Eliminating restrictions of time-delayed feedback control using equivariance

Pyragas control is a widely used time-delayed feedback control for 
 the stabilization of periodic orbits in dynamical systems. 
 In this paper we investigate how we can use equivariance to 
 eliminate restrictions of Pyragas control, both to select periodic 
 orbits for stabilization by their spatio-temporal pattern and to render 
 Pyragas control possible at all for those orbits. 
 Another important aspect is the optimization of equivariant Pyragas control, 
 i.e. to construct larger control regions. 
 The ring of $n$ identical Stuart-Landau oscillators coupled diffusively in a 
 bidirectional ring serves as our model.

1. Introduction. A particularly successful method of time-delayed feedback control has been introduced in 1992 by Pyragas [10]. A short summary of the huge amount of experimental and theoretical results following the original publication can be found in [11]. A widely open subject of research is Pyragas control for networks of coupled oscillators, where the solutions have different spatio-temporal symmetries besides synchrony.
The time delayed feedback control as introduced by Pyragas is noninvasive, i.e. it vanishes, on the periodic orbit z * (t) with minimal period p, solvingż(t) = F (z(t)), z ∈ C n . This is established in delayed feedback systems of the forṁ z(t) = F (z(t)) + B − z(t) + z(t − N p) where N ∈ N, and B is either a complex control parameter or a matrix. The control term is noninvasive on the periodic orbit, since we have −z * (t) + z * (t − N p) = 0 by periodicity.
For the network type presented in this paper, the only periodic orbit which can be stabilized by standard Pyragas control is the synchronized one. Hence, the modification to equivariant Pyragas control is necessary to eliminate restrictions of time-delayed feedback control.
Therefore, we use a modification of Pyragas control for stabilizing periodic orbits with prescribed spatio-temporal patterns on networks. This modification has been discussed previously in [13,9] for general systems near equivariant Hopf bifurcation. For general F , a new delayed feedback system is introduced as follows: Here h and Θ(h) describe the spatio-temporal pattern of the periodic orbit, for details see Fiedler [3] and section 2 below. The new control term is also noninvasive on the periodic orbit. For a detailed discussion on how to choose h and Θ(h), see sections 3 and 6.
In first works on small networks consisting of two and three Stuart-Landau oscillators, such as [5,12], it was shown that the control method (1) can indeed stabilize unstable periodic orbits with prescribed symmetry near equivariant Hopf bifurcation. An important aspect is that the control, if chosen correctly, can select the desired periodic orbit, even if several periodic orbits exist with the same period. Such a control term B − z(t) + hz(t − Θ(h) p) is called pattern-selective if it is noninvasive on exactly one periodic orbit with a prescribed spatio-temporal pattern.
However, many restrictions remained in [5,12]: For example stabilization is impossible if the coupling between oscillators is too strong, or if the cubic term of the Hopf normal form expansion does not fulfill certain requirements. Successful attempts to overcome these restrictions for two coupled oscillators have been presented in [1,2].
In the present work, we want to apply equivariant Pyragas control as in equation (1) to a specific network type, which consists of n identical Stuart-Landau oscillators coupled in a ring. Specifically, we want to select exactly one periodic orbit with prescribed symmetry and stabilize it -we will see that, for this network type, all but one of the periodic orbits are unstable. The main task consists of finding a suitable description h and Θ(h) of the required symmetry type. Secondly, we want to overcome previously existing restrictions to the application of equivariant Pyragas control. A third, closely related aspect is the optimization of the control region, i.e. to construct larger control regions.
All three tasks lead back to the following important questions: How should we describe the symmetry of a periodic orbit and how can it be utilized to optimize stabilization?
This paper is organized as follows: In section 2, we present the model of n identical oscillators coupled diffusively in a ring. We state our main results, introducing the modified Pyragas control adapted to the equivariant case in section 3. The proof of the Main Theorem follows in section 4, where we discuss the stabilization regions, and in section 5, where we finalize the proof. The full rotational symmetry of the system is used in section 6 to further improve the stabilization results, i.e. to obtain even larger control regions and to use arbitrary time delay for the control. We include multiple time delays into the control term and discuss the consequences in section 7. Finally, we recapitulate the paper in section 8.

2.
Model and periodic solutions with spatio-temporal symmetry. We consider n identical Stuart-Landau oscillators diffusively coupled in a ring: Each oscillator z k is symmetrically coupled to its two nearest neighbors z k−1 and z k+1 , i.e. we consider local couplinġ where z k ∈ C ∼ = R 2 , a > 0 is the positive coupling parameter and i.e. the normal form of Hopf bifurcation truncated at third order, with the Hopf frequency scaled to unity. This can always be achieved by rescaling time. λ ∈ R is a real bifurcation parameter, γ ∈ C is fixed. [10] [10] We define z −k := z n−k and z n := z 0 in order to cope with the indices. The system (2), (3) is equivariant with respect to the group D n × S 1 , where D n = ρ, κ is the dihedral group symmetry induced by the coupling between the single oscillators. The rotations ρ are generated by the index shift (ρz) k = z k−1 , and the reflection κ is given by (κz) k = z −k with k mod n and κ 2 = (κρ) 2 = Id. S 1 is the rotational symmetry of the truncated Hopf normal form, i.e. e i θ f (z) = f (e i θ z) for all angles θ ∈ [0, 2π].
Following [3], we describe the symmetry of periodic orbits z * (t) of the D n × S 1equivariant system (2), (3) by triplets (H, K, Θ). The isotropy subgroup H ≤ D n × S 1 leaves the periodic orbit {z * (t) : t ∈ R} fixed as a set, while K ≤ H leaves it fixed pointwise. The isotropy subgroups of D n × S 1 can for example be found in [7]. Θ is a group homomorphism which is defined uniquely by time-shift for all t: For our system, we have H = Z n and Θ(e 2π i m/n ) = ms/n, where s, m ∈ N, corresponding to discrete rotating waves.
Remark 1. The wave with index j bifurcates at the same point λ j as the wave with index n − j, see also Figure 2 for n = 4 and n = 5 oscillators. If all oscillators are identical, the diffusive coupling term a(z k−1 − 2z k + z k+1 ) vanishes and we can observe standard Hopf bifurcation at λ 0 = 0 leading to the synchronized periodic orbit.
Most periodic orbits of the system (2), (3) are unstable, making it suitable for the investigation of equivariant Pyragas control: Proposition 2. For j = 0, the bifurcating discrete rotating waves (4), enumerated by j, are unstable, both in the sub-and the supercritical case. For j = 0, i.e. the synchronized case, the periodic solution is unstable in the subcritical and stable in the supercritical case.
Both propositions can be verified by direct calculation: We choose appropriate coordinates x 0 , x 1 , . . . , x n−1 ∈ C, adapted to the equivariant nature of the coupled oscillator system, in which the linearization of the system at the trivial equilibrium z 0 ≡ · · · ≡ z n−1 ≡ 0 decouples: where we use the abbreviation σ = 2π/n. This leads to the inverse coordinate transformation The most important step of this analysis is to find n dynamically invariant subspaces In the subspace X j the system (2), (3) can be reduced to the two-dimensional equationẋ x j , which corresponds to a shifted Hopf normal form. Therefore, we can conclude that a Hopf bifurcation occurs at λ j = 2a(1 − cos(2πj/n)), where the symmetry of the bifurcation periodic orbit mirrors the symmetry of the subspace X j . The amplitude r j and the minimal period p j can be directly verified within the subspace X j . Conveniently, the linearization of the system (2), (3) decouples in the new coordinates X j , yielding the same Hopf bifurcation points.
We have thus proven Proposition 1. Note that the simple Hopf bifurcation occurring at λ 0 = 0 corresponds to the synchronized periodic orbit. For n even there is another simple Hopf bifurcation at λ n/2 = 2a(1 − cos π) = 4a, corresponding to an antisymmetric periodic orbit. All other Hopf bifurcations are double.
The periodic orbit of standard Hopf bifurcation is stable in the supercritical and unstable in the subcritical case. In both cases, the trivial equilibrium becomes unstable for λ > 0. Hence none of the periodic orbits for j > 0 can be stable: 3. Equivariant Pyragas control -main result. For a general systemż(t) = F (z(t)), the system including equivariant Pyragas control is given as follows [13]: Here B ∈ C n×n is a complex feedback parameter or matrix. For our concrete example (2), (3), the group element h is given by an index shift (hz) k = (ρz) k = z k−m between oscillators z k , and Θ corresponds to the phase shift of the discrete rotating wave.
The discrete rotating waves, as discussed in section 2, are numbered by the index j. We now select one of these waves for the pattern-selective equivariant Pyragas stabilization and denote it by the index j = s (for "selected"). Thus we aim at stabilizing the unstable periodic orbits with spatio-temporal pattern i.e. an index shift by m corresponds to a phase shift Θ = ms/n. p s is the minimal period of the selected rotating wave. For the delayed control term as above, it is therefore suitable to use a delay time Note that using a delay time larger than p s would be possible. However, we find that a larger time delay leads to smaller or even vanishing control regions, see section 6 for a detailed discussion.
In the present publication, we aim at stabilizing the discrete rotating waves (5) for s > 0, which are always unstable, see Proposition 2. For s = 0, see [4].
Since we want our control term to be pattern-selective, we require that it is noninvasive, i.e. vanishes, only on our selected periodic orbit, and that it is invasive on every other periodic orbit. Therefore we choose m co-prime to n, to identify the periodic orbit uniquely by its symmetry.
In contrast to previous publications on Pyragas control of Stuart-Landau oscillators, we use a complex control matrix B which commutes with the index shift with coefficients B k ∈ C, i.e. matrices with constant diagonals. We can also define the matrix elements B 0 , . . . , B n−1 , via n control parameters b 0 , . . . , b n−1 , which will be helpful for later analysis: In conclusion, we apply the control term as follows, with τ = (msp s )/n mod p s : where we again use the shift-representation ρz k = z k−1 , and we define f on the The main stabilization result then reads: Main Theorem. Consider the Hopf bifurcation of discrete rotating waves of the Stuart-Landau rinġ with λ ∈ R, a > 0 and γ ∈ C \ R + . Then for every combination of s and m, with s, m ∈ {1, . . . , n−1} and m co-prime to n, there exists a positive constant a m,s such that the following conclusion holds for all real diffusion constants 0 < a < a m,s , and near λ s = 2a(1 − cos(2πs/n)): There exist open regions of complex control parameters b 0 , . . . , b n−1 such that in the delayed feedback systeṁ The stabilization is noninvasive and pattern-selective.
The proof that a m,s indeed gives a sharp upper threshold for the stabilization using the above control scheme can be found in section 4, where we establish the parameter regions for stabilization.
Remark 3. For m = 1 it is possible to find the minimum by considering j ∈ [0, n] ⊂ R. Differentiating A j /|2 cos(2πj/n) − 2 cos(2πs/n)| with respect to j gives an expression on the (real) j for which the derivative is zero. By rounding we find the j ∈ {0, . . . , n − 1} for which the minimum is obtained. Standard Pyragas control corresponds to Θ = 1 and m = 0. Hence we need to solve the following system, for which we obtain the trivial solutions (A j , ω j ) = (0, 0) for every j and hence a 0,s = 0 for all s > 0. Theorem 3.1 therefore implies that, indeed, equivariant Pyragas control is necessary to stabilize the discrete rotating waves for s = 0: with s = 0 cannot be stabilized by standard Pyragas control, i.e. in a delayed feedback system of the forṁ for any B ∈ C n×n as above and any a > 0.   (2), (3) controlled as in (7) with index shift m = 1, for the third Hopf bifurcation (i.e. s = 2). The coupling parameter is a = 0.08 in (a) to (e) and a = 0.01 in (f). The Hopf curves from the invariant subspace with index j = s ((a) and (b)) are drawn in red (color online) while the green curves ((c) to (e)) correspond to Hopf curves with indices j = s, which all have eigenvalues with positive real part in the uncontrolled system. In (f) the yellow region is the region for which a stabilization is successful when only one complex control parameter b is used.

4.
Parameter regions for stabilization. By introducing equivariant Pyragas control, we select exactly one of the bifurcating discrete rotating waves. This is established by choosing a control term which is noninvasive only on the wave with the prescribed spatio-temporal pattern. The control is therefore pattern-selective, which is the most important property of equivariant Pyragas control, and also the reason why it can succeed for the ring of coupled oscillators, where standard Pyragas control fails (except for the synchronized solution z 0 = z 1 = · · · = z n−1 ). The goal of the proof is to reduce the problem to standard Hopf bifurcation. This facilitates the following stability analysis: It is possible to determine the stability of the selected bifurcating periodic orbit using standard exchange of stability in a two dimensional center manifold [6].
The proof consists of two parts: In the first part, we find the stabilization region, i.e. the region where the trivial equilibrium is stable at the selected Hopf bifurcation point. In the second part (see section 5), we prove that, under the given conditions, the selected Hopf bifurcation is supercritical and thus stable.
Using characteristic equations, we can determine the stability of the trivial equilibrium. We linearize the coupled oscillator system, including the control term, in the new coordinates x 0 , . . . , x n−1 : Note that the linearized equations decouple corresponding to the invariant subspaces X j , even when adding the delayed feedback term. By an exponential ansatz we   (2), (3) with coupling constant a = 0.2, controlled as in (7) with index shift m = 1, for the second Hopf bifurcation (i.e. s = 1). The Hopf curve corresponding to the invariant subspace with j = s = 1 is red (color online), in (a) and (b). The green curve (c) corresponds to j = 0, which has a pair of eigenvalues with positive real part in the uncontrolled system (λ 0 < λ 1 ). Blue ((d) and (e)) is used for the subspaces without unstable eigenvalues in the uncontrolled system, j = 2 and j = 3, where λ j > λ 1 . The black curve in (f) is used for the subspace X 4 where in the uncontrolled system the bifurcation occurs simultaneously to the selected one (i.e. λ 4 = λ 1 ). Note that this curve goes through the origin, as there is an Hopf bifurcation in the original system, i.e. B = 0. See also Figure 2 (c) and (d).
obtain the characteristic equations for complex eigenvalues : The control matrix B is chosen carefully in such a way that every characteristic equation χ j contains its own control parameter b j , which makes it possible to choose them individually. Similar to [5] we define the unstable dimension E(b 0 , . . . , b n−1 ) at the selected Hopf bifurcation point λ s as the number of eigenvalues η with strictly positive real part, depending on the control parameters b 0 , . . . , b n−1 , counting multiplicity. Each characteristic equation χ j contributes its unstable dimensions E j (b j ) independently to the total unstable dimension E(b 0 , . . . , b n−1 ) = n−1 j=0 E j (b j ). The total unstable dimension of the uncontrolled system E(b 0 = · · · = b n−1 = 0) is given by E(b 0 , . . . , b n−1 = 0) = 2s for s ≤ n/2 2(n − s) for s ≥ n/2 . Now consider b j = 0. We search for purely imaginary eigenvalues η, which we parametrize by ω, more precisely η = i(1 + ω). We denote the corresponding curves, on which the purely imaginary eigenvalues lie, by b j (ω). They are obtained by evaluating the characteristic equations at the Hopf bifurcation point λ s = 2a(1−

ISABELLE SCHNEIDER AND MATTHIAS BOSEWITZ
cos(2πs/n)) where p s (λ s ) = 2π: These curves denote the stability changes by Hopf bifurcation of the trivial equilibrium, i.e. the unstable dimension increases or decreases by 2 if the control parameter b j crosses the corresponding line b j (ω). Note that the double eigenvalue η = 0 can only occur for ω = 0 and thus lies on the already determined Hopf curves. Further note that the curve b s (ω), which corresponds to the selected Hopf bifurcation, is symmetric with respect to the real axis, crossing it only at b s (0) = 1/(2πΘ). Moreover we find that the curve b n−s (ω) goes through the origin: b n−s (0) = 0.
Wherever the Hopf curves b j , j = 0, . . . , n − 1, are complex differentiable they preserve complex orientation. This enables us to identify regions with different unstable dimensions, as to the right of the oriented curve b j (ω) the dimension is bigger by 2 than to the left of the curve. For b j = 0 the unstable dimension E j (b j ) is known and is either 0 (λ j ≥ λ s ) or 2 (λ j < λ s ).
Next we determine the region with E j (b j ) = 0. For λ j ≥ λ s the existence is trivial, as the origin is included.
For λ j < λ s , as the curves are all given explicitly, we find that the curve b j forms a "loop". Following the curve for increasing ω, we find that the region inside this loop has no unstable eigenvalues, see also Figures 3 and 4. This follows from complex differentiability which implies orientation preservation.
The existence of the loop can be seen as follows: lies to the right of the imaginary axis. Also the tangent b j (0) divides the complex plane into to halves, where the curve is always to the left of it (following the complex orientation, and between the two poles around 0 in ω): The curvature of b j (ω) is strictly positive for all ω between the poles: if a is small enough for the loop to exist. For Ω j tending to the pole, b j (ω) tends to infinity in the real as well as in the imaginary part. Therefore, the area inside must have E j (b j ) = 0. However complex differentiability is not given for all coupling parameters a. If a becomes too large, the loop shrinks and finally disappears. For the parameter a where the loop disappears, the curve b j is not complex differentiable.
If λ s − λ j > A j for some j, no loop exists and therefore no stabilization is possible near the selected Hopf bifurcation. This concludes the proof of Theorem 3.1.
In particular, as stated in Corollary 1, it follows that we cannot use standard Pyragas control for the stabilization of the selected wave with index s. The equations in standard Pyragas control with Θ = 1, and m = 0 read: sin (π(1 + ω j )) cos (π(1 + ω j )) = −ω j π sin 2 (π(1 + ω j )) = A j π from which follows immediately that A j = 0 for all j and therefore the stabilization is impossible for any a > 0, which proves Corollary 1.
For further analysis, we will call the regions in b j , where E j (b j ) = 0 holds, B j .

5.
Proof of the main stabilization theorem. In section 4, we have achieved linear stability E(b 0 , . . . , b n−1 ) = 0 at the selected Hopf bifurcation point λ s . We fix the complex control parameters b j in the regions B j where the characteristic equations (9) produce only eigenvalues with strictly negative real part (with exception of the pair of purely imaginary eigenvalues of the selected Hopf bifurcation), see section 4 for details.
In the last step of the proof of stabilization, we must guarantee that we only encounter standard supercritical Hopf bifurcation at the Hopf point. Standard Hopf bifurcation for nonzero control amplitude is ensured because we only encounter eigenvalues with nonzero real part. Consequently, it remains to show that the selected periodic orbit lies on the side of the Hopf bifurcation where the trivial equilibrium has unstable dimension two, i.e. that the bifurcation is supercritical. This is achieved by counting the unstable dimensions in the (λ, τ )-plane for fixed control parameters b 0 , . . . , b n−1 . The Hopf curves in this plane tell us where the stability changes. The Pyragas curve determines the position of the periodic orbit.
The Pyragas curve τ P (λ) is given by where Θ = (ms)/n mod 1. Note that the Pyragas curve τ P does not depend on the control matrix B. By the normalized Hopf frequency, we know that p s (λ s ) = 2π. Furthermore, the continuation of the Pyragas curve is differentiable at λ = λ s : Next, we determine the curves τ j (λ), j = 0, . . . , n − 1, at b j = |b j | exp(i β j ) from the characteristic equations χ j (η) = 0: τ j (λ) = ± arccos cos β j − (λ − λ s )/|b j | + β j + 2πmj/n + 2πN , for j = 0, . . . , n − 1, with integer N , enumerating the solutions of the arccosfunction. These curves determine the Hopf bifurcations in the (λ, τ )-plane. Similar Hopf curves are obtained in [13], therefore we do not repeat the calculation here. Note that the Hopf curves τ j depend on the respective control parameter b j . For further calculations, we linearize the characteristic equations with respect to λ = λ s +λ, η = iω, and τ = 2πΘ +τ . Of particular interest is the linearization of the characteristic equation for j = s, i.e. corresponding to the selected Hopf bifurcation. Linearizing for j = s and separating into real and imaginary part yields the following expression: Rearranging these equations, we obtain Therefore we can conclude that the derivative of τ s with respect to λ at the selected Hopf bifurcation point λ s is given by By the following orientation considerations we can determine the resulting total unstable dimensions E(λ, τ ) of the trivial equilibrium x 0 = · · · = x n−1 = 0 in the domains complementary to the Hopf curves. We once more linearize the characteristic equation (9) for j = s, but now with respect to τ and λ, ϕ(λ, τ ) = λ − η τ b s exp(2π i ms/n − τ η) = ξ, and also with respect to η, ψ(η) = 1 + η τ b s exp(2π i ms/n − τ η) = ξ. Now we find the expression where ψ is orientation preserving because it is holomorphic. To determine the orientation of ϕ, we need to calculate its determinant at η = i, λ = λ s , τ = 2πΘ: Thus, depending on the control parameter b s , we conclude that ϕ is either orientation reversing (Re b s > 0) or orientation preserving (Re b s < 0). The τ s -curve is oriented downwards in both cases. In the orientation reversing case, it follows that the region with E(λ, τ ) = 2 can be found to the left of the Hopf curve τ s in the (λ, τ )-plane, while in the orientation preserving case, this region can be found to the right.
In the following, we will carry out the analysis for the orientation reversing case, the other case being analogous.
Whether the Pyragas curve exists to the right or to the left of λ = λ s depends on whether the original bifurcation (without control) is subcritical (λ < λ s ) or supercritical (λ > λ s ).
Supercritical case. The Pyragas curve exists for λ > λ s . If τ s (λ s ) < 0 then we find that τ P enters the region with unstable dimension 2 whenever On the other hand, if τ s (λ s ) > 0 we find that it enters the region with unstable dimension 2 whenever Note that the control parameter b s can also be chosen to be real. Indeed, if the control parameter b s is chosen on the real line, then the Hopf curve is oriented vertically downwards and we can stabilize for all possible values of p (λ s ).
Subcritical case. The Pyragas curve exists for λ < λ s . In this case, if τ s (λ s ) < 0 then we find that τ P enters the region with E(λ, τ ) = 2 whenever Note that the inequality sign changes, compared to the supercritical case. On the other hand, if τ s (λ s ) > 0 we find that τ P enters the region with unstable dimension 2 whenever This concludes the proof of the Main Theorem.
6. Taking advantage of the full rotational symmetry. The ring of coupled Stuart-Landau oscillators (2), (3) offers an additional rotational symmetry which has not been used yet for the construction of the equivariant control term. Including this rotating wave property, we may choose arbitrary delay time by introducing an additional rotational operator. Again we select the wave with index j = s and we aim to stabilize by equivariant Pyragas control as in the following system: The new control term is also noninvasive on the selected rotating wave, since after time Θp, each individual oscillator has rotated by an angle of 2πΘ. The parameter Θ, and consequently also the time delay τ = Θp, can be chosen arbitrary.
The analysis from sections 4 and 5 can be carried out in the same manner as before. We obtain the characteristic equations (j = 0, . . . , n − 1) The stability of the trivial equilibrium changes by 2 whenever one of the curves is crossed with the complex control parameter b j . Note that these equations also depend on the weight Θ of the time delay. Moreover the investigation of the supercriticality of the Hopf bifurcation can be carried out analogously to section 5, and is therefore not repeated. This control ansatz results in larger control regions for smaller time delay. As we can choose Θ arbitrarily small, there is in particular no upper bound on the coupling parameter a. Theorem 6.1. Consider Hopf bifurcation of discrete rotating waves as in the Main Theorem with fixed a > 0 and s ∈ {1, . . . , n − 1}. Let m ∈ N be co-prime to n. Choose Θ > 0 small enough.
Then there exist open regions of complex control parameters b 0 , . . . , b n−1 such that in the delayed feedback systeṁ the discrete rotating wave solution is stabilized selectively and noninvasively for a time delay τ = Θp.
To prove the remaining part of Theorem 6.1, we investigate again the vanishing point of the loop of the curve b j . Analogously to section 4, we obtain the equations sin (πm(j − s)/n − πΘω j ) cos (πm(j − s)/n − πΘω j ) = −ω j πΘ sin 2 (πm(j − s)/n − πΘω j ) = A j πΘ.
Since j = s and m co-prime to n, it follows that sin 2 (πm(j − s)/n − πΘω j ) converges to a fixed, nonzero value in the limit Θ → 0.
The case sin 2 (πm(j − s)/n − πΘω j ) = 0 corresponds to the poles, where the curve is not complex differentiable. These points are thus not of interest here. Therefore it follows that A j → ∞ for Θ → 0, which proves Theorem 6.1.

7.
Linear combinations of control terms. In addition to the control terms discussed in the previous sections, it is of course possible to include more than one noninvasive control term into the equivariant Pyragas control. However, arbitrary linear combinations of noninvasive equivariant control terms will in general not lead to non-empty control regions, even if the individual controls do separately. It seems to be a necessary condition for the existence of a control region that the control is invasive on all the other bifurcations which occur for λ ≤ λ s , but up to date, no proof exists.
Another suggestion is to include the sum of all noninvasive control terms, with appropriate time delays τ m = (ms/n)p s mod p s . For a single complex control parameter b and for a = 0 this has been explored in [13]. These results also hold for the control matrix B as introduced above and small coupling parameter a. One example can be seen in Figure 5. Note that, for increasing n, the control region also grows in comparison with a single control term. It can therefore be suitable to introduce a control of this form. 8. Summary and discussion. The main goal of this paper was to show how we can use equivariance to eliminate severe restrictions of Pyragas control. A network of n identical Stuart-Landau oscillators coupled in a bidirectional ring was investigated, whose symmetry group is given by D n ×S 1 , i.e. the direct product of dihedral group D n and the circle group S 1 . This coupled network system contains n periodic orbits, which can be distinguished by their different spatio-temporal patterns, i.e. the interplay of index-shifts and phase-shifts between oscillators. Depending on the system parameters, only the synchronized periodic orbit, or even none of the orbits is stable. Due to these characteristic features, the ring of n coupled Stuart-Landau oscillators is an ideal candidate for the application of equivariant Pyragas control. Using equivariant control terms, we are able to select one of these unstable periodic orbits, and stabilize it. A stabilization with standard Pyragas control is restricted to the synchronized periodic orbit only, which is proven in section 4. Thus the main aim is to adapt Pyragas control for the non-synchronized periodic orbits. This is achieved by using the equivariance of the periodic orbit and including the spatio-temporal pattern in the time-delayed control term. As the main result, we show that, indeed, a control is now possible for every periodic orbit, regardless of its spatio-temporal symmetry. Also note that the cubic term of the Hopf normal form may now take any complex value except for γ ∈ R + . A sharp upper bound on the coupling parameter a is also established.
By including additionally the rotational symmetry into the control term, as demonstrated in section 6, the stabilization can also be achieved for arbitrary strong coupling parameter a. In fact, the rotational symmetry now also allows us to use arbitrary time delay, which can be expected to be useful in experimental realizations.
In contrast to previous publications concerning Pyragas stabilization of coupled Stuart-Landau oscillators, we have included a complex matrix into the control term which also incorporates the prescribed symmetry. The case of a single control parameter is included in this control matrix. However, using only one control parameter diminishes the chances of stabilization drastically, since it is necessary to find an overlap of control regions B j .
Linear combinations of noninvasive control terms were briefly discussed in section 7. Such linear combinations often provide control regions. However, necessary and sufficient conditions for the stabilization by linear combinations of noninvasive control terms are presently unknown.
The sum of all possible control terms does indeed give us large control regions for small coupling parameters a, see Figure 5. An upper bound on a needs yet to be established.
In conclusion, this publication shows that equivariant Pyragas control succeeds in situations where the well-established Pyragas control fails. Such situations include the equivariance of a system, too strong coupling parameters, restrictions on the cubic term in the Hopf normal form and a fixed time-delay. In the present setting of n diffusively coupled Stuart-Landau oscillators, we are able to give explicit analytic necessary and sufficient conditions leading to stabilization for different equivariant time-delayed control schemes.