## Abstract

The modest aim of this case study is the non-invasive and pattern-selective stabilization of discrete rotating waves (‘ponies on a merry-go-round’) in a triangle of diffusively coupled Stuart–Landau oscillators. We work in a setting of symmetry-breaking equivariant Hopf bifurcation. Stabilization is achieved by delayed feedback control of Pyragas type, adapted to the selected spatio-temporal symmetry pattern. Pyragas controllability depends on the parameters for the diffusion coupling, the complex control amplitude and phase, the uncontrolled super-/sub-criticality of the individual oscillators and their soft/hard spring characteristics. We mathematically derive explicit conditions for Pyragas control to succeed.

## 1. Introduction

Following a broadly applicable idea of Pyragas [1], periodic orbits *z*_{*}(*t*) of minimal period *p*>0 can be stabilized by differences *z*(*t*−*τ*)−*z*(*t*) which involve a feedback term with time delay *τ*>0. The difference vanishes, i.e. the feedback becomes non-invasive, if *τ*=*np* is an integer multiple *n* of the minimal period *p*. For example, this may happen in systems of the form
1.1with control matrix *b* and *z*_{*}(*t*) solving .

In the present paper, we address the case of equivariant systems, i.e. of systems with a (linear) group action *z*↦*gz*, such that *gz*(*t*) is a solution whenever *z*(*t*) is, for all elements *g* of an equivariance group *G*. The *N*-gon of diffusively coupled oscillators
1.2*k* mod *N*, with parameters will serve as the principal example of our case study. The dihedral group *D*_{N}=〈*ρ*,*κ*〉 of symmetries of a regular *N*-gon acts by index shift here. The rotations are generated by (*ρz*)_{k}:=*z*_{k−1} and *κ* is the reflection (*κz*)_{k}=*z*_{−k} with *k* mod *N*, *κ*^{2}=(*κρ*)^{2}= Id. Frequently, the group *G*=*D*_{N} itself is also described by its standard complex representation *ρw*=e^{2πi/N}*w* and on —not to be confused with the above action by index shift.

Following the study of Fiedler [2], the symmetry of a periodic orbit *z*_{*}(*t*) of a *G*-equivariant system is given by triplets (*H*,*K*,*Θ*). Here, *H*≤*G* leaves the orbit fixed as a set. The possibly strict subgroup *K*≤*H*≤*G* leaves each *z*_{*}(*t*) fixed, pointwise, for each *t*. The group homomorphism is defined uniquely by time shift
1.3for all *t*. Note that *Θ* is well defined because the normal subgroup of *H* is the kernel, *Θ*(*K*)=0. In local settings, this construction has first been introduced by Golubitsky *et al*. [3] and Golubitsky & Stewart [4].

Discrete *rotating waves* in oscillator *N*-gons (1.2), for example, are characterized by and *Θ*(e^{2πi/N})=*m*/*N*, usually with *m* coprime to *N*, i.e. *K*={Id}. *Standing waves*, in contrast, have *H*=〈*κ*〉 and *Θ*(*κ*)=0. Another possibility, realized in human walking, for example, is the alternating wave *H*=〈*κ*〉 and . Again *K*={Id}.

The main question of the present paper is the following: *How should one adapt the idea of delayed feedback control to selectively stabilize periodic orbits of prescribed symmetry type ( H,K,Θ)?* In short: how does one achieve non-invasive but pattern-selective feedback stabilization?

For the case of two oscillators, *N*=2, *G*=〈*κ*〉, this question has been addressed in [5], numerically. For a more detailed mathematical analysis, see [6]. The crucial idea there were feedback terms of the form
1.4For *N*=2, *H*=〈*κ*〉 and *h*=*κ* the delay *τ* therefore becomes the half period *p*/2, owing to , rather than the full period. Indeed control (1.4) becomes invasive on *z*(*t*), unless
1.5for all *t*; see (1.3).

In the present paper, we pursue the same question in a more general equivariant context. In order not to get lost, neither in mathematical generalities nor in exuberant detail, we restrict ourselves to the modest paradigm of discrete rotating waves in a triangle of *N*=3 oscillators (1.2). More precisely, we study the specific control
1.6for *k* mod 3, real diffusion coupling *a*>0, complex control *b* and complex *z*_{k}. For mathematical convenience, we consider only complex Stuart–Landau nonlinearities
1.7with real parameter *λ* and fixed complex *γ*. For some recent justification of this choice in a non-equivariant local centre manifold and normal form setting, see [7]. To further facilitate our computations, we study only local symmetry-breaking Hopf bifurcation of (1.6) and (1.7) in the two-parameter plane (*λ*,*τ*). But we invoke exchange of stability results in the full system. Thus, our results are not restricted to any centre manifold. See also [6,8–10].

In §2, we summarize our main results: stabilization of discrete rotating waves in the full system (1.6) and (1.7) by non-invasive and pattern-selective delayed feedback. We distinguish sub- and supercritical cases, depending on the sign of Re*γ*. We also distinguish the cases of soft and hard springs, depending on the sign of Im*γ*. In §3, we illustrate the control domains of successful stabilization of the trivial equilibrium *z*=0 at Hopf bifurcation. Section 4 briefly sketches the proofs and §5 summarizes our conclusions. All results are based on [11].

## 2. Main results

In this section, we consider a triangle of coupled Stuart–Landau oscillators (1.6) and (1.7), *N*=3. We first study local Hopf bifurcation of discrete rotating waves
2.1for *k* mod 3 and real *t*, with minimal period *p*>0 and parameter *λ*; see proposition 2.1. Without feedback, *b*=0, trivially, spatially homogeneous Hopf bifurcation with *z*_{0}(*t*)≡*z*_{1}(*t*)≡*z*_{2}(*t*) occurs at *λ*=0. This contributes real unstable dimension 2 to the spatially inhomogeneous Hopf bifurcation at *λ*=3*a*, of real dimension 4, which is addressed in proposition 2.1. Theorems 2.2–2.4 summarize our main results on full stabilization of the bifurcating discrete rotating waves, at *λ*=3*a*, by pattern-selective and non-invasive delayed feedback.

### Proposition 2.1

*Consider the coupled oscillator triangle (1.6) and (1.7). Hopf bifurcation of discrete rotating waves (2.1) occurs at the parameter value λ*=3*a*. *The rotating waves are harmonic*,
2.2*for k*=0,1,2, *and are phase shifted by* 2*π*/3 *between oscillators. Amplitude r and minimal period p are given explicitly by*
2.3*In particular, the Hopf bifurcation is supercritical, i.e. towards λ*>3*a*, *for* Re*γ*<0 *and subcritical for* Re*γ*>0. *The minimal period p grows with amplitude (soft spring) if* Im*γ*<0 *and decreases (hard spring) if* Im*γ*>0.

The proposition can be verified easily, by direct calculation. Note however that the imaginary eigenvalue +i of the linearization of (1.6), (1.7) is complex double at the Hopf bifurcation point *z*=0, *λ*=3*a*, rather than simple. This real dimension 4 of the Hopf eigenspace complicates the stability analysis. It also leads to the simultaneous bifurcation of further branches of periodic standing waves, *z*_{1}(*t*)≡*z*_{2}(*t*), and of alternating waves, *z*_{0}(*t*+*p*/2)=*z*_{0}(*t*) and *z*_{1}(*t*+*p*/2)≡*z*_{2}(*t*), of possibly different periods *p* and bifurcation directions. In particular, supercritical bifurcation need not lead to stable periodic orbits for two reasons. First, the four-dimensional Hopf eigenspace can comfortably accommodate further unstable Floquet exponents. Second, the onset of homogeneous Hopf bifurcation *z*_{0}(*t*)≡*z*_{1}(*t*)≡*z*_{2}(*t*) at *λ*=0 forces the trivial equilibrium *z*=0 itself to possess unstable dimension 2, for 0<*λ*<3*a*. This is promptly inherited by the discrete rotating waves, and by all periodic orbits, which bifurcate at *λ*=3*a*.

Nevertheless, we obtain the following stability results, for small enough amplitudes *r*>0 near *λ*=3*a*, where we distinguish between the super- and subcritical cases and between the model for soft and hard springs. We recall that the hard spring model, Im*γ*>0, is characterized by decreasing period *p* as the amplitude *r* increases, whereas for the soft spring model, Im*γ*<0, the period *p* increases with amplitude.

### Theorem 2.2

*Consider the supercritical case* Re*γ<0 for Hopf bifurcation of discrete rotating waves (2.1) of the Stuart–Landau triangle (1.6) and (1.7), near λ=3a.*

*Then there exists a positive constant a*_{+}*≈0.0974 and continuous strictly monotone functions
*2.4*independent of γ, such that the following conclusion holds for all real diffusion constants a and real control amplitudes b with
*2.5*and for all sufficiently small λ−3a>0.*

*The delayed feedback control (1.5) under one-third period
*2.6*stabilizes the discrete rotating wave solutions of proposition 2.1 for the above range of diffusion couplings a, control amplitudes b and parameters λ. The stabilization is non-invasive and pattern-selective. Moreover, the following limits hold for the control range:*
2.7

In the supercritical case, Re*γ*<0, real controls *b* suffice. We comment on possible complex control variants in the supercritical case at the end of §4.

The subcritical cases Re*γ*>0 necessarily require complex controls . Moreover, the amplitude dependence of period, as measured by |Im*γ*|, has to be strong enough in proportion to subcriticality, as measured by Re*γ*.

### Theorem 2.3

*Consider the subcritical case* Re*γ>0 for soft springs*, Im*γ<0, with the setting and notation of proposition 2.1 and theorem 2.2 under delay feedback (2.6) of one-third period.*

*Then there exists a positive constant a*_{+}*≈0.0974 and a continuous, strictly monotone function
*2.8*independent of γ, such that the following conclusion holds for all diffusion constants a and spring nonlinearities* *which satisfy
*2.9*There exists an open region of complex controls* *which depends on a and γ, such that the subcritically bifurcating discrete rotating waves are stabilized for all sufficiently small |λ−3a|. Moreover,
*2.10

### Theorem 2.4

*Consider the subcritical case* Re*γ>0 for hard springs*, Im*γ>0, with the above setting under delay feedback (2.6) of one-third period.*

*Then there exist positive constants a*_{−}*≈0.2960,* *independent of γ, such that the conclusion of theorem 2.3 holds for all diffusion constants a and spring nonlinearities* *which satisfy
*2.11

In summary, stabilization of discrete rotating waves at Hopf bifurcation succeeds for sufficiently weak diffusive coupling *a*>0. In all three theorems, the assumption |*λ*−3*a*| sufficiently small guarantees that the positive Floquet exponent originating from the homogeneous Hopf bifurcation is small enough for stabilization; see also [12].

## 3. Domains of stability

In this section, we linearize the coupled oscillator triangle at the trivial equilibrium *z*=0 and derive the characteristic equations. For complex controls *b*, we also plot stability regions *b*∈*Λ*_{a} of the remaining non-imaginary spectrum for the trivial equilibrium *z*=0 at the Hopf point *λ*=3*a*. We consider delay *τ*=*p*/3 at the limiting minimal period *p*=2*π*, associated with the complex double eigenvalue +i. This is a necessary, but not sufficient, prerequisite for pattern-selective and non-invasive stabilization of the bifurcating discrete rotating wave by delayed feedback. The consequences for the bifurcating discrete rotating waves, which prove theorems 2.2–2.4, are considered in §4.

The linearization of (1.6) and (1.7) at the trivial equilibrium *z*=0 commutes with the symmetry group *G*=*D*_{3} of the oscillator triangle. By Schur's lemma the linearization therefore diagonalizes in coordinates adapted to the irreducible representations of the linear *D*_{3}-action on (1.2) and (1.6). The pertinent coordinates are
3.1

The diagonal form of the linearization, in these coordinates, decouples the characteristic equation *χ*(*η*)=0 for exponentials *x*(*t*)=e^{ηt}(*x*_{0},*x*_{1},*x*_{2}) into a product of three factors, *χ*=*χ*_{0}⋅*χ*_{1}⋅*χ*_{2}=0. The complex spectrum is therefore given by those which solve at least one of the three characteristic equations
3.2

Let *E*(*b*) denote the strict unstable dimension of *z*=0 at the Hopf point *λ*=3*a* with Pyragas delay *τ*=*p*/3=2*π*/3, i.e. the total number of solutions *η* with Re*η*>0 for the characteristic equation *χ*(*η*)=0, counting algebraic multiplicities. Of course, we seek domains where *E*(*b*)=0 gives the bifurcating discrete rotating waves a chance to be born stable.

It is easy to determine the complex Hopf curves *b*_{j}(*ω*), where an additional Hopf eigenvalue changes the strict unstable dimension *E*(*b*) via :
3.3

In figure 1, we illustrate the resulting strict unstable dimensions *E*(*b*) of the trivial equilibrium *z*=0 for *a*=0.06. The stability domain
3.4disappears at the critical point
3.5where the complex derivative *b*′_{0}(*ω*) of the complex analytic function *ω*↦*b*_{0}(*ω*) vanishes. In fact, it is easy to determine the stability domain. We know *E*(0)=2, by direct analysis of the uncontrolled oscillators. Moreover, analyticity of implies that *E*(*b*) is larger, by two, on the right-hand side of the Hopf curves *b*_{j}(*ω*) when that curve is oriented towards increasing real *ω*, i.e. along the imaginary axis (figure 1).

In fact, the stability region *Λ*_{a} is bounded by *b*_{1}(*ω*) from below and by *b*_{0}(*ω*) from above, for 0<*a*<*a*_{*}. Note that Re*b*>0 on *Λ*_{a}. For *a*<*a*_{*} near *a*_{*}, we observe Im*b*<0. Only when
3.6the loop of *b*_{0}(*ω*), which defines the relevant upper boundary of the stability domain *Λ*_{a}, touches the Re*b* axis and enables stabilization by real controls, as in theorem 2.2. For 0<*a*<*a*_{+}, the interval of (2.4) is given by the intersection of the loop of *b*_{0}(*ω*) with the real axis (figure 1). The monotonicity claims on can be derived from the fact that the map preserves local orientation. The limits *a*↗*a*_{+} in (2.7) indicate where *b*_{0}(*ω*) touches the real axis at *a*=*a*_{+}. The limits for *a*↘0 follow by direct analysis of *b*_{0}(*ω*) at *a*=0.

We emphasize that the above analysis asserts only linear stability Re*η*<0 of the remaining non-imaginary eigenvalues at the Hopf point *λ*=3*a*, *τ*=*p*/3, *p*=2*π* itself, for *b*∈*Λ*_{a}. In particular, the nonlinearity coefficient does not affect *Λ*_{a}.

## 4. Proof of theorems 2.2–2.4

We summarize the mathematical proofs of theorems 2.2–2.4. For complete details, see also [11].

The basic strategy of proof is the same, for all three theorems. For given 0<*a*<*a*_{+},*a*_{−}, respectively, we first fix the control parameter in the region *Λ*_{a} of (3.4) where the characteristic equation (3.2) produces strictly stable non-Hopf eigenvalues Re*η*<0, only, at *λ*=3*a*, *τ*=*p*/3, *p*=2*π*. The complex double Hopf eigenvalue *η*=+i is generated by a simple zero of each of the remaining factors *χ*_{1},*χ*_{2} of the characteristic equation *χ*(*η*)=*χ*_{0}⋅*χ*_{1}⋅*χ*_{2}=0; see (3.2). In particular, *χ*_{0} is strictly stable, at *λ*=3*a*, *τ*=2*π*/3, and remains strictly stable, locally.

In the (*λ*,*τ*) plane, we then determine the Hopf bifurcation curves *τ*_{j}(*λ*) where *χ*_{j}(*η*)=0 for some purely imaginary *η*=i*ω* and some remaining factor *χ*_{j}, *j*=1,2, of the characteristic equation.

Orientation considerations similar to §3 determine the resulting total unstable dimensions *E*(*λ*,*τ*) of the trivial equilibrium *z*=0 in the domains complementary to the Hopf curves. See figure 2 for numerical examples.

Locally near *λ*=3*a*, we next restrict the control delay *τ* to the (smoothly extended) *Pyragas curve* *τ*_{p}(*λ*)=*p*(*λ*)/3 of our proposed feedback, see (2.6). Explicitly,
4.1by (2.3). Note how the Pyragas curve *τ*_{p} depends on the spring nonlinearity *γ*, but not on the control *b*. Conversely, the Hopf curves *τ*_{j} depend on *b*, but not on *γ*.

The additional assumptions of theorems 2.2–2.4 now guarantee that we only encounter a *standard supercritical Hopf bifurcation*, when we restrict all considerations to parameters (*λ*,*τ*) on the one-dimensional Pyragas curve *τ*_{p}(*λ*). ‘Standard’ means that we only encounter transverse crossing of one simple imaginary eigenvalue, across the imaginary axis. The unique bifurcating Hopf branch then consists of exactly the original harmonic discrete rotating waves of proposition 2.1, owing to the non-invasive property of delayed feedback control along the Pyragas curve. ‘Supercritical’ means that the periodic orbits bifurcate to the part of the Pyragas curve (*λ*,*τ*_{p}(*λ*)) in the domain where *E*(*λ*,*τ*)=2. Because *E*(*b*)=0 and *because the standard Hopf bifurcation is guaranteed to be always supercritical along the Pyragas curve, standard exchange of stability therefore guarantees stability of the bifurcating discrete rotating waves*. See also [13].

To prove theorems 2.2–2.4, it remains only to check supercriticality along with the Pyragas curve *τ*_{p}(*λ*). The Hopf curves *τ*_{j}(*λ*) at introduced earlier can be calculated explicitly to be
4.2with integer *n*.

From §3, we already know the strict unstable dimension *E*(*b*)=0 at *λ*=3*a*,*τ*=*p*/3, *p*=2*π* for our choice of *b*∈*Λ*_{a}. From (4.2), we conclude that (*λ*,*τ*_{1}(*λ*)) does not enter a small neighbourhood of (*λ*,*τ*)=(3*a*,2*π*/3). Neither does (*λ*,*τ*_{2}(*λ*)), unless *n*=0 and the negative sign of ± is chosen. We also recall Re*b*>0 and hence |*φ*|<*π*/2. Therefore,
4.3

### (a) Supercritical case

To address the supercritical case Re*γ*<0 of theorem 2.2, near *λ*=3*a*, we observe that our real choice Im*b*=0 of the control *b* causes vertical slope of the Hopf curve *τ*_{2}(*λ*) at *λ*=3*a*; see also figure 2*a*. Moreover, the region *E*(*λ*,*τ*)=2 is found towards larger *λ*, i.e. to the right of the Hopf curve *τ*_{2}. The original uncontrolled Hopf bifurcation was assumed supercritical, i.e. to the right. Therefore, the controlled Hopf bifurcation remains supercritical, and theorem 2.2 is proved.

### (b) Subcritical soft spring case

To address the subcritical soft spring case Re*γ*>0> Im*γ* of theorem 2.3, near *λ*=3*a*, we also calculate the negative slope
4.4of the Pyragas curve, at *λ*=3*a*, *τ*=2*π*/3.

We first consider the case Im*b*<0 in the loop *Λ*_{a}. Then Re*b*>0 in *Λ*_{a} and (4.3) and (4.4) imply *τ*′_{2}(3*a*)>0>*τ*′_{p}(3*a*); figure 2*b*. In particular, *E*(*λ*,*τ*_{p}(*λ*))=0 for *λ*<3*a* and the Hopf bifurcation along the Pyragas curve is subcritical, rather than supercritical. Hence Pyragas stabilization fails for Im*b*<0.

Next consider the case Im*b*>0 in the loop *Λ*_{a}. Again the stability region *E*(*λ*,*τ*)=2 is located above the Hopf curve *τ*_{2}, locally (figure 2*c*). The subcritical Hopf bifurcation becomes supercritical, along the Pyragas line *τ*_{p}(*λ*), if and only if the region *λ*<3*a* of periodic orbits lies inside the dashed region *E*(*λ*,*τ*)=2, along the Pyragas curve. This, in turn, holds true if and only if the negative Pyragas slope *τ*′_{p}(*λ*) at *λ*=3*a* satisfies
4.5This proves theorem 2.3 with the choice Im*b*>0 in the loop *Λ*_{a} for
4.6

### (c) Subcritical hard spring case

We conclude with the case of subcritical hard springs, i.e. positive Re*γ*, Im*γ* near *λ*=3*a*, as addressed in theorem 2.4. If Im*b*>0 in *Λ*_{a} then Re*b*>0, (4.3) and (4.4) imply *τ*′_{2}(3*a*)<0<*τ*′_{p}(3*a*), this time (figure 2*d*). Hence Pyragas stabilization fails by arguments quite analogous to those given for theorem 2.3, Im*b*<0.

Next, consider Im*b*<0 in *Λ*_{a} (figure 2*e*). Then, it is sufficient to require
4.7This proves theorem 2.4, (2.11) with
4.8In fact, does not depend on *a*, owing to the precise geometry of the curves *b*_{0}(*ω*) and *b*_{1}(*ω*) in figure 1.

This completes the proofs of theorems 2.2–2.4.

### (d) Remarks on the supercritical case

The above analysis for complex controls *b* also applies to the supercritical case of theorem 2.2, Re*γ*<0, of course. We consider the same complex control region *Λ*_{a} as for the subcritical cases (see also figure 1). We then obtain stabilization for an extended range
4.9of diffusion couplings. The price we have to pay, however, is restrictions on the nonlinearity quotient Im*γ*/Re*γ*, as follows.

First consider a control with Im*b*>0. Then stabilization is possible for
4.10where *β*(*a*) is given by
4.11In the soft spring case, stabilization is always possible for 0<*a*<*a*_{+}≈0.0974 and suitable *b*∈*Λ*_{a}, Im*b*>0. Specific restrictions on the nonlinearity only arise for hard springs.

Next consider a control with Im*b*<0. In this case, stabilization is possible for
4.12where *β*(*a*) is given by
4.13It is the hard spring, this time, which is stabilizable for all 0<*a*<*a*_{−}≈0.2960 and suitable *b*∈*Λ*_{a} with Im*b*<0. The soft spring case requires the above constraint.

## 5. Conclusions

The modest scope of our case study was the non-invasive and pattern-selective stabilization of discrete rotating waves in a triangle of diffusively coupled oscillators at symmetry-breaking Hopf bifurcation. The oscillators were assumed to be in Stuart–Landau normal form. Feedback was of delay type, reminiscent of Pyragas control, but adapted to be pattern-selective.

For explicit intervals of small enough diffusion coupling, we have determined explicit domains of the control coefficient *b*, and explicit constraints on the spring nonlinearity, such that delayed feedback control succeeds.

In the supercritical case of theorem 2.2, real control *b* succeeded for diffusion coupling 0<*a*<*a*_{+}≈0.0974 and arbitrary nonlinearities. Control also succeeds for 0<*a*<*a*_{*}≈0.2960 and suitable complex controls *b*, without constraints on the nonlinearities for hard springs, but with specific restrictions for soft springs.

In the subcritical cases of theorems 2.3 and 2.4, complex controls *b* were necessary, along with constraints on the nonlinearities. These constraints depended on the diffusion coupling 0<*a*<*a*_{+}, in the case of soft springs. For hard springs, the required constraints turned out to be independent of 0<*a*<*a*_{−}≈0.2960.

## Funding statement

This work was partially supported by SFB 910 ‘Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application’, project A4: ‘Design of Self-Organizing Spatio-Temporal Patterns’ of the Deutsche Forschungsgemeinschaft.

## Acknowledgements

The author is very grateful to Prof. Dr Bernold Fiedler for his valuable help and suggestions and thanks Dr Stefan Liebscher for fruitful discussions. The author thanks also Prof. Dr Eckehard Schöll, PhD, coordinator of the SFB 910, for the opportunity to talk at the International Conference on Delayed Complex Systems, Palma de Mallorca, 2012.

## Footnotes

One contribution of 15 to a Theme Issue ‘Dynamics, control and information in delay-coupled systems’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.