## Abstract

We propose a necessary condition for the successful stabilization of a periodic orbit, using the extended version of time-delayed feedback control. This condition depends on the number of real Floquet multipliers larger than unity and is therefore related to the well-known odd-number limitation in non-autonomous systems. We show that the period of the orbit that is induced by mismatching the delay time of the control scheme and the period of the uncontrolled orbit plays an important role in the formulation of the odd-number limitation in the autonomous case.

## 1. Introduction

By their nature, chaotic systems are extremely sensitive to external perturbations, which makes it difficult to predict their future evolution. However, this sensitivity also allows for the surprising possibility of *controlling*a chaotic system. As first demonstrated by Ott *et al.* [1], an external perturbation can be grafted in such a way that one of the unstable periodic orbits of the chaotic attractor becomes stable. This discovery has triggered a large research activity centred around the oxymoronic term *chaos control* [2,3].

A simple but highly efficient scheme of chaos control was introduced by Pyragas [4]. In order to stabilize a periodic orbit of period *τ*, the original system is converted into a system of time-delayed differential equations by adding terms that involve the difference *x*(*t*−*τ*)−*x*(*t*). Here, *x*(*t*) denotes a point in the phase space of the original system. Control terms of this form vanish whenever a periodic orbit with period *τ* is reached, and therefore the Pyragas control scheme is automatically *non-invasive*. The time-delayed feedback control was extended by Socolar *et al.* [5] by using additional control terms of the form *x*(*t*−*kτ*)−*x*(*t*−(*k*−1)*τ*) for integer values *k*>1. Again, terms of this form vanish on a periodic orbit of period *τ*, and therefore enable *non-invasive*control. This *extended time-delayed feedback control* (EDFC) scheme is important for practical applications, because it can significantly increase the range of periodic orbits that can be stabilized [6].

Unfortunately, not all periodic orbits can be stabilized by time-delayed feedback control, which severely limits its applicability. In particular, in a non-autonomous system, a hyperbolic periodic orbit with an odd number of Floquet multipliers larger than unity can never be stabilized by time-delayed feedback control. This has become known as the *odd-number limitation* and was proved in [7] for the original Pyragas control scheme and in [8] for a fairly general class of EDFCs. While the proofs concerning non-autonomous systems presented in [7,8] are correct, the autonomous case was not treated correctly. In particular, footnote 2 in Nakajima [7] claims that all results regarding the odd-number limitation can be proved for the autonomous case ‘with a slight revision’. More importantly, theorem 2 in Nakajima & Ueda [8] claims explicitly that the odd-number limitation applies to the autonomous case. However, the proof of this theorem contains an error in the expansion of a determinant and is therefore wrong. The non-autonomous case was also discussed in [9] and it was stated that the Pyragas method for non-autonomous systems should only be able to stabilize orbits with ‘finite torsion’, but not orbits with a single Floquet multiplier larger than unity. Although it was not explicitly claimed in [9] that this statement can be extended to the autonomous case, the casual reader might not have noticed this subtle point. Based on [7–9], there was, therefore, a general belief among members of the chaos-control community that the odd-number limitation also holds in the autonomous case.

This changed with the work by Fiedler *et al.* [10], who gave an example of a two-dimensional autonomous system, with precisely one Floquet multiplier larger than one, which can be stabilized using the original Pyragas control scheme. This immediately showed that the odd-number limitation does not hold for autonomous systems and opened up the exciting possibility that the time-delayed feedback control could be far more powerful than previously thought.

While Fiedler *et al.* [10] provide a counterexample for the original odd-number limitation, it raises the question of whether the odd-number limitation is outright wrong, or holds at least under certain additional conditions. After all, numerical and experimental evidence never showed any problem with the odd-number limitation before the publication of Fiedler *et al.* [10], which suggests that for many systems or control schemes the odd-number limitation might indeed be true. Motivated by this possibility, we showed, in our recent work [11], that a modified version of the odd-number limitation holds for Pyragas-type control. Like the original odd-number limitation, our modified version also involves the number of Floquet multipliers greater than unity, but, in addition, also depends on an analytical expression that involves an integral of the control force along the desired periodic orbit. Interestingly, this analytical expression can also be obtained by studying the period of the orbit that is induced if the system is forced with a delayed feedback term where the delay time does not match the period of the orbit in the unforced system. Our modified version of the odd-number limitation correctly predicts the stability boundaries of the previous counterexample presented in [10].

In this study, we generalize our previous results on the odd-number limitation of Pyragas control to the case of EDFC. While we closely follow the arguments laid out in [11], we intend to keep the presentation self-contained. The remainder of the paper is organized as follows: in §2, we introduce the notation and state the main theorem, which is then proved in §3. In §4, we discuss the significance and practical implication of our results.

## 2. Statement of the theorem

We consider a dynamical system of the form
2.1with and , and assume that there exists a *τ*-periodic solution *x**(*t*)=*x**(*t*+*τ*) of (2.1). With this periodic orbit, we associate a (principal) *fundamental matrix* *Φ*(*t*) that fulfils the matrix equation
2.2where *Df*(*x**(*t*)) denotes the Jacobian of *f* evaluated at the point *x**(*t*) along the periodic orbit. For *t*=*τ*, the fundamental matrix *Φ*(*τ*) is often called the *monodromy matrix,* and the generalized eigenvalues *μ*_{1},…,*μ*_{n} of *Φ*(*τ*) are known as *Floquet multipliers* (or characteristic multipliers) of the periodic orbit *x**(*t*). Taking the time derivative of (2.1) and taking into account that the function *f* does not explicitly depend on time, we observe that
Comparing with (2.2), we can therefore identify and in particular note that
It therefore follows that the monodromy matrix *Φ*(*τ*) for a periodic orbit in an autonomous system has at least one eigenvalue equal to one, and we choose in the following *μ*_{1}=1. As *Φ*(*τ*) is a real matrix, the set of the remaining Floquet multipliers {*μ*_{2},…,*μ*_{n}} is composed of either real numbers or complex conjugate pairs of complex numbers. The significance of the Floquet multipliers lies in the fact that they allow us to characterize the periodic orbit in question. For example, if there exists at least one Floquet multiplier *μ*_{k} such that |*μ*_{k}|>1, then the periodic orbit is unstable. If there exists at least one Floquet multiplier *μ*_{k} with *k*>1 such that |*μ*_{k}|=1, then the orbit is called *non-hyperbolic*. In the following, we will assume that the periodic orbit *x**(*t*) is hyperbolic, i.e. none of the Floquet multipliers other than *μ*_{1} is located on the unit circle. We also define the symbol *m* to denote the number of Floquet multipliers that are real and larger than one. For *m*>0, the periodic orbit is unstable; however, the converse is not true.

Following the ideas of Socolar *et al.* [5], we now introduce an extended time-delayed feedback term by modifying the system (2.1) as
2.3where and *τ* is a positive parameter for the delay time. Both the control matrix *K* and the memory matrix *R* are *n*×*n* matrices, where *R* fulfils the condition
The solution of (2.3) for *t*>0 requires the knowledge of the two (left-continuous) functions *x*(*t*) and *y*(*t*) on the interval *t*∈(−*τ*,0]. Note that formally the system (2.3) is not a delay *differential* equation, because no time derivative of *y*(*t*) appears. For *R*=0, we recover the traditional Pyragas control scheme. Instead of (2.3), many other essentially equivalent formulations of EDFC are used, for example

The system (2.3) possesses the obvious *τ*-periodic solution
2.4However, the stability of this solution may have been affected by the control scheme. In the following, we would like to gain some insights into the stability properties of this solution.

Before we formulate our main theorem, we slightly change the time-delay parameter *τ* appearing in (2.3) to a new value as follows:
2.5If is sufficiently close to *τ*, then we can assume that there exists a periodic solution for (2.5) that is close to the original solution (2.4). However, we expect that the period of this new solution will be, in general, different from both *τ* and , and we denote this period by the new symbol . Continuity requires that . We can now formulate our main theorem.

### Theorem 2.1

*Let x***(t) be a τ-periodic orbit of (2.1) with m real Floquet multipliers greater than unity and let* *be the period of the induced periodic orbit of (2.5) with* *. Then, the orbit x***(t) is an unstable solution of (2.3) if the condition
*2.6*is fulfilled.*

## 3. Proof of the theorem

In the proof, we follow the techniques developed in [7,8,11], but we aim to keep the following presentation as self-contained as possible. We first formulate a lemma, which allows us to connect the linear stability of system (2.3) with the solution of a time-dependent linear equation.

### Lemma 3.1

*If the equation*
3.1*possesses a solution of the form*
3.2*for real* *ν*>1, *then the periodic orbit* *x**(*t*) *is an unstable solution of (2.3)*.

### Proof.

First, note that, because of the requirement , the eigenvalues of *Rν*^{−1} are contained in the unit circle of the complex plane, and the matrix [1−*Rν*^{−1}] is indeed non-singular for all *ν*≥1. It is then straightforward to check that the ansatz
and
fulfils (2.3) to first order in *δx*(*t*). Because of (3.2), we have therefore explicitly constructed an exponentially growing linear perturbation of the solution (2.4), which implies that *x**(*t*) is an unstable solution of (2.3). This concludes the proof of lemma 3.1. □

We now have to show that, under the conditions of the theorem, there exists a *ν*>1 such that the linear equation (3.1) allows for a solution fulfilling (3.2). We first introduce the fundamental matrix for (3.1) via
3.3and
3.4If we now find a *ν*>1 such that
3.5then there exists a that fulfils *Ψ*_{ν}(*τ*)*δx*_{0}=*νδx*_{0}. Using *δx*(*t*)=*Ψ*_{ν}(*t*)*δx*_{0}, we can then construct a solution of (3.1) that is of the form required by (3.2). This motivates the introduction of the function [7]
3.6and, from the above discussion, we can conclude that the proof of our theorem is complete if we are able to show that there exists a *ν*_{c}>1 such that *F*(*ν*_{c})=0. For *ν*=1, we see from (3.3) that *Ψ*_{1}(*t*)=*Φ*(*t*). One of the eigenvalues of *Φ*(*τ*) is however equal to unity, and we therefore find that *F*(1)=0.

For a further discussion of *F*(*ν*), we write the fundamental matrix in the form
3.7which can easily be checked by direct differentiation. All the terms appearing in this expression for *Ψ*_{ν}(*t*) remain finite for and therefore (3.6) yields , or, in other words, *F*(*ν*) diverges as *ν*^{n} for large *ν*. We can therefore summarize the discussion in the last two paragraphs in lemma 3.2.

### Lemma 3.2

*If, for a given periodic orbit* *x**(*t*), *the condition*
3.8*holds, then the orbit is an unstable solution of (2.3)*.

### Proof.

As *F*(*ν*) is continuous, *F*(1)=0 and *F*′(1)<0, there exists a *ν*_{n}>1 such that *F*(*ν*_{n})<0. However, as *F*(*ν*) diverges as *ν*^{n} for large *ν*, it follows by the intermediate value theorem that there exists *ν*_{c}>1 with *F*(*ν*_{c})=0. For this *ν*_{c}, we can then construct the function *δx*(*t*) required for lemma 3.1, and the orbit is unstable. Therefore, lemma 3.2 is proved.

In view of lemma 3.2, we now need to study the conditions for which *F*^{′}(1)<0 holds. Before we proceed, it is now useful to introduce the matrix *W* that diagonalizes *Φ*(*τ*), i.e.
Here, the asterisks denote either 0 or 1, depending on the Jordan block associated with a particular eigenvalue. This allows us to formulate the following lemma.

### Lemma 3.3

*A periodic orbit with* *m* *Floquet multipliers greater than unity is an unstable solution of the EDFC scheme (2.3) if the condition*
3.9*holds*.

### Proof.

The idea of the proof is to show that (3.9) implies (3.8) and then use lemma 3.2. To calculate the sign of , we expand *F*(1+*ϵ*) to first order in *ϵ*. We obtain
3.10where the matrices *M*^{0} and *M*^{1} collect the terms in zeroth and first order of *ϵ*. We obtain
and therefore
and
Because of the special form of *M*^{0}, only one term contributes to the determinant in (3.10) in first order of *ϵ*,
For the sign of *F*′(1), we therefore conclude that
3.11where in the last step we used that every real Floquet multiplier greater than unity contributes a minus sign to the product. Comparing with (3.9), we observe that if condition (3.9) is fulfilled then condition (3.8) is also fulfilled because of (3.11). Therefore, lemma 3.3 now follows by evoking lemma 3.2.

As a final step in the proof of the main theorem, it now remains to be shown that (2.6) implies condition (3.9). We need to establish how the detuning between the period *τ* of the uncontrolled orbit and the delay time of the EDFC scheme (2.5) influences the period of the orbit that is induced by this detuning. For the Pyragas control, this problem was solved by Just *et al.* [12], and for the case of EDFC by Novičenko & Pyragas [13]. In our notation, the result of equation (28) from [13] can be written in the form
or
It therefore follows that the condition (3.9) in lemma 3.3 and the condition (2.6) in the main theorem are equivalent, and this concludes the proof of the theorem.

## 4. Discussion

Our main theorem provides a limitation on the applicability of EDFC, which involves the number of real Floquet multipliers (*m*) and a combination of the period of the uncontrolled system (*τ*), a detuned delay time and the resulting period of the induced orbit . Let us now have a closer look at the condition (2.6) appearing in the theorem. We see that *m* only appears in the form (−1)^{m}, which is negative for odd *m* and positive otherwise. The second factor involves a ratio *r* of the form
4.1and we are asked to evaluate the sign of this ratio in the limit where goes to *τ*. If this ratio was always positive, then our theorem would simply reduce to the statement of the old odd-number limitation, i.e. orbits with odd number *m* cannot be controlled. Our theorem now modifies this statement as follows: orbits with odd *m* can be stabilized, but only if the control terms are implemented in such a way that *r* is negative. We stress that our theorem gives only a necessary condition for control, i.e. a violation of condition (2.6) will not guarantee that an orbit will be stabilized. Intuitively, the case of negative *r* seems to be unusual, as it implies that, if we slightly increase the delay time to a value larger than *τ*, then the period of the induced orbit needs to be even larger than itself. This intuitively strange situation might be one of the reasons why the violation of the original odd-number limitation in the autonomous case was not observed earlier. In the case of the counterexample given in [10], *r* is indeed negative, as was shown in [11].

We also remark that lemma 3.3 provides useful insights in its own right. We can slightly rewrite the condition (3.9) as follows:
Here, *z*^{T}(*t*) is the first row of the matrix *W*^{−1}*Φ*(*t*)^{−1}, and is also known as the dual vector of the zero mode. If we now write the control matrix with a scalar prefactor *k* in the form
and introduce *κ* via
then the condition (3.9) can be written in the form
4.2The period of the induced orbit now depends on the scalar coupling strength *k* and the delay time in the following way [12,13]:
4.3This allows us to conveniently determine the parameter *κ* through
4.4If we focus our interest again on the case where the stabilization for odd *m* is possible, we can conclude from (4.2) that this requires a positive *κ*. From (4.4), we can obtain the value and in particular the sign of *κ* from a ‘measurement’ of in the low coupling regime. From (4.3), we observe that for positive *κ* the period of the induced orbit is always outside the interval for all positive values of *k*, and diverges at the point where (4.2) is first violated.

## Funding statement

This work was supported by Science Foundation Ireland (grant no. 09/SIRG/I1615).

## Acknowledgements

We thank B. Fiedler, V. Flunkert, P. Hövel, W. Just, K. Pyragas and E. Schöll for fruitful discussions.

## 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.