0γ0 = LM(xℓ)i,j

= λ(i,j)ℓ (M)

= −Cij(xℓ) (A.14)

N∑k=1

∑1≤µ≤ν≤n

c0,(k,µ,ν)γ(µ,ν)k + c0,0γ0 = c0∥f(x0)∥4 (A.15)

for 1 ≤ ℓ ≤ N , 1 ≤ i ≤ j ≤ n. The coefficients c·,· form a symmetric (see below) matrix of

size N n(n+1)2 + 1.

Let us express the c(ℓ,i,j),(k,µ,ν) in terms of the previously calculated b(ℓ,i,j),(k,µ,ν).Noting that

N∑k=1

n∑µ,ν=1

b(ℓ,i,j),(k,µ,ν)β(µ,ν)k + b(ℓ,i,j),0β0 =

N∑k=1

∑1≤µ≤ν≤n

c(ℓ,i,j),(k,µ,ν)γ(µ,ν)k

+c(ℓ,i,j),0γ0 (A.16)

N∑k=1

n∑µ,ν=1

b0,(k,µ,ν)β(µ,ν)k + b0,0β0 =

N∑k=1

n∑1≤µ≤ν≤n

c0,(k,µ,ν)β(µ,ν)k

+c0,0γ0 (A.17)

24

as well as the definition of γ(µ,µ)k and γ0, we have from the first equation for all (ℓ, i, j)

b(ℓ,i,j),0 = c(ℓ,i,j),0 (A.18)

N∑k=1

n∑µ,ν=1

b(ℓ,i,j),(k,µ,ν)β(µ,ν)k =

N∑k=1

n∑µ=1

b(ℓ,i,j),(k,µ,µ)β(µ,µ)k

+

N∑k=1

∑1≤µ<ν≤n

(b(ℓ,i,j),(k,µ,ν)β(µ,ν)k + b(ℓ,i,j),(k,ν,µ)β

(ν,µ)k )

=N∑k=1

n∑µ=1

b(ℓ,i,j),(k,µ,µ)γ(µ,µ)k

+

N∑k=1

∑1≤µ<ν≤n

1

2(b(ℓ,i,j),(k,µ,ν) + b(ℓ,i,j),(k,ν,µ))γ

(µ,ν)k .(A.19)

Comparing (A.19) to (A.16) gives, using (A.10)

c(ℓ,i,i),(k,µ,µ) = b(ℓ,i,i),(k,µ,µ)

c(ℓ,i,i),(k,µ,ν) =1

2

(b(ℓ,i,i),(k,µ,ν) + b(ℓ,i,i),(k,ν,µ)

)= b(ℓ,i,i),(k,µ,ν)

c(ℓ,i,j),(k,µ,µ) = b(ℓ,i,j),(k,µ,µ) =1

2

(b(ℓ,i,j),(k,µ,µ) + b(ℓ,j,i),(k,µ,µ)

)c(ℓ,i,j),(k,µ,ν) =

1

4

(b(ℓ,i,j),(k,µ,ν) + b(ℓ,j,i),(k,ν,µ) + b(ℓ,i,j),(k,ν,µ) + b(ℓ,j,i),(k,µ,ν)

)=

1

2

(b(ℓ,i,j),(k,µ,ν) + b(ℓ,i,j),(k,ν,µ)

)(A.20)

where we assume µ < ν and i < j.From(A.17) we have

c0,0 = b0,0 (A.21)

N∑k=1

n∑µ,ν=1

b0,(k,µ,ν)β(µ,ν)k =

N∑k=1

n∑µ=1

b0,(k,µ,µ)β(µ,µ)k

+N∑k=1

∑1≤µ<ν≤n

(b0,(k,µ,ν)β(µ,ν)k + b0,(k,ν,µ)β

(ν,µ)k )

=N∑k=1

n∑µ=1

b0,(k,µ,µ)γ(µ,µ)k

+N∑k=1

∑1≤µ<ν≤n

1

2(b0,(k,µ,ν) + b0,(k,ν,µ))γ

(µ,ν)k . (A.22)

Comparing (A.22) to (A.17) gives, using (A.12)

c0,(k,µ,ν) =1

2

(b0,(k,µ,ν) + b0,(k,ν,µ)

)= b0,(k,µ,ν), (A.23)

where we assume µ < ν. The matrix c·,· is symmetric due to (A.11) and (A.13).

25

Summarising, for the computations we calculate the coefficients b·,· using (A.6), (A.7),(A.8) and (A.9), and then the symmetric matrix c·,· using (A.20), (A.21) and (A.23). Then

we determine γ(µ,ν)k and γ0 by solving (A.14) and (A.15) and compute βk ∈ Sn×n from γk;

recall that β(j,i)k = β

(i,j)k = 1

2γ(i,j)k if i < j and β

(i,i)k = γ

(i,i)k as well as β0 = γ0. M(x) and

LM(x) are then given by (A.2) and (A.3).

Appendix B. Computation of the condition.Once the approximation M is calculated, we seek to show that M(x) is positive definite

and LM(x) is negative.

To check that M(x) is positive definite, we calculate the characteristic polynomial of

−M(x):

χ−M(x)(λ) = a0 + a1λ+ . . .+ an−1λ

n−1 + λn.

Now we can apply the Lienard-Chipart criterion to the coefficients to ensure that −M(x)is negative definite.

To check that LM(x) is negative, we need to show that LM(x) = V T M(x)+M(x)V (x)+

M ′(x) where V (x) = Df(x)− f(x)f(x)T (Df(x)+Df(x)T )∥f(x)∥2 satisfies vTLM(x)v < 0 for all v with

∥v∥ = 1 and v ⊥ f(x). We consider the symmetric real-valued matrix PTx LM(x)Px. Thismatrix has one eigenvalue 0 with eigenvector f(x) and for the negativity we need to showthat all other eigenvalues are negative. This follows from the fact that there exists anorthonormal basis of eigenvectors.

We calculate the characteristic polynomial of PTx LM(x)Px:

χPT

x LM(x)Px(λ) = b0 + b1λ+ . . .+ bn−1λ

n−1 + λn.

Since 0 is an eigenvalue, we have b0 = (−1)n det(PTx LM(x)Px) = 0 and we can writethe characteristic polynomial as

χPT

x LM(x)Px(λ) = λ(b1 + b2λ+ . . .+ bn−1λ

n−2 + λn−1).

We now apply the Lienard-Chipart criterion to the coefficients of

b1 + b2λ+ . . .+ bn−1λn−2 + λn−1.

In two-dimensional systems, we have a0 = det(−M(x)) = det(M(x)) and

a1 = − trace(−M(x)) = trace(M(x)) and we require a0, a1 > 0; similarly b1 =

− trace(PTx LM(x)Px) and we require b1 > 0. In the examples, we calculate the func-tion sign(a0(x))+ sign(a1(x)) and determine the area where it has the value 2 and similarlywhere sign(b1(x)) has the value 1.

In three-dimensional systems we have a0 = −det(−M(x)) = det(M(x)), a1 =∑3i=1 det(−M(x))i =

∑3i=1 det(M(x))i, where Ai denotes the 2 × 2 matrix obtained by

deleting the i-th row and i-th column of A, as well as a2 = − trace(−M(x)) = trace(M(x)).We require a2, a0 > 0 as well as a2a1 − a0 > 0.

Moreover, we have b1 =∑3i=1 det(P

Tx LM(x)Px)i as well as b2 = − trace(PTx LM(x)Px)

and require b2, b1 > 0.

26

REFERENCES

[1] G. Borg. A condition for the existence of orbitally stable solutions of dynamical systems, volume 153.Elander, 1960.

[2] M. Buhmann. Radial basis functions: theory and implementations, volume 12 of Cambridge Mono-graphs on Applied and Computational Mathematics. Cambridge University Press, Cambridge,2003.

[3] W. A. Coppel. Stability and Asymptotic Behaviour of Differential Equations. Heath, Boston, 1965.

[4] F. Forni and R. Sepulchre. A differential Lyapunov framework for contraction analysis. IEEE Trans-actions on Automatic Control, 59:614–628, 2014.

[5] P. Giesl. Necessary conditions for a limit cycle and its basin of attraction. Nonlinear Anal., 56:643–677,2004.

[6] P. Giesl. Converse theorem on a contraction metric for a periodic orbit. arXiv 1808.02689, 2018.

[7] P. Giesl. On a matrix-valued PDE characterizing a contraction metric for a periodic orbit. arXiv1808.02691, 2018.

[8] P. Giesl and S. Hafstein. Construction of a CPA contraction metric for periodic orbits using semidefiniteoptimization. Nonlinear Anal., 86:114–134, 2013.

[9] P. Giesl and S. Hafstein. Computation and Verification of Lyapunov functions. SIAM J. Appl. Dyn.Syst., 14:1663–1698, 2015.

[10] P. Giesl and N. Mohammed. Verification Estimates for the Construction of Lyapunov Functions usingMesh-free Collocation. Discrete Contin. Dyn. Syst. Ser. B, 2019. in press.

[11] P. Giesl and H. Wendland. Meshless collocation: error estimates with application to Dynamical Sys-tems. SIAM J. Numer. Anal., 45(4):1723–1741, 2007.

[12] P. Giesl and H. Wendland. Kernel-based Discretisation for Solving Matrix-Valued PDEs. SIAM J.Numer. Anal., 56(6):3386–3406, 2018.

[13] P. Hartman. Ordinary Differential Equations. Wiley, New York, 1964.

[14] P. Hartman and C. Olech. On global asymptotic stability of solutions of differential equations. Trans.Amer. Math. Soc., 104:154–178, 1962.

[15] A. Yu. Kravchuk, G. A. Leonov, and D. V. Ponomarenko. Criteria for strong orbital stability oftrajectories of dynamical systems. I. Differentsialnye Uravneniya, 28(9):1507–1520, 1652, 1992.

[16] G. A. Leonov, I. M. Burkin, and A. I. Shepelyavyi. Frequency Methods in Oscillation Theory. Ser.Math. and its Appl.: Vol. 357, Kluwer, 1996.

[17] I. R. Manchester and J.-J. E. Slotine. Transverse contraction criteria for existence, stability, androbustness of a limit cycle. Systems Control Lett., 63:32–38, 2014.

[18] F. J. Narcowich, J. D. Ward, and H. Wendland. Sobolev bounds on functions with scattered zeros,with applications to radial basis function surface fitting. Math. Comp., 74(250):743–763, 2005.

[19] M. J. D. Powell. The theory of radial basis function approximation in 1990. In Advances in numericalanalysis, Vol. II (Lancaster, 1990), Oxford Sci. Publ., pages 105–210. Oxford Univ. Press, NewYork, 1992.

[20] R. Schaback and H. Wendland. Kernel techniques: from machine learning to meshless methods. ActaNumer., 15:543–639, 2006.

[21] B. Stenstrom. Dynamical systems with a certain local contraction property. Math. Scand., 11:151–155,1962.

[22] H. Wendland. Error estimates for interpolation by compactly supported Radial Basis Functions ofminimal degree. J. Approx. Theory, 93:258–272, 1998.

[23] H. Wendland. Scattered data approximation, volume 17 of Cambridge Monographs on Applied andComputational Mathematics. Cambridge University Press, Cambridge, 2005.

27