Equivariant Pyragas control of discrete waves Babette de Wol Abstract

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Equivariant Pyragas control of discrete waves

Babette de Wolﬀ∗

Abstract

Equivariant Pyragas control is a delayed feedback method that aims to stabilize spatio-temporal

patterns in systems with symmetries. In this article, we apply equivariant Pyragas control to discrete

waves, which are periodic solutions that have a ﬁnite number of spatio-temporal symmetries. We prove

suﬃcient conditions under which a discrete wave can be stabilized via equivariant Pyragas control. The

result is applicable to a broad class of discrete waves, including discrete waves that are far away from

a bifurcation point. Key ingredients of the proof are an adaptation of Floquet theory to systems with

symmetries, and the use of characteristic matrix functions to reduce the inﬁnite dimensional eigenvalue

problem to a one dimensional zero ﬁnding problem.

AMS Subject classiﬁcation: 34K20, 34K35, 93C23, 37C81.

Key words: delayed feedback control, equivariant Pyragas control, spatio-temporal patterns.

1 Introduction

In [Pyr92], Kestitutis Pyragas introduced a delayed feedback control scheme (now known as ‘Pyragas control’)

that aims to stabilize periodic motion. Pyragas considers a system without feedback that is described by an

ordinary diﬀerential equation (ODE)

˙x(t) = f(x(t)), t ≥0 (1.1)

with f:RN→RN. Pyragas then introduces a feedback term that measures the diﬀerence between the

current state and the state time pago, and then feeds this diﬀerence (multiplied by a matrix) back into the

system. Concretely, the system with feedback control becomes

˙x(t) = f(x(t)) + B[x(t)−x(t−p)] (1.2)

with gain matrix B∈RN×N. For a periodic solution with period p, the diﬀerence between the current state

and the state time pago is zero. Hence a p-periodic solution of the original system (1.1) is also a solution

of the feedback system (1.2). However, the global dynamics of the systems (1.1) and (1.2) are radically

diﬀerent, and we can try to choose the matrix B∈RN×Nin such a way that an unstable p-periodic solution

of (1.1) is a stable solution of (1.2).

If the original system (1.1) has built-in symmetries, its periodic solutions can satisfy additional spatio-

temporal relations. In this case, one can adapt the Pyragas control scheme so that it vanishes on solutions

with a prescriped spatio-temporal relation. As in [FFS10], we write the feedback system as

˙x(t) = f(x(t)) + B[x(t)−hx(t−θh)] (1.3)

with time delay θh>0 and h∈RN×Na linear, spatial transformation. The control term now feeds back

the diﬀerence between the current state and a spatio-temporal transformation of the state, and vanishes on

spatio-temporal patterns of the form hx(t) = x(t+θh). The feedback scheme (1.3) has the advantage that it

is able to select a prescribed spatio-temporal pattern amongst a family of periodic solutions with the same

∗Vrije Universiteit Amsterdam, Department of Mathematics, b.wolﬀ@vu.nl

arXiv:2210.02211v1 [math.DS] 5 Oct 2022

period, and in such situations can indeed be more succesful in stabilizing a speciﬁc pattern than Pyragas

control [SB16, PBS13]. Since symmetries of the uncontrolled system (1.1) are often described in terms of

equivariance relations, we refer to the control scheme (1.3) as equivariant Pyragas control.

Implementation of Pyragas control requires knowledge of the period of the targeted solution, but uses

no additional information on the original system (1.1). This ‘model-independence’ makes Pyragas control

widely applicable; for example in semiconductor lasers [SHW+06, SWH11], CO2-lasers [BDG94] and enzy-

matic reactions [LFS95]; the paper [Pyr92] has currently (October 2022) more than 3100 citations. The

equivariant control scheme (1.3) has recently been experimentally implemented in networks of chemical

oscillators [HGTS].

While Pyragas control has many experimental realizations, proving mathematically rigorous results on

Pyragas control is challenging. This is mainly because, from a mathematical perspective, the controlled

systems (1.2) and (1.3) are delay diﬀerential equations (DDE) that generate inﬁnite dimensional dynamical

systems. In order to associate to (1.2) (resp. (1.3)) a well-posed initial value problem, we have to provide a

function on the interval [−p, 0] (resp. [−θh,0]) as initial condition. So the state space is a function space (that

has to be speciﬁed more precisely) and the associated dynamical system is inﬁnite dimensional. Although

the abstract theory of DDE is well developed [HV93, DvGVW95], the inﬁnite dimensional nature of DDE is

still demanding when we want to perform an explicit stability analysis in concrete examples.

This article is concerned with equivariant Pyragas control of discrete waves, which are periodic solutions

that have a ﬁnite number of spatio-temporal symmetries. The main result of this article, Theorem 2.1,

provides suﬃcient conditions under which a discrete wave can be stabilized via equivariant Pyragas control.

The suﬃcient conditions are formulated in terms of eigenvalue properties of the uncontrolled system, and

the result is applicable to a broad class of discrete waves.

The results in this article in particular apply to periodic orbits that are not close to a bifurcation point and

that are ‘genuine’ periodic orbits, i.e. they cannot be transformed to a ring of equilibria of an autonomous

system. This is signiﬁcant, since in the literature so far, most analytical results on succesful stabilization by

(equivariant) Pyragas control either concern periodic orbits that bifurcate from an equilibrium [HKRH19,

FLR+20, VdW17, HBKR17] or concern rotating waves, i.e. periodic orbits that can be transformed to

equilibria of autonomous systems [PPK14, FFG+08, SB16, FFS10, SdWD22]. In both these cases, the

stability analysis simpliﬁes, because we can determine the stability of the periodic orbit by determining the

stability of an equilibrium in an autonomous system. These simpliﬁcations cannot be made in the setting

considered in Theorem 2.1 and consequently the stability problem becomes more involved.

The stability analysis we perform here is based on a combination of equivariant Floquet theory with

the theory of characteristic matrix functions. In systems without symmetry, we determine the stability of

ap-periodic solution using the monodromy operator, which involves solving the linearized equation over

a time step p. We show that in equivariant settings, we can work with the twisted monodromy operator,

which involves solving the linearized equation over only a fraction of the period. We then prove that the

twisted monodromy operator has a characteristic matrix function, a concept that was recently introduced

in [KV22]. A characteristic matrix function captures the spectrum of a bounded linear operator in a matrix

valued function, and using this concept we rigorously prove that the eigenvalues of the twisted monodromy

operator are zeroes of a scalar valued function. This translates the inﬁnite dimensional eigenvalue problem

of the twisted monodromy operator to a one dimensional zero ﬁnding problem, which means a signiﬁcant

dimension reduction. As the ﬁnal step, we analyze the scalar valued function and prove suﬃcient conditions

under which the control scheme (1.3) succesfully stabilizes a discrete wave.

This article is structured as follows. In Section 2, we state the main result (Theorem 2.1) in mathe-

matically precise form, after having introduced the necessary terminology. In Section 3, we describe the

symmetry relations of the controlled system (1.3); we then prove that the stability of discrete wave solutions

of (1.3) is determined by the spectrum of the twisted monodromy operator. In Section 4, we introduce

the concept of a characteristic matrix function; and show that the eigenvalues of the twisted monodromy

operator can be computed as zeroes of a scalar-valued function. We analyze this scalar-valued function in

Section 5 and subsequently prove Theorem 2.1.

Acknowledgements

The contents of this article are based upon contents of the authors doctoral thesis [dW21], written at the

Freie Universit¨at Berlin under the supervision of Bernold Fiedler. The author is grateful to Bernold Fiedler

and Sjoerd Verduyn Lunel for useful discussions and encouragment; and to Jia-Yuan Dai, Bob Rink and

Isabelle Schneider for comments on earlier versions.

2 Setting and statement of the main result

Throughout the rest of this article, we assume that the ODE (1.1) is equivariant with respect to a compact

Lie group Γ. This means that there exists a group homomorphism

ρ: Γ →GL(N, R)

(called a representation of Γ) such that

f(ρ(γ)x) = ρ(γ)f(x) (2.1)

for all x∈RNand all γ∈Γ. If now x(t) is a solution of (1.1) and γis an element of Γ, then ρ(γ)x(t)

is a solution of (1.1) as well. So the group Γ (or rather the group {ρ(γ)|γ∈Γ}) is indeed a group of

symmetries of the solutions of (1.1). In many examples, symmetries of an ODE can be eﬀectively described

using compact Lie groups; see for example the monographs [GSS88, GS02].

A compact Lie group Γ always has a orthogonal representation, i.e. there always exists a group homo-

morphism

ρ: Γ →O(N),

cf. [GSS88, p. 31]. In the rest of this article, we directly view Γ as a subgroup of the orthogonal group O(N),

instead of viewing it as an abstract compact Lie group with an orthogonal representation. Consequently we

also supress the representation in the notation, e.g. we now write the equivariance condition (2.1) as

f(γx) = γf (x)

with γ∈Γ⊆O(N) and x∈RN.

The symmetries of the ODE (1.1) naturally induce two symmetry groups on a given periodic solution.

Suppose that x∗is a periodic solution of (1.1) with minimal period p > 0; denote by O={x∗(t)|t∈R}its

orbit. Then the group

K:= {γ∈Γ|γx∗(0) = x∗(0)}

leaves the initial condition x∗(0) invariant, and the group

H:= {γ∈Γ|γO=O}

leaves the orbit Oinvariant; cf. [Fie88]. If k∈K, then kx∗(t) and x∗(t) are two solutions of (1.1) with the

same intial condition, and hence

kx∗(t) = x∗(t)

for all t∈R. So elements of the group Kleave the orbit of x∗ﬁxed pointwise and hence we refer to the

group Kas the group of spatial symmetries of x∗. For any an element h∈H, there exists a time-shift

θh∈[0, p) such that hx∗(0) = x∗(θh). But then hx∗(t) and x∗(t+θh) are both solutions of (1.1) with the

same initial condition, and hence

hx∗(t) = x∗(t+θh) (2.2)

for all t∈R. So every element of Hinduces a spatio-temporal relation of the form (2.2) on x∗, and hence

we refer to the group Has the group of spatio-temporal symmetries of x∗.

If h, g ∈Hare two spatio-temporal symmetries of x∗, then ghx∗(t) = x∗(t+θh+θg) and hence θhg =

θh+θgmod p. Thus the map

H→S1∼

=R/p Z

h7→ θh

is a group homomorphism. Since the group Kis exactly the kernel of the map H3h7→ θh, it is in particular

a normal subgroup of H, and the quotient group H/K is a subgroup of S1. This implies that

(H/K ∼

=Znfor some n∈N,or

H/K ∼

=S1,

where Zndenotes the cyclic group of order n. If H/K 'S1, the periodic solution x∗is called a rotating

wave; if H/K 'Znthe periodic solution x∗is often called a discrete wave, cf. [Fie88]. For discrete waves,

the time-shift θhassociated to spatio-temporal symmetry h∈His always rationally related to the minimal

period pof the orbit. Indeed, if h∈Hand H/K 'Zn, then necessarily hn∈K. So nθh= 0 mod pand

therefore there exists an integer m∈ {1, . . . , n}such that

θh=m

np, (2.3)

i.e. θhand pare rationally related.

Throughout the rest of this article, we focus on the stabilization of discrete waves. For future reference,

we collect the relevant assumptions on the ODE (1.1) in a seperate hypothesis.

Hypothesis 1.

1. f:RN→RNis a C2function;

2. system (1.1) has a periodic solution x∗with minimal period p > 0;

3. system (1.1) is equivariant with respect to a compact symmetry group Γ⊆O(N), i.e.

f(γx) = γf (x)for all x∈RNand γ∈Γ.(2.4)

4. The periodic solution x∗is a discrete wave, i.e. H/K 'Znfor some n∈N.

To determine whether the p-periodic discrete wave x∗is a stable solution of (1.1), we consider the linearized

equation

˙y(t) = f0(x∗(t))y(t),(2.5)

which is non-autonomous and p-periodic in its time argument. We denote by Y(t)∈RN×Nthe fundamental

solution of (2.5) with Y(0) = I, i.e. Y(t) is the matrix-valued solution of the initial value problem

dtY(t) = f0(x∗(t))Y(t), Y (0) = I.

Floquet theory implies that the eigenvalues of the monodromy operator Y(p)∈RN×Ndetermine whether

x∗is a stable solution of (1.1). The equivariance assumption in Hypothesis 1 allows us to reﬁne Floquet

theory for discrete waves. We do this in detail in Section 3; for now we just mention that in the stability

analysis of discrete waves, the operator

Yh:RN→RN, Yh=h−1Y(θh) with h∈H, (2.6)

plays an important role; we call the operator (2.6) the twisted monodromy operator (associated to h).

The twisted monodromy operator Yhalways has an eigenvalue 1 ∈C, which we call the trivial eigenvalue.

This is because diﬀerentiating the relation ˙x∗(t) = f(x∗(t)) with respect to time implies that ˙x∗(t) is a

solution of the linearized equation (2.5). So Y(t) ˙x∗(0) = ˙x∗(t) and together with (2.2) this implies that

h−1Y(θh) ˙x∗(0) = h−1x∗(θh) = ˙x∗(0).

In Section 3, we additionally prove that if Yhhas an eigenvalue |µ|>1, then x∗is an unstable solution of

(1.1).

With these preparations, we are now ready to state this article’s main result.

Theorem 2.1. Consider the ODE (1.1) satisfying Hypothesis 1. Assume that the discrete wave x∗has

a spatio-temporal symmetry h∈Hsuch that the twisted monodromy operator Yhdeﬁned in (2.6) has the

following properties:

1. The eigenvalue 1∈σ(Yh)is algebraically simple and Yhhas no other eigenvalues on the unit circle;

2. If µ∈σ(Yh)and |µ|>1, then

−e2<µ<−1.(2.7)

Then there exists an open interval I⊆(−∞,0) such that for b∈I,x∗is a stable solution of

˙x(t) = f(x(t)) + b[x(t)−hx(t−θh)] .(2.8)

Theorem 2.1 addresses equivariant Pyragas control with scalar control gain, i.e. in the control term

b[x(t)−hx(t−θh)]

the factor bis a real number rather than a matrix. The fact that Theorem 2.1 achieves stabilization with

a scalar control gain is remarkable since non-equivariant Pyragas control with a scalar control gain fails to

stabilize a rather large class of periodic solutions. Indeed, if x∗is a p-periodic solution of the ODE (1.1),

and the monodromy operator Y(p) has at least one real eigenvalue µ > 1, then x∗is an unstable solution of

the controlled system

˙x(t) = f(x(t)) + b[x(t)−x(t−p)] (2.9)

for every choice of b∈R[dWS21]. Although Theorem 2.1 makes assumptions on the eigenvalues of the twisted

monodromy operator Yh, it does not make assumptions on the eigenvalues of the monodromy operator Y(p).

Given a periodic orbit x∗, it is possible that its monodromy operator Y(p) has an eigenvalue µ > 1, while its

twisted monodromy operator Yhsatisﬁes the assumptions of Theorem 2.1. In this situation, the equivariant

control scheme (2.8) overcomes a limitation to Pyragas control in the sense that the periodic solution can

be stabilized using the control (2.8) but is always an unstable solution of (2.9).

A concrete example of this situation occurs in the Lorenz equation











˙x1=−σx1+σx2,

˙x2=−x1x3+λx1−x2,

˙x3=x1x2−x3,

(2.10)

with x1, x2, x3∈Rand with parameters σ, , λ ∈R. System (2.10) is symmetric with respect to the group

Z2={e, γ}, where eis the identity element of the group and we represent γon R3as

(x1, x2, x3)7→ (−x1,−x2, x3).

In [WS06], Wulﬀ and Schebes numerically show that for parameter values σ= 10,  = 8/3 and λ= 312

system (2.10) has a periodic solution with H=Z2and K={e}. The authors continue this periodic

orbit with respect to the parameter λ, while keeping the parameters σand ﬁxed. They ﬁnd that for

λ≈312.97, the periodic orbit undergoes a ﬂip-pitchfork bifurcation (a term introduced in [Fie88]), which

means that there exist parameter values close to λ≈312.97 where the twisted monodromy operator Yhhas

an eigenvalue µ < −1 and satisﬁes the assumptions of Theorem 2.1, whereas the monodromy operator Y(p)

has an eigenvalue larger than 1. So in this parameter regime the periodic orbit can be stabilized using the

equivariant control scheme (2.8), although it cannot be stabilized using Pyragas control of the form (2.9).

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EquivariantPyragascontrolofdiscretewavesBabettedeWol*AbstractEquivariantPyragascontrolisadelayedfeedbackmethodthataimstostabilizespatio-temporalpatternsinsystemswithsymmetries.Inthisarticle,weapplyequivariantPyragascontroltodiscretewaves,whichareperiodicsolutionsthathaveanitenumberofspatio-tempora...

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