# Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

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## Abstract

We suggest a delayed feedback control scheme with arbitrary delay for stabilizing a periodic orbit, while maintaining the noninvasiveness of the controller. Since the constraint on the delay to be adjusted to the period of the unstable periodic orbit is not imposed, a richer structure of the dynamics can be observed: Not only weakly unstable, but also strongly unstable periodic orbits are stabilized and even stabilization of orbits with infinite period is achieved. The control mechanism is elucidated for the generic model of a subcritical Hopf bifurcation. A complete bifurcation analysis for the fixed point as well as the periodic orbit is presented and the stability domains are identified. Furthermore, we study the effects of distributed delayed feedback on the stabilization of periodic orbits, and show that larger variance of the delay distribution considerably enlarges the stabilization region in the parameter space. We extend the control scheme to a network of Hopf normal forms coupled with heterogeneous delays. By tuning the coupling parameters, different synchronization patterns, i.e., in-phase, splay, and clustering, can be selected. The characteristic equation for Floquet exponents of the heterogeneous delay network is derived in an analytical form, which reveals the coupling parameters for successful stabilization. The equation takes a unified form for both subcritical and supercritical Hopf bifurcations regardless of the synchronization patterns. Analysis of Floquet exponents and direct numerical simulations show that the heterogeneity in the delays drastically facilitates stabilization and provides an enlarged parameter region for successful control. Finally, we consider the thermodynamic limit in the framework of a mean field approximation and show that heterogeneous delays offer an enhanced performance of control.

## Keywords

Time-delayed feedback control Synchronization Networks Heterogeneous delay Hopf normal form## 1 Introduction

Time delayed feedback control (DFC), proposed by Pyragas [1], is a simple and convenient method to stabilize unstable periodic orbits (UPOs) occurring in a *single* dynamical system. Since DFC uses only the difference of the current and the delayed state, where the time delay is given by the period of the UPO, the control is noninvasive. Furthermore, the control is applicable to systems whose equations of motion are unknown. Due to this convenience, the algorithm of DFC has been applied to quite diverse experimental systems, and many theoretical advances have also been made [2, 3, 4, 5]. However, it was commonly believed that torsion-free UPOs cannot be stabilized by DFC [6]. To overcome this limitation, modified control schemes, like an oscillating feedback [7], a half-period delay [8] and the introduction of an unstable controller [9, 10, 11] were proposed.

This alleged odd-number theorem has been refuted [12] by a counterexample, using the normal form of a subcritical Hopf bifurcation that is a representative system for the odd-number limitation. The normal form of a sub- or supercritical Hopf bifurcation is also called Stuart-Landau oscillator, and is often used as a generic model for oscillators. Experimental evidence for the predicted DFC of odd-number orbits generated by a subcritical Hopf bifurcation has recently been presented [13, 14]. For a state-of-the-art general discussion of the odd-number limitation see Ref. [15, 16]. The normal form of a system near a subcritical Hopf bifurcation also allows for an analytical treatment of the stability of the UPO, including the calculation of the Floquet exponents [17, 18, 19, 20]. However, since the constraint is imposed that the delay time should be adjusted to the period of the UPO, strongly unstable periodic orbits are difficult to stabilize.

Meanwhile, *coupled systems* ranging from a few elements to large networks have become a central issue in nonlinear science [21, 22, 23, 24]. In particular, the study of the synchronization of networks has evolved into a rapidly expanding field [25, 26]. Time delays are always present in coupled systems due to the finite signal propagation time. These time lags give rise to complex dynamics and have been shown to play a key role in the synchronization behavior of systems [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42], see also the review [43]. In Ref. [20, 44, 45], we have shown that UPOs in networks of Hopf normal forms can be stabilized to exhibit in-phase synchronization in the sub- and supercritical case, and that additionnally more general splay or cluster synchronization patterns can be stabilized in the supercritical case. Here we extend these results in various ways; among other issues, we show that cluster synchronization can also be stabilized in networks of subcritical Hopf normal forms, and that one can deliberately switch in a unique and noninvasive form, including the case of heterogeneous delays, between different synchronous states in a network of supercritical Hopf normal forms by adjusting the coupling phase parameter.

In practice, the exact values of the system parameters might be unknown, and they might be heterogeneous, and moreover, one cannot always adjust the delays to a specified value, e.g., to the period of the local node dynamics. In particular, it has been pointed out in the biological context that heterogeneous delays lead to more realistic models [46]. Nevertheless, most studies have assumed that all the interactions occur with the same time delay and, up to now, little is known about stabilizing UPOs and controlling the synchrony patterns in networks coupled with *heterogeneous* delays. For instance, the dynamics of an array of chaotic logistic maps coupled with *random* delay times [47], the effects of *heterogeneous* delays in the coupling of two excitable neural systems [48, 49] or a neural network [50], and amplitude death in the Stuart-Landau system coupled with *distributed* delays [51, 52, 53] or periodically modulated delay [54] were investigated.

In this paper, we suggest a DFC scheme with arbitrary delay for stabilizing a UPO, but maintain the noninvasiveness property, for the normal form of a subcritical Hopf bifurcation. Unlike the conventional DFC, where the delay-time is adjusted to match the period of the target orbit, a rotation of the feedback term is used to obtain noninvasive control. Since the constraint on the delay to be adjusted to the period is not imposed in the present study, rich dynamics can be found for different delays. A complete bifurcation analysis for the fixed point as well as for the periodic orbit is presented, and the stability domains are identified. We will show that both weakly and strongly unstable periodic orbits can be stabilized. The former have a global basin of attraction and the latter have only a local basin. Furthermore, stabilization of orbits with infinite period is possible. We also introduce a nonlinear feedback control scheme, which is more efficient than the linear one, with respect to both the control aim and the theoretical analysis, and discuss effects of distributed delays in the feedback.

We extend the feedback control scheme to a network of Hopf normal forms coupled with heterogeneous delays. By tuning the coupling rotations, only one state among different synchronization patterns, i.e., in-phase, splay, and clustering, can be chosen in noninvasive form and their stabilizations can be efficiently achieved. Adopting the rotation of the delay terms as a control parameter, rather than the coupling phase, makes it possible to remove multistabilities of the collective amplitude and frequency that were undesirable effects in the previous studies [44, 45]. Our control scheme might be viewed as a natural way to model the network coupled with delay proportional to the distance and with rotation of the input signal proportional to the distance. They could thus be particularly suitable for networks of semiconductor lasers and in neuroscience. Similar extensions of the Pyragas control scheme based on the symmetry-breaking of equivariant Hopf bifurcation [55, 56] have recently been suggested and applied to noninvasive and pattern-selective stabilization of three coupled Stuart-Landau oscillators (ponies on a merry-go-round) [55].

The characteristic equation for Floquet exponents of the heterogeneous delay network is derived in an analytical form, which reveals the coupling parameters for successful stabilization. The equation takes a unified form regardless of the Hopf bifurcation forms including the subcritical and supercritical case and regardless of the synchronization patterns. As a result we can show that UPOs (limit cycles) in many types of networks can be stabilized in cluster form by heterogeneous delayed coupling. Our results of controlling the periodic orbits are demonstrated also by direct numerical simulations. The effects of heterogeneous delays on the cluster stabilization of periodic orbits in noninvasive form are discussed, and it is shown that the heterogeneity in the delays can *greatly* enlarge the set of parameters for successful control. We consider the thermodynamic limit of the network system in the frame of mean-field approximation, and derive the characteristic equations for the stability of UPOs, relating them to the distribution of the heterogeneous delays. Our results show that heterogeneous delays can greatly enhance the control of UPOs.

The paper is organized as follows. In Sect. 2 we consider delayed-feedback control with arbitrary delay for a *single* Hopf normal form. Two coupling schemes, namely linear and nonlinear, for DFC of the normal form of a subcritical Hopf bifurcation are proposed in Sect. 2.1. Sections 2.2, 2.3 and 2.4 are devoted to a complete bifurcation analysis for the fixed point as well as the periodic orbit. Section 3 describes the effect of distributed delays. In Sect. 4, we consider networks of Hopf normal forms coupled with heterogeneous delays and provide the example of a bidirectional ring in Sect. 4.1. The characteristic equations for Floquet exponents of networks coupled with *heterogeneous* delays are derived and the effects of heterogeneous delays on the clustering are presented in Sect. 4.2. Section 4.3 describes the special cases and the numerical results. In Sect. 4.4, the thermodynamic limit in the framework of a mean-field approximation is considered. Finally, we conclude with Sect. 5. In the Appendix, the normal form of a Hopf bifurcation of the fixed point is constructed using the method of multiple scales.

## 2 Self-feedback control with arbitrary delay in a single system

By construction, the control scheme given by Eq. (2) is noninvasive: the control force vanishes at successful stabilization, and the periodic orbit itself remains untouched by the control terms. In general, however, it is difficult to find a transformation matrix \(S(t,\tau )\) satisfying Eq. (1) unless full information of the UPO is given. If we consider, however, the normal form for Hopf bifurcation, then we can overcome this limitation by taking the \(S^1\) equivariance of the system into account.

### 2.1 Time-delayed feedback scheme

*any*delay time \(\tau \) by construction of the control system (3). This is in contrast to the conventional DFC, in which the delay time should be adjusted to the period of the UPO, \(\tau =T\), defining the Pyragas curve

Throughout our analytical study the coupling phase \(\beta \) is confined to an interval of \([0,\pi /2]\) and the control amplitude \(K\) takes positive values or, equivalently, the \(K\) values satisfy \(K\cos \beta >0\) with \(\beta \in [0,\pi ]\). In our numerical diagrams we will usually fix the three parameters, \(\omega _0, \gamma \), and \(\beta \), to \(\omega _0=1, \gamma =-10\), and \(\beta =\pi /4\).

Obviously, the linear feedback system, Eqs. (3) and (4), is reduced to Pyragas control when the delay time \(\tau \) coincides with the period \(T\) since \(e^{i\Omega _0 T}=1\). On the other hand, the choice in Eq. (5) means that the delayed-feedback in Eq. (3) might still be influenced by the phase of the variable, but not by the amplitude. Nevertheless, it is worth to mention that with such a choice the parameter region for successful control becomes remarkably larger than with the linear technique and, moreover, a thorough analytical study for the stabilization of the UPO becomes possible, as will be shown in Sect. 2.3.

### 2.2 Stability analysis of the fixed point

Stabilization of the UPO does not necessarily require destabilization of the fixed point. There could, for instance, be a parameter range with multistability in the presence of an additional delay-induced orbit. However, the stabilization could be accelerated and achieved with a larger basin of attraction, i.e., the whole region inside the UPO. Moreover, the properties of the two schemes are closely related since the feedback term includes the frequency of the UPO.

#### 2.2.1 Hopf curves of the fixed point

*any*\(\tau \). Thus, \(\lambda =0\), i.e., the \(\tau \) axis in the \((\lambda , \tau )\)-plane also forms another part of Hopf curves besides the curves given by Eq. (14a) (see the lines denoted by \(H^{+}\) and \(H^{-}\) in Fig. 1b).

#### 2.2.2 Codimension-2 bifurcations on Hopf curves

#### 2.2.3 Numerical diagrams

We note that the primary domain for destabilizing the fixed point, bounded by \(\tau _0^{+}\) and \(\tau _0^{-}\), is much larger than that of the conventional DFC given by Eq. (6) and, moreover, contains not only the primary domain, but also most of the secondary one of conventional DFC. The domains of the destabilized fixed point, bounded by the Hopf curves, are marked by light gray shading (yellow online), which becomes darker in color with the increasing dimension of the unstable manifold.

The light gray curves denote Hopf curves with Pyragas delay, which was considered in [12] and [17]. The Pyragas curve given by Eq. (6) is seen in cyan color (light). When moving along the Pyragas curve, we clearly observe that these three curves cross each other in a point, reflecting that our control system is reduced to the Pyragas system as the delay time is chosen to be equal to the period of the UPO.

#### 2.2.4 Nonlinear feedback

It is obvious that the control system (3) with the nonlinear feedback (5) yields the same characteristic equation of the fixed point as that of the linear case, Eq. (8), since \(\left| z(t)\right| =\left| z(t-\tau )\right| \) at the Hopf bifurcation. Therefore, the Hopf curves and the parameter region for destabilizing the fixed point are determined by the same equations as the linear feedback case.

### 2.3 Stability analysis of the UPO

As mentioned above, an unstable fixed point is neither a necessary nor a sufficient condition for stabilization of the UPO. It affects only the global properties of the basin for the stabilization of the UPO: an overlapping region of parameters for simultaneously destabilizing the fixed point and stabilizing the UPO yields a global attractor basin for the domain inside the UPO, while the stabilization of the UPO alone guarantees its local stability.

#### 2.3.1 Floquet exponents

*Linear Feedback*In polar coordinates \(z=r e^{i\varphi }\), Eq. (3) with the linear feedback control (4) reads

*any*positive value.

*Nonlinear feedback*For the nonlinear control (5), Eq. (3) can be written as follows:

#### 2.3.2 Transcritical bifurcation of the UPO

*Point of Transcritical Bifurcation*Note that \(\Lambda =0\) is always a solution of Eq. (27), which corresponds to the Goldstone mode of the periodic orbit, i.e., the trivial Floquet mode with Floquet exponent zero. A second, but nontrivial vanishing eigenvalue is generated when the transcritical bifurcation of periodic orbits occurs. If we confine \(\Lambda \) within real eigenvalues, then the condition for a transcritical bifurcation in the control system is provided by a

*double*zero of \(0=\chi _1(\Lambda ;K,\tau ,\lambda )\), i.e.,

*Turning Point of the Stability Direction*We calculate the analytical conditions for the change of the stabilization direction that occurs on the transcritical curve, the codimension-2 bifurcation. Eliminating the Goldstone mode, a transcendental function \(G_1(\Lambda )=\chi _1(\Lambda )/\Lambda \) is considered instead of \(\chi _1(\Lambda )\). Taking into account the derivatives

#### 2.3.3 Hopf bifurcation of the UPO

The transcritical stable region could be cut off by an appearance of torus bifurcation, i.e., Hopf bifurcation of the UPO.

*Linear Feedback*One can find the Hopf bifurcation of the UPO (in a semi-analytic way) by inserting \(\Lambda =i\Omega _H\) into Eq. (27) and separating the equation into real and imaginary parts as follows:

*fixed point*along the \(\tau \) axis, i.e., , \(B_{1},B_{2},\ldots ,\) given by Eq. (18b), which can be seen in Fig. 2a.

*Nonlinear Feedback*In case of the nonlinear feedback control, it is possible to thoroughly perform the analytical study. With abbreviations \(a=-2\lambda , b=K\cos \beta \) and \(c=-2\lambda K(\cos \beta +\gamma \sin \beta )=-2K\delta \), Eq. (28) is rewritten as

#### 2.3.4 Intersection of the transcritical and Hopf curves

There exists a region of \(\tau \) for stabilizing the UPO in Eq. (45) only in the case of \(\tau ^{Tr}<\tau _0^{H}\). The intersection of the two curves occurs at the point in parameter space where Eqs. (27) and (28) have a triple real root.

Thus, the intersection of the transcritical curve and Hopf curve occurs at a point that coincides with the point \(C\) of the codimension-2 bifurcation on the transcritical curve (see Figs. 2, 3).

### 2.4 Numerical bifurcation diagrams

Figure 3 depicts a two-dimensional bifurcation diagram of both the fixed point and, in pronounced form, the UPO in the \((K, \tau )\)-plane for \(\gamma =-10, \beta =\pi /4\), and \(\lambda =-0.005\): In Fig. 3a, a transcritical bifurcation of the UPO [solid blue (dark)], Hopf bifurcation of the UPO for linear feedback [solid green (light)], a Hopf bifurcation of the UPO for nonlinear feedback [dotted green (light)], stable [solid cyan (light)] and unstable [dashed cyan (light)] Pyragas curves can be seen. Blue (light gray) shading marks the domains of stabilization of the target orbit using the linear feedback, and the gray shading corresponds to the enlarged part of the domain with the nonlinear feedback. Figure 3b is enlarged part close to the crossing point \(C\) of the transcritical and Hopf bifurcation curves of the UPO, determined by \((K, \tau )=(K^{*}, \tau ^{*})\). The dotted blue (dark) line indicates the transcritical bifurcation curve for further destabilizing the orbit for the \(\tau \)-axis direction and the dotted black (dark) curve corresponds to the spurious (unphysical) solution for Hopf bifurcation. Insets (i), (ii), and (iii) of Fig. 3b display the real part of the leading Floquet exponents in dependence on the control amplitude \(K\) for the delay time given by \(\tau ^{*}+20, \tau ^{*}\) and \(\tau ^{*}-20\) with \(\tau ^{*}=249.4\), respectively. The blue (dark) and magenta (light) lines in the insets correspond to the transcritical mode (with zero imaginary part) and Hopf mode (with non-zero imaginary part), respectively. Note that the gradient of the transcritical mode curve at \(\Lambda =0\) varies from positive to infinity, and to negative in the insets (i), (ii), and (iii), respectively.

Time series obtained by numerical calculations are displayed in Fig. 4. Parameter sets of Fig. 4a–c correspond to the black crosses \(N_1, N_2\), and \(N_3\) in Fig. 2, and are given by \((\lambda ,\tau )=(-0.04,0.9\pi ), (\lambda ,\tau )=(-0.2,0.4\pi )\) and \((\lambda ,\tau )=(-0.1,0.9\pi )\), respectively. The numerical simulations show that the parameter regime for simultaneous destabilization of the fixed point and stabilization of the target orbit, e.g., \(N_1\), yields a global attractor basin including the whole area inside the UPO, while the stabilization of the UPO alone, e.g., at the point \(N_2\), guarantees only local stability around the UPO. Note that the points of operation \(N_1\) and \(N_2\) are inside and outside the primary region for the unstable fixed point, which corresponds to weakly and strongly UPOs, respectively, while \(N_3\) indicates the point, where the period of the UPO becomes infinite. We note that, with the parameter set corresponding to the point \(N_{2}\), the rotations around the fixed point and the UPO are in opposite direction of each other, since the angular frequency \(\Omega _0 =\omega _0-\gamma \lambda <0\) for \(\omega _0 = 1 > 0\). In any case, the control signal vanishes when the target orbit is stabilized, demonstrating the noninvasiveness of our method.

It is worth noting that, in spite of the common belief that strongly unstable orbits are notoriously difficult to tackle by the conventional DFC, our control scheme allows us to extend remarkably the range of stabilization up to the point \(C\), i.e., \(\lambda ^{*}\) determined by Eqs. (32) and (36) for linear and nonlinear control schemes, respectively. In particular, we see in Fig. 4c that even orbits with infinite period can be stabilized.

## 3 Delayed-feedback control with distributed delays

*distributed*delayed feedback. This is common in a variety of systems where the delay times are given by a continuous distribution [48, 52, 53, 54]. The model can be written as follows:

*any*distribution function \(g(\tau )\).

### 3.1 Floquet exponents

### 3.2 Enlargement of the stability region

The dependence of the stability range on the distribution of delays is determined by Eq. (53), which is reduced to Eq. (27) with constant time delay for the case \(g(u)=\delta (u-{\tau })\).

To make further analytical progress, it is instructive to specify a particular choice of the delay distribution function. As the first example, we consider a uniform distribution

*polynomial*equation of \(2p+2\) degrees, rather than a transcendental one.

## 4 Noninvasive control of synchrony in networks coupled with heterogeneous delays

Our suggested control scheme with arbitrary delay-time lends itself for controlling the dynamics of a network coupled with heterogeneous delays. Also, it is possible to describe the supercritical as well as the subcritical case in a unified form.

### 4.1 Model

Heterogeneous delays and the rotational coupling might give a natural way to model memory effects in interacting systems if we conceive the heterogeneous time delays and the rotation angle as proportional to the *distance* between nodes. If \(a_{jj}=1\) for all \(j\), this corresponds to a network with delayed self-feedback, while \(a_{jj}=0\) characterizes a network without self-feedback.

### 4.2 Calculation of Floquet exponents

#### 4.2.1 Variational equations

#### 4.2.2 Control by tuning the parameter \(\psi _{jn}\)

Note that Eq. (62) makes no distinction between super- and subcritical Hopf bifurcation and between the cluster numbers.

*heterogeneous*delays can be determined in the self-consistent form by the transcendental equation

We note that the characteristic equation for the Floquet exponents, Eq. (67), does *not* distinguish the bifurcation forms, i.e., the supercritical and subcritical case. Also, note that Eq. (67) determining the criteria for controlling the synchrony, does *not* include any parameters defining the synchronization patterns, e.g., the cluster number \(m\). Therefore, the stability of the synchronous periodic orbits in the state of zero-lag, splay, and cluster is determined by the *same* equation, revealing the *same* parameter region.

### 4.3 Special cases and numerical simulations

#### 4.3.1 The case \(\tau _{jn}=\hbox {const.}\)

*only*in-phase synchronization for the subcritical case and controlling synchrony in the

*non-unique*and

*invasive*form for the supercritical case were investigated.

For \(k=0\), corresponding to the synchronization manifold, Eq. (68) is reduced to Eq. (27) with the rescaled coupling strength \(\bar{K}=\mu K\), since the relation \(\nu _0=\mu \) holds for any network structure with a constant row sum \(\mu \). The eigenvalues of coupling matrices \(A\) without self-feedback and for \(k\ne 0\) can be briefly summarized as follows: for all-to-all coupling, i.e., \(a_{ij}=1\) for \(i\ne j\) and zero otherwise, the eigenvalues are given by \(\nu _{k}=-1\). Uni- and bidirectionally coupled rings yield \(\nu _{k}=e^{2ik\pi /N}\) and \(\nu _{k}=2\cos {(2\pi k/N)}\), respectively. In the presence of self-feedback, a replacement \(\nu _k\longmapsto \nu _k+1\) is needed.

*any*value of \(K\) and \(\tau \), for the

*various*network structure (not only unidirectional ring but also bidirectional ring and all-to-all coupling). Thus, the \(m\)-clustering is stabilized in a

*unique*and

*noninvasive*form (59) for the general network topology by tuning the coupling rotation with Eq. (60), which is in a striking contrast to our previous studies [44, 45].

#### 4.3.2 Slightly nonidentical delay times

#### 4.3.3 The case of \(N=2\) with different delay times

*algebraic*equation:

#### 4.3.4 Numerical calculations

Solid dark (blue) and dotted (red) curves correspond to \(\mathrm {Re}(\Lambda )\) of the synchronization manifold and transversal modes for the heterogeneous delays, respectively. The gray curves denote the results for the discrete delays, determined by Eq. (68). We see that there exists an interval of \(K\) in which all of the real parts of Floquet exponents are negative. That is, the four UPOs are stabilized and the \(m\)-cluster is synchronized. Note that the interval for heterogeneous delay is larger than the discrete one. Furthermore, for \(k=0\), the transcritical branch shows little difference for both the discrete and heterogeneous cases, while the Hopf branch for discrete delay becomes positive faster than the heterogeneous case. The transversal modes with \(k=1\) and \(k=3\) for the discrete delay configuration are overlapping since the bidirectional ring configuration yields the eigenvalues \(\nu _k=2\cos (2\pi k/N)\) that are twice degenerate, while the heterogeneous delay configuration shows clearly separated lines for \(k=1\) and \(k=3\).

### 4.4 Thermodynamic limit

The dependence of the stability range on the distribution of heterogeneous delays is determined by Eq. (79). Note that Eq. (79) coincides with Eq. (53), the characteristic equation for the *distributed-delay feedback* control, except for the rescaled coupling strength \(\bar{K}=\mu K\).

## 5 Conclusion

We have proposed a delayed feedback control scheme with arbitrary delay for stabilizing the unstable periodic orbit in the normal form of a subcritical Hopf bifurcation.

In our control scheme, the constraint on the delay to be adjusted to the period of the target orbit is not imposed, while still retaining the condition of noninvasive control. Therefore, by including the delay as a variable bifurcation parameter, we have gained extensive insight into the bifurcation scenarios of the controlled dynamics: a complete bifurcation analysis for the fixed point as well as for the periodic orbit was presented, and the stability domains were identified. Not only weakly unstable periodic orbits but also strongly unstable orbits can be stabilized; the former have a global basin of attraction and the latter only a local basin. Even stabilization of the orbit with infinite period is achieved. A nonlinear control scheme has been developed and studied analytically, which has an advantage over the linear one, since the regime of stabilization of the target orbit is larger.

We have also studied the effects of distributed delayed feedback on the stabilization of the unstable periodic orbit, and we have shown that the regime of stabilization increases with increasing width of the distribution.

We have extended the control scheme to a network of Hopf normal forms coupled with heterogeneous delays. By tuning the parameters of coupling rotations, one of different synchronization patterns, i.e., in-phase, splay or clustering, can be chosen in a unique and noninvasive form, and their stabilizations can be efficiently achieved. The characteristic equation for the Floquet exponents of the heterogeneous delay network has been derived in an analytical form, which reveals the coupling parameters for successful stabilization. The equation takes a unified form regardless of the Hopf bifurcation forms including the subcritical and the supercritical case and regardless of the synchronization patterns. Analysis of Floquet exponents and direct numerical simulations have shown that the heterogeneity in delay makes the stabilization much easier, and provides an enlarged parameter region for successful control.

The thermodynamic limit of a network was considered in the framework of mean-field approximation, and the characteristic equations for the stability of UPOs was derived. Our results show that heterogeneous delays offer greatly enhanced performance for controlling periodic orbits.

From the point of view of applications, it looks very promising to apply our control scheme to semiconductor laser systems with arbitrary delay, i.e., control of UPOs and synchrony by tuning the feedback phase or coupling phase rather than the delay time.

## Notes

### Acknowledgments

Chol-Ung Choe acknowledges support from Alexander von Humboldt Foundation. This work was also supported by DFG in the framework of Sfb 910. Philipp Hövel acknowledges support by BMBF (Grant No. 01Q1001B) in the framework of Bernstein Center for Computational Neuroscience Berlin.

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