Control and Observability Aspects of Phase Synchronization

TL;DR

This paper investigates control and observability issues in phase synchronization of oscillators, proposing a feedback control framework, analyzing coordinate choices, and addressing practical challenges in phase-based control.

Contribution

It introduces a novel control framework for phase synchronization, compares coordinate effects, and proposes a method to handle indirect phase error control.

Findings

Observability is less critical for phase synchronization than for complete synchronization.

Coordinate choice impacts the control design but not the core synchronization ability.

A practical method is proposed to control phase when phase error isn't directly used in control.

Abstract

This paper addresses important control and observability aspects of the phase synchronization of two oscillators. To this aim a feedback control framework is proposed based on which issues related to master-slave synchronization are analyzed. Comparing results using Cartesian and cylindrical coordinates in the context of the proposed framework it is argued that: i)~observability does not play a significant role in phase synchronization, although it is granted that it might be relevant for complete synchronization; and ii)~a practical difficulty is faced when phase synchronization is aimed at but the control action is not a direct function of the phase error. A procedure for overcoming such a problem is proposed. The only assumption made is that the phase can be estimated using the arctangent function. The main aspects of the paper are illustrated using the Poincar\'e equations, van der Pol and R\"ossler oscillators in dynamical regimes for which the phase is well defined.

Tables

Table 1. Table 1: Observability coefficients computed as in (Letellier et al.,, 2005 ) . Index α 𝛼 \alpha indicates the percentage of situations within the investigated ranges for which PS was verified. The investigated ranges are ω ∈ [ 0.85 ,  1.15 ] 𝜔 0.85 1.15 \omega\in[0.85,\,1.15] and k s ∈ [ 0 ,  1 ] , s ∈ { x , y , ρ , ϕ } formulae-sequence subscript 𝑘 𝑠 0 1 𝑠 𝑥 𝑦 𝜌 italic-ϕ k_{s}\in[0,\,1],~{}s\in\{x,\,y,\,\rho,\,\phi\} for the Poincaré and van der Pol systems, and k s ∈ [ 0 ,  0.5 ] , s ∈ { x , y , z , ρ , ϕ } formulae-sequence subscript 𝑘 𝑠 0 0.5 𝑠 𝑥 𝑦 𝑧 𝜌 italic-ϕ k_{s}\in[0,\,0.5],~{}s\in\{x,\,y,\,z,\,\rho,\,\phi\} for the Rössler system.

System	Cartesian		Cylindrical
Poincaré	${\delta }_x=0.340$	${\alpha }_x=75\%$	${\delta }_{\rho }=0$	${\alpha }_{\rho }=8\%$
	${\delta }_y=0.338$	${\alpha }_y=75\%$	${\delta }_{\varphi }=0$	${\alpha }_{\varphi }=98\%$
van der Pol	${\delta }_x=1$	${\alpha }_x=70\%$	${\delta }_{\rho }=0.128$	${\alpha }_{\rho }=57\%$
$\mu =1$	${\delta }_y=0.125$	${\alpha }_y=82\%$	${\delta }_{\varphi }=0.114$	${\alpha }_{\varphi }=98\%$
van der Pol	${\delta }_x=1$	${\alpha }_x=85\%$	${\delta }_{\rho }=0.036$	${\alpha }_{\rho }=14\%$
$\mu =10$	${\delta }_y=0.002$	${\alpha }_y=33\%$	${\delta }_{\varphi }=0.001$	${\alpha }_{\varphi }=77\%$
Rössler	${\delta }_x=0.022$	${\alpha }_x=77\%$	${\delta }_{\rho }=0.019$	${\alpha }_{\rho }=7\%$
	${\delta }_y=0.133$	${\alpha }_y=76\%$	${\delta }_{\varphi }=0.001$	${\alpha }_{\varphi }=98\%$
	${\delta }_z=0.006$	${\alpha }_z=4\%$	${\delta }_z=0.003$	${\alpha }_z=4\%$

Table 2. Table 2: Synchronization regions for the three oscillators. Parameter values and ranges for gain and frequency are as for Table  1 but with the CA-MV mismatch problem solved using the procedure proposed in Sec.  3 , hence in Cartesian coordinates. The α x subscript 𝛼 𝑥 \alpha_{x} are computed with m x = − y 1 ​ k ϕ ​ [ r ϕ 2 ​ ( t ) − ϕ 1 ​ ( t ) ] subscript 𝑚 𝑥 subscript 𝑦 1 subscript 𝑘 italic-ϕ delimited-[] subscript 𝑟 subscript italic-ϕ 2 𝑡 subscript italic-ϕ 1 𝑡 m_{x}=-y_{1}k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)] and m y = 0 subscript 𝑚 𝑦 0 m_{y}=0 ; the α y subscript 𝛼 𝑦 \alpha_{y} are computed with m y = x 1 ​ k ϕ ​ [ r ϕ 2 ​ ( t ) − ϕ 1 ​ ( t ) ] subscript 𝑚 𝑦 subscript 𝑥 1 subscript 𝑘 italic-ϕ delimited-[] subscript 𝑟 subscript italic-ϕ 2 𝑡 subscript italic-ϕ 1 𝑡 m_{y}=x_{1}k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)] and m x = 0 subscript 𝑚 𝑥 0 m_{x}=0 ; and the α x ​ y subscript 𝛼 𝑥 𝑦 \alpha_{xy} were computed with m x = − y 1 ​ k ϕ ​ [ r ϕ 2 ​ ( t ) − ϕ 1 ​ ( t ) ] subscript 𝑚 𝑥 subscript 𝑦 1 subscript 𝑘 italic-ϕ delimited-[] subscript 𝑟 subscript italic-ϕ 2 𝑡 subscript italic-ϕ 1 𝑡 m_{x}=-y_{1}k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)] , m y = x 1 ​ k ϕ ​ [ r ϕ 2 ​ ( t ) − ϕ 1 ​ ( t ) ] subscript 𝑚 𝑦 subscript 𝑥 1 subscript 𝑘 italic-ϕ delimited-[] subscript 𝑟 subscript italic-ϕ 2 𝑡 subscript italic-ϕ 1 𝑡 m_{y}=x_{1}k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)] . In the case of the Rössler m z = 0 subscript 𝑚 𝑧 0 m_{z}=0 always.

System	Synchronization
	Regions
	${\alpha }_x=94\%$
Poincaré	${\alpha }_y=94\%$
	${\alpha }_{xy}=97\%$
	${\alpha }_x=94\%$
van der Pol	${\alpha }_y=94\%$
$\mu =1$	${\alpha }_{xy}=97\%$
	${\alpha }_x=92\%$
van der Pol	${\alpha }_y=94\%$
$\mu =10$	${\alpha }_{xy}=97\%$
	${\alpha }_x=65\%$
Rössler	${\alpha }_y=74\%$
	${\alpha }_{xy}=92\%$

Equations

$${\cal O}_{1}:\left\{\begin{array}[]{lll}\dot{x}_{1}&=f_{x}(x_{1},\,y_{1})+m_{x}(t)\\ \dot{y}_{1}&=f_{y}(x_{1},\,y_{1})+m_{y}(t),\\ \end{array}\right.$$

$${\cal O}_{1}:\left\{\begin{array}[]{lll}\dot{\rho}_{1}&=g_{\rho}(\rho_{1},\,\phi_{1})+m_{\rho}(t)\\ \dot{\phi}_{1}&=g_{\phi}(\rho_{1},\,\phi_{1})+m_{\phi}(t).\end{array}\right.$$

$$\dot{{\varphi }_1}$$

$$\displaystyle\frac{\partial~{}\tan^{-1}(y/x)}{\partial\mbox{\boldmath$x$}}=\left[\frac{-y}{x^{2}+y^{2}}~{}\frac{x}{x^{2}+y^{2}}\right]^{\scriptscriptstyle\rm T}.$$

$$\dot{{\varphi }_1}=\nu (\mathbf{x})+\left(\frac{-y_1}{x_1^2+y_1^2}\right)k_{\varphi }\left[r_{{\varphi }_2}(t)-{\varphi }_1(t)\right].$$

$$\dot{{\varphi }_1}=\nu (\mathbf{x})+\left(\frac{x_1}{x_1^2+y_1^2}\right)k_{\varphi }\left[r_{{\varphi }_2}(t)-{\varphi }_1(t)\right],$$

$$m_x$$

$$m_y$$

$$\dot{{\varphi }_1}=\nu (\mathbf{x})+{\tilde{k}}_{\varphi }(t)[r_{{\varphi }_2}(t)-{\varphi }_1(t)],{\tilde{k}}_{\varphi }(t)>0,$$

$$\left\{\begin{array}[]{lll}\dot{x}_{1}&=-\omega y_{1}-\lambda(x_{1}^{2}+y_{1}^{2}-p^{2})x_{1}+m_{x}(t)\\ \dot{y}_{1}&=\omega x_{1}-\lambda(x_{1}^{2}+y_{1}^{2}-p^{2})y_{1}+m_{y}(t),\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}\dot{\rho}_{1}&=\lambda\rho_{1}(p^{2}-\rho_{1}^{2})\\ \dot{\phi}_{1}&=\omega+m_{\phi}(t).\end{array}\right.$$

$$\left\{\begin{array}[]{lll}\dot{x}_{1}&=\omega y_{1}+m_{x}(t)\\ \dot{y}_{1}&=-\omega x_{1}-\mu(\beta x_{1}^{2}-1)y_{1}+m_{y}(t),\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}\dot{\rho}_{1}&=-\mu(\beta\rho_{1}^{2}\cos^{2}\phi_{1}-1)\rho_{1}\sin^{2}\phi_{1}\\ \dot{\phi}_{1}&=-\omega-\mu(\beta\rho_{1}^{2}\cos^{2}\phi_{1}-1)\sin\phi_{1}\cos\phi_{1}+m_{\phi}(t),\end{array}\right.$$

$$\left\{\begin{array}[]{lll}\dot{x}_{1}&=-\omega y_{1}-z_{1}+m_{x}(t)\\ \dot{y}_{1}&=\omega x_{1}+ay_{1}+m_{y}(t)\\ \dot{z}_{1}&=b+z_{1}(x_{1}-c)+m_{z}(t),\\ \end{array}\right.$$

$$\left\{\begin{array}[]{lll}\dot{\rho}_{1}&=a\rho_{1}\sin^{2}\phi_{1}-z_{1}\cos\phi_{1}\\ \dot{\phi}_{1}&=\omega+a\sin\phi_{1}\cos\phi_{1}+(z_{1}/\rho_{1})\sin\phi_{1}+m_{\phi}(t),\\ \dot{z}_{1}&=b+z_{1}(\rho_{1}\cos\phi_{1}-c).\end{array}\right.$$

$$R=\sqrt{⟨\sin |{\varphi }_2(t)-{\varphi }_1(t)|{⟩}^2+⟨\cos |{\varphi }_2(t)-{\varphi }_1(t)|{⟩}^2},$$

$$\displaystyle{\cal O}_{\rho}=\left[\begin{array}[]{cc}1&0\\ \lambda(p^{2}-3\rho^{2})&0\end{array}\right],\hskip 14.22636pt{\cal O}_{\phi}=\left[\begin{array}[]{cc}0&1\\ 0&0\end{array}\right],$$

$$\displaystyle{\cal O}_{x}(\mbox{\boldmath$x$})$$

$$\displaystyle{\cal O}_{y}(\mbox{\boldmath$x$})$$

$${\mathcal{O}}_{\rho }(\rho ,\varphi )$$

$${\mathcal{O}}_{\varphi }(\rho ,\varphi )$$

$$\displaystyle{\cal O}_{x}(\mbox{\boldmath$x$})$$

$$\displaystyle{\cal O}_{y}(\mbox{\boldmath$x$})$$

Full text

**Control and Observability Aspects of Phase Synchronization **

Luis A. Aguirre

Departamento de Engenharia Eletrônica, Programa de Pós Graduação em Engenharia Elétrica da Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, 31.270-901 Belo Horizonte, MG, Brazil. aguirre@ufmg.br

Leandro Freitas

Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais, Campus Betim, 32.677-564 Betim MG; Programa de Pós-Graduação em Engenharia Elétrica da Universidade Federal de Minas Gerais — Av. Antônio Carlos 6627, 31.270-901 Belo Horizonte MG, Brazil. leandrofabreu@yahoo.com.br

1 Introduction

The field of synchronization of nonlinear oscillators has witnessed a sharp increase in interest and number of papers in the last three decades or so. By now, a number of types of synchronization and ways to achieve them have been reported (Boccaletti et al.,, 2002; Osipov et al.,, 2007).

It is possibly safe to say that in the earliest days of the synchronization of chaotic oscillators the emphasis was on complete synchronization (CS) of identical oscillators and the standard way of achieving it was via multivariate dissipative coupling (Fujisaka and Yamada,, 1983). Roughly a decade later, it was shown that under some conditions CS could be achieved by dissipatively coupling the oscillators via a single variable (Kapitaniak,, 1994). Another major step in the field was the report of phase synchronization (PS) as a weaker form of synchronization than CS (Rosenblum et al.,, 1996). It was shown that PS very often happened by means of weak  dissipative coupling of a single variable in nonidentical oscillators. Hence, in many papers, especially the earlier ones, PS was seen simply as an early stage of synchronization, almost a “route to CS”.

Not before long, it became clear that PS was a relevant type of synchronization per se. In fact, in many engineering applications it is PS that matters (Piqueira,, 2011), however the “traditional” dissipative coupling is still commonly used. It has been pointed out in (Belykh et al.,, 2005) that such a way of coupling might not be effective in PS applications.

An interesting and related problem in the context of synchronization attained by monovariable coupling is that of deciding which variable to use. To the best of the authors’ knowledge there is no general choice. Nonetheless, it was shown that for some investigated systems the variables used in bidirectional dissipative coupling that achieved synchronization (CS and PS) for lower values of coupling gain often coincided with those variables that provided greater observability of the dynamics, although the results are also influenced by the dynamical regime (Letellier and Aguirre,, 2010). A similar approach was recently extended to investigate how controllability might influence the establishment of CS and PS to an external signal, however no general relationship was found (Aguirre and Letellier,, 2016). In a similar vein, possible relations between classes of synchronization and observability have been investigated in (Sendiña-Nadal et al.,, 2016), where the authors conclude that although some relation exists, observability only explains synchronization partially.

The relationship between observability and the ability to synchronize two oscillators by monovariable dissipative coupling can be understood in the following terms. For the sake of argument, let us assume that \mbox{\boldmathx}=[x,\,y,\,z]^{\scriptscriptstyle\rm T} and that the coupling is done using the $x$ variable, hence, for the slave oscillator a (linear) dissipative coupling term of the form $k_{1}[x_{2}(t)-x_{1}(t)]$, where the subscripts indicate the different oscillators (1 for the slave oscillator and 2 for the master), will be added to the first equation of the slave oscillator 111In case of bidirectional coupling, the term $k_{2}[x_{1}(t)-x_{2}(t)]$ will be added to the first equation of the second oscillator, and the oscillators are no longer called master or slave.. Now assume that CS is achieved, then $|x_{2}(t)-x_{1}(t)|\rightarrow 0$, $|y_{2}(t)-y_{1}(t)|\rightarrow 0$ and $|z_{2}(t)-z_{1}(t)|\rightarrow 0$, although the only  signal used from the master oscillator was $x_{2}(t)$. Hence, somehow the knowledge of $x_{2}(t)$ is sufficient to figure out the full state of the second oscillator. Hence the first oscillator plays the role of an observer of the second one. This is typically the case when $x$ conveys good observability of the dynamics. Conversely, if $x$ is poor from an observability point of view, to synchronize the oscillators by coupling using such a variable is more difficult (Letellier and Aguirre,, 2010). Hence there seems to be an acknowledged connection between observability and synchronization at least in some contexts (Parlitz et al.,, 1996; Maybhate and Amritkar,, 1999; Freitas et al.,, 2005; Sedigh-Sarvestani et al.,, 2012).

How does all this translate to the case of PS? As for the coupling, it has already been pointed out that dissipative coupling is not always effective to achieve PS (Belykh et al.,, 2005), but why? Also, could it be that a more fundamental issue in synchronization problems – at least in the case of PS – is not so much the level of (state) observability provided by the variable used in the coupling but rather the amount of information concerning the phase conveyed by such a variable? In order to address such questions PS is set in the context of a control problem. This framework – the lack of which has been pointed out in (Belykh et al.,, 2005) – is used to investigate PS and to point out the need for new tools. Using this framework it will be argued that a practical problem is that of a mismatch that arises between the controlled and manipulated variables. A procedure to overcome this shortcoming is proposed.

This paper is organized as follows. In Section 2 the proposed feedback control framework is presented. The main points and definitions of the paper are provided in that section. In particular, the control action and manipulated variable mismatch  problem is pointed out in Sec. 2.2.3, and a solution to that problem is proposed in Sec. 3. The numerical results using benchmark oscillators in two systems of coordinates – Cartesian and cylindrical – are presented in Section 4. The results are discussed in Section 5 where the main conclusions and suggestions for future work are also provided.

2 Feedback Control Framework

In order to motivate the main point of this paper we start by describing the typical control loop shown in Figure 1. The input $r(t)$ is the reference or set point. The feedback signal $h(t)$ is compared to the reference and an “error” signal $e(t)$ is produced based on which the controller ${\cal D}$ yields the control action (CA) $m(t)$ to drive the plant ${\cal O}_{1}$, by changing the corresponding manipulated variable (MV) within ${\cal O}_{1}$. The signal at the righthand of the loop, $c(t)$, is the controlled variable.

In most control problems, the controlled variable $c(t)$ is fedback and compared to the reference. In Figure 1 this can be attained by placing the switch ${\cal S}$ in position 2, and hence $h(t)=c(t)$. In such a situation, the controller ${\cal D}$ is supposed to drive the plant until $e(t)=r(t)-c(t)$ remains within acceptable bounds. In the sequel the basic control scheme just described will be used to analyze two different types of synchronization.

2.1 Complete synchronization

Consider a generic 2D periodic oscillator with control inputs $m_{x}(t)$ and $m_{y}(t)$

[TABLE]

Assume that we choose $r_{x}(t)=x_{2}(t)$ and $m_{y}=0$, where $x_{2}(t)$ is the first state variable from another oscillator (the master). Suppose we want $x_{1}(t)$ to follow the reference, then we choose $c(t)=x_{1}(t)$, place ${\cal S}$ in position 2, and use, say, dissipative (or diffusive) coupling $m_{x}(t)=k_{x}[r_{x}(t)-x_{1}(t)]$. Often there is a range of values of $k_{x}$ (and $m_{y}=0$) for which $|e(t)|\rightarrow 0$ is achieved. In principle, coupling could also be achieved using the $y$ variable, that is taking $k_{x}=0$, $r_{y}(t)=y_{2}(t)$, $c(t)=y_{1}(t)$ and $m_{y}(t)=k_{y}[r_{y}(t)-y_{1}(t)]$ it is often possible to get $|e(t)|\rightarrow 0$ over a range of values of $k_{y}$.

The scheme just described has been pursued since the early days of synchronization of chaotic – 3D – oscillators (Fujisaka and Yamada,, 1983), where coupling was dissipative, as described in the preceding paragraph, although controlling all state variables, that is, the control aim was to make the full state vector \mbox{\boldmathc}(t)=\mbox{\boldmathx}_{1}(t) follow the reference \mbox{\boldmathr}(t)=\mbox{\boldmathx}_{2}(t). An early paper in which only one state variable was fedback (like shown in Figure 1) is (Kapitaniak,, 1994) where complete synchronization was also the goal.

Remark 1. An important aspect to notice is that in (1) the control action $m_{s}(t)$ is added to the corresponding equation, that is, to the equation that defines the dynamics of the manipulated variable, $s\in\{x,\,y\}$. When the control action is a function of the error in the manipulated variable, as in this case, we have a favorable situation from a control point of view.

Remark 2. Another common problem is the PS to an external force – usually periodic – which is simply added to one of the equations of the oscillator ${\cal O}_{1}$. Such is an open-loop  type of operation in which the switch ${\cal S}$ remains opened, hence $e(t)=r(t)$. An early example of this was reported in (Stone,, 1992) for the case of the Rössler system with $r(t)$ an impulsive force acting on the $y$ component at a constant frequency.

2.2 Phase synchronization

In 1996, Rosenblum and co-workers reported a different type of synchronization: phase synchronization (Rosenblum et al.,, 1996). They showed that the Rössler system with $m_{x}(t)=k_{x}[r_{x}(t)-x_{1}(t)]$ and $m_{y}(t)=0$ would satisfy the condition for phase synchronization, also known as non-strict phase locking (Osipov et al.,, 2007): $|\phi_{2}(t)-\phi_{1}(t)|<{\rm const}$, even though $e(t)=r_{x}(t)-x_{1}(t)$ did not  converge to zero, but rather $r_{x}(t)$ and $x_{1}(t)$ remained practically uncorrelated. In the aforementioned paper the authors reported on two cases: i) $r_{x}(t)$ was the $x$ variable of another Rössler oscillator, and ii) $r_{x}(t)$ was the $u$ variable of a bidirectionally coupled Mackey-Glass differential-delay system.

In what follows we discuss different choices for the reference signal and therefore for the control action, and this related to the mathematical representation of oscillator ${\cal O}_{1}$. We start with the most commonly found situation.

2.2.1 Cartesian coordinates

This is the most common situation found in the literature and it is not difficult to see why. As pointed out in Sec. 1, this situation emerges from CS problems where the aim is to attain \mbox{\boldmathx}_{1}=\mbox{\boldmathx}_{2}. Since the PS was reported, many works have focused on such a phenomenon, though retaining the original and more general setting. So here we analyze PS in this “CS-oriented” setting, that is, $r_{s}(t)=s(t),~{}s\in\{x,\,y$}. In what follows we choose $r_{x}(t)=x_{2}(t)$, although $r_{y}(t)=y_{2}(t)$ could be used instead.

In PS investigations, it would be natural to chose the phase as the controlled variable, that is, $c(t)=\phi_{1}(t)$ although the reference signal is a state variable $r_{x}(t)=x_{2}(t)$ and not a phase signal, because of the (inherited) general CS setting. In order to produce an error signal that makes sense,222Ideally this signal should be a function of the phase error, e.g. $e(t)=\phi_{2}(t)-\phi_{1}(t)$. ${\cal S}$ is connected to 1 and the first state variable of ${\cal O}_{1}$ is fedback, hence $h(t)=x_{1}(t)$, rendering $e(t)=x_{2}(t)-x_{1}(t)$. One of the simplest controllers is proportional to the error, and therefore the control action is $m_{x}(t)=k_{x}[x_{2}(t)-x_{1}(t)]$, although it might not be the most effective choice (Belykh et al.,, 2005). According to the nomenclature defined $m_{x}(t)$ is added to the first equation of ${\cal O}_{1}$ (as in Eq. 1) and thus the welcome matching between the control action and the manipulated variable is attained. However, the resulting control problem in this case is somewhat unconventional, with ${\cal S}$ connected to 1 and with the reference and error of a different type than the controlled variable.

Remark 3. The condition for PS is that $|\phi_{2}(t)-\phi_{1}(t)|<{\rm const}$. However the control actuates in a way to reduce the error $e(t)=x_{2}(t)-x_{1}(t)$. Therefore the key to the success of pursuing PS using a CS-oriented setting is that $x_{2}(t)$ and $x_{1}(t)$ should be good proxy variables  for $\phi_{2}(t)$ and $\phi_{1}(t)$, respectively.

2.2.2 Cylindrical coordinates

In order to focus on the phase dynamics of an oscillator a common procedure is to reduce the oscillator to a phase oscillator, which has an explicit equation for the phase, as in the famous model proposed by Kuramoto (Kuramoto,, 1975; Dörfler and Bullo,, 2014). Josić and Mar, (2001) showed that it is possible to reduce a chaotic oscillator to a phase oscillator by means of an appropriate change of coordinates, which exists for most phase-coherent systems. Hence the use of phase oscillators solves two important problems. First, because the phase is readily available, there is no need to use proxy variables for it. Secondly, the control action can be added directly to the equation that describes the phase dynamics thus avoiding any mismatch between the control action and the manipulated variable. This will be detailed below.

Now let us assume that the aim is to achieve PS of two oscillators. For the sake of argument we only address the case of unidirectional coupling and refer the reader to Figure 1. From an ideal control  point of view in this problem the reference should be a phase signal, hence $r_{\phi}(t)=\phi_{2}(t)$, the controlled variable should be the corresponding phase of ${\cal O}_{1}$, hence $c(t)=\phi_{1}$, switch ${\cal S}$ should be connected to terminal 2, and $e(t)$ then becomes the phase error. In this framework the control goal is that $|e(t)|=|\phi_{2}(t)-\phi_{1}(t)|<{\rm const}$ for all $t>t_{0}$. In the case of dissipative (or diffusive) coupling, the controller can be a pure gain $k_{\phi}$, such that the control action is $m_{\phi}(t)=k_{\phi}[r_{\phi}(t)-\phi_{1}(t)]$, where $r_{\phi}(t)=\phi_{2}(t)$.333A common nonlinear and $2\pi$-periodic control action is $m_{\phi}(t)=k_{\phi}\sin[r_{\phi}(t)-\phi_{1}(t)]$. In this paper we use a $2\pi$-periodic control action $m_{\phi}(t)=k_{\phi}{\tt wraptopi}[r_{\phi}(t)-\phi_{1}(t)]$, where wraptopi wraps angles to the range $(-\pi,\,\pi]$.

In order to clearly see this ideal framework, let us rewrite system (1) in cylindrical444In 2D this boils down to polar coordinates. Because in this paper 2D and 3D oscillators are considered, the term cylindrical coordinates will be used. coordinates:

[TABLE]

The advantage of representing ${\cal O}_{1}$ as in (2) is twofold: first, numerical integration directly provides the phase, which in this problem is the controlled variable $c(t)$; second, it permits actuating directly on the equation that describes the phase dynamics. If PS is intended we choose $c(t)=\phi_{1}(t)$, $r_{\phi}(t)=\phi_{2}(t)$ and $m_{\rho}(t)=0$, where $\phi_{2}(t)$ is the phase from the master oscillator. Connecting ${\cal S}$ to position 2, and using dissipative coupling as $m_{\phi}(t)=k_{\phi}[r_{\phi}(t)-\phi_{1}(t)]$ this scheme is quite successful in attaining PS, as will be shown.

2.2.3 Control action and manipulated variable mismatch

For the sake of completion, we investigate a rather awkward situation: ${\cal O}_{1}$ will be represented in Cartesian coordinates but the use of proxy variables is avoided by choosing the reference $r_{\phi}(t)=\phi_{2}(t)$, and hence the control action will be $m(t)=k_{\phi}[\phi_{2}(t)-\phi_{1}(t)]$. A second difficulty is that $m(t)$ is connected to the phase error, but because there is no equation for the phase dynamics, regardless to which equation $m(t)$ is added, there will be a control action and manipulated variable (CA-MV) mismatch. The one positive aspect of this strange configuration is that no proxy variables are employed.

In a sense this control problem is a transition  between the ones described in sections 2.2.1 and 2.2.2. Here we still aim at PS and to emphasize this we still assume that $c(t)=\phi_{1}(t)$ and $r_{\phi}(t)=\phi_{2}(t)$ and we do not  resort to the helpful cylindrical coordinates (2) but rather consider the system to be in Cartesian coordinates as in (1). Since $\phi_{1}$ is not a state variable in this representation, we do not explicitly feedback that signal, instead we place the switch ${\cal S}$ in position 1 and feedback a function of the states of the system. It is assumed that $h(t)=\tan^{-1}(y_{1}/x_{1})$ is a good estimate for the phase555The arctangent function is considered with two arguments: atan2($y,\,x$), where the angle information can be obtained in the four quadrants. and hence the error $e(t)$ is the phase difference. This is almost equivalent to the case investigated in Sec. 2.2.2, if it was not for the fact that here we face a CA-MV mismatch.

2.3 Other synchronization schemes

Here we briefly mention that the proposed framework can be used to understand other synchronization schemes found in the literature. In what follows two PS are considered as illustrative examples.

As a first example, we quote the phase-locked loop (known by the acronym PLL) which first appeared early in the 20th century (Hsieh and Hung,, 1996). The main elements of a PLL are a phase detector, a low-pass filter and a voltage-controlled oscillator (Piqueira,, 2011). This scheme has two of the principal benefits argued so far, namely, given two sinusoidal oscillations, the phase error is detected and  the oscillator frequency is changed (controlled) proportionally to such an error, hence avoiding the CA-MV mistmatch.

Another interesting scheme to guarantee PS and that closely resembles a PLL has been proposed in (Belykh et al.,, 2005) where the controller output is a function of the phase difference, i.e. $m(t)=F[e(t)],~{}e(t)=\phi_{2}(t)-\phi_{1}(t)$, that is added to frequency-related parameters, which is a way of avoiding the CA-MV mismatch although it might be difficult to implement in practice.

In summary, the success of the two PS schemes mentioned in this section stems mainly from two facts related to the control action i) it is a function of the phase difference, and ii) it directly influences the instantaneous frequency of ${\cal O}_{1}$, avoiding CA-MV mismatch.

2.4 Observability requirements

The observability of linear dynamical systems was introduced by Rudolf Kalman (Kalman,, 1960) and has been extended and investigated for nonlinear system in many works, as for instance (Hermann and Krener,, 1977). This concept has proven relevant in a number of problems in nonlinear dynamics (Parlitz et al.,, 2014) where observability coefficients $\delta_{s}$ have been proposed as average condition numbers of observability matrices ${\cal O}_{s}$ for the system when only variable s(t)=h(\mbox{\boldmathx}) is recorded from the system (Letellier et al.,, 2005).

The concept of observability can be understood as follows. Assume that the dynamical system under study is \dot{\mbox{\boldmathx}}(t)=\mbox{\boldmathf}(\mbox{\boldmathx}(t)), where $t$ is the time, \mbox{\boldmathx}\in\mbox{\bbfnt R}^{n} the state vector and $f$ the nonlinear vector field. Let $s(t)$ be a scalar time series obtained using a measurement function $h:\mbox{\bbfnt R}^{n}\rightarrow\mbox{\bbfnt R}$, that is $s(t)=h(\mbox{x}(t))$. The system is said to be observable  if, given $s(t)$, it is possible to find the initial condition \mbox{\boldmathx}_{0} for finite $t$. In more practical terms, if the pair (\mbox{\boldmathf},\,s) is observable, then it is possible to determine the state \mbox{\boldmathx}(t) from the recorded time series $s(t)$.

Now it becomes clear that because the goal in CS is to attain \mbox{\boldmathx}_{1}=\mbox{\boldmathx}_{2} and if only one state variable is used, say $x$, then it makes sense to require that (\mbox{\boldmathf}_{2},\,x_{2}) be observable, because from the reference signal $r_{x}(t)=x_{2}(t)$ the controlled oscillator ${\cal O}_{1}$ should be able to “see” the full state of the master oscillator \mbox{\boldmathf}_{2} in order to follow it. No wonder there is significant correlation between observability and CS via dissipative bidirectional coupling (Letellier and Aguirre,, 2010) or by complete substitution (Pecora and Carroll,, 1990), as pointed out in (Aguirre and Letellier,, 2016).

On the other hand, if PS is aimed at and if the framework in Sec 2.2.2 is considered, there is no need to require that (\mbox{\boldmathf}_{2},\,\phi_{2}) be observable  because there is no need to “see” (reconstruct) the entire state vector from the recorded $\phi_{2}(t)$. Hence the requirements on observability in synchronization problems will have some relevance if the aim is CS or possibly lag synchronization (LS) (Rosenblum et al.,, 1997), but might not be the right feature to ask for if the goal is PS. This “decoupling” between PS and observability will be illustrated with simulations in Section 4, after proposing a solution the the CA-MV mismatch problem.

3 A Solution to the CA-MV mismatch problem

In this section a solution to the CA-MV mismatch problem pointed out in Sec. 2.2.3 is proposed. Without loss of generality it is supposed that the slave oscillator is described in 2D Cartesian coordinates with possible control actions $m_{x}$ and $m_{y}$ actuating directly on $\dot{x}_{1}$ and $\dot{y}_{1}$ equations, respectively. It is assumed that the phase of the slave oscillator can be adequately estimated by $\phi_{1}=\tan^{-1}(y_{1}/x_{1})$ and that $m_{x}$ or $m_{y}$ should be such as to reduce the phase error $e(t)=r_{\phi_{2}}(t)-\phi_{1}(t)$, where the reference signal $r_{\phi_{2}}(t)=\tan^{-1}(y_{2}/x_{2})$ is the phase of the master oscillator.

The ideal situation occurs when the CA actuates directly on the desired manipulated variable (MV) dynamics, that is $\dot{\phi}_{1}$. This is not  the case here as the slave oscillator is assumed to be in Cartesian coordinates. The rationale behind the solution to be developed is to find suitable control actions $m_{x}$ and $m_{y}$ in Cartesian coordinates that would effectively influence the phase dynamics. To achieve this, the phase dynamics of the master oscillator is written thus

[TABLE]

where the variables $(\dot{x},\,\dot{y})=(f_{x}(x_{1},\,y_{1}),\,f_{y}(x_{1},\,y_{1}))$ are the time derivatives of the uncoupled oscillator (see Eq. 1) and the term in brackets – called $\nu(\boldsymbol{x})$ – is its natural frequency. In the second line the following result was used

[TABLE]

To see the consequences of the CA-MV mismatch, two situations will be considered in turn.

• (a)

$m_{x}=k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)]$ and $m_{y}=0$. From (3) the phase dynamics can be described thus:

[TABLE]

Assuming $k_{\phi}>0$, whenever the term in parenthesis is negative there will be a positive feedback loop, reinforcing the phase error.

• (b)

$m_{y}=k_{\phi}[r_{\phi_{2}}(t)-\phi_{1}(t)]$ and $m_{x}=0$. As before, from (3) the phase dynamics are:

[TABLE]

and feedback will be positive whenever the term in parenthesis is negative.

Hence, as the trajectory travels through state space, there will be regions of positive feedback where such a simple scheme will fail to keep phase error small. A solution to this scenario, which is a consequence of the CA-MV mismatch, is to choose

[TABLE]

Hence, the phase dynamics will be of the form

[TABLE]

which always results in negative feedback. In (7) $\tilde{k}_{\phi}(t)$ includes the state variables $x_{1}(t)$, $y_{1}(t)$ and the constant gains $k_{\phi_{x}}$ and $k_{\phi_{y}}$. If $k_{\phi_{x}}=k_{\phi_{y}}=k_{\phi}$, then $\tilde{k}_{\phi}(t)=k_{\phi}$.

The only assumption that leads to (3) is that the phase can be described by arctangent function of a quotient of two state variables. The procedure can be applied to 3D oscillators as long as the phase can be computed taking the arctangent of the quotient of two of the state variables. Hence such a procedure is applicable to a large class of systems. Results for the Rössler oscillator will be presented in Sec. 4.5.

4 Numerical Results

In this section we numerically investigate the PS of oscillator ${\cal O}_{1}$ with the reference signal $r(t)$ provided by a master oscillator in order to support the analysis and claims put forward in the previous sections. The oscillators considered are: Poincaré, Van der Pol and Rössler, for which representation in Cartesian and cylindrical coordinates are readily available in the literature and also because they display phase coherent dynamics. This is important to test the proposed framework because other relevant practical problems, such as defining and estimating the phase for phase noncoherent dynamics, are avoided.

In what follows potential control actions (coupling terms) will be indicated in the system equations. In transforming the equations to cylindrical coordinates, only the autonomous part of the original equations were considered and the potential control action was added later. In all simulations the transients were removed from the analysis.

The equations for the Poincaré system are 666As for the origin of this system the following quotation is relevant “Since the first description of two-dimensional radially symmetric differential equations with limit cycles was given by Poincaré, we propose that these systems be called Poincaré oscillators” (Glass and Mackey,, 1988, pp. 25). However, It is pointed out that the denomination is used for a class of two-dimensional differential equations, and the specific oscillator in (8) is not considered in that reference. (Belykh et al.,, 2005):

[TABLE]

with $(\omega,\,p,\,\lambda)=(1,\,1,\,0.5)$ and which can be represented in cylindrical coordinates as:

[TABLE]

The van der Pol oscillator (van der Pol,, 1927) can be described as:

[TABLE]

with $(\omega,\,\mu,\,\beta)=(1,\,0.1,\,1)$, in which $\mu$ indicates the nonlinearity of the system. Defining $x_{1}=\rho_{1}\cos\phi_{1}$ and $y_{1}=\rho_{1}\sin\phi_{1}$, the oscillator can be described in cylindrical coordinates as:

[TABLE]

which is valid \forall\mbox{\boldmathx}\neq{\bf 0}, and the phase is assumed to be $\phi_{1}=\tan^{-1}(y_{1}/x_{1})$.

The Rössler system (Rössler,, 1976) in Cartesian coordinates with control inputs is described as:

[TABLE]

and $(a,\,b,\,c,\,\omega)=(0.398,\,2.0,\,4.0,\,1.0)$. For these parameters, considering the control action in the $x$ variable, $m_{y}(t)=m_{z}(t)=0$, with $m_{x}(t)=k_{x}[r_{x}(t)-x_{1}(t)]$ and $k_{x}=1.5$, $|e(t)|\to 0$ is achieved. The Rössler system can be rewritten in cylindrical coordinates with control action in the second equation as (Rosenblum et al.,, 1997):

[TABLE]

For phase coherent dynamics, $\phi_{1}$ in (13) has a clear interpretation of phase, as it can be computed thus $\phi_{1}=\tan^{-1}(y_{1}/x_{1})$.

These oscillators will be considered in the light of Sections 2.2.1 to 2.2.3. To evaluate the quality of PS the mean phase coherence  will be used. This has been defined as (Mormann et al.,, 2000):

[TABLE]

where $\langle\cdot\rangle$ is time average. $R=1$ is achieved for strict phase locking, that is, for $|\phi_{2}(t)-\phi_{1}(t)|=$const.

4.1 Cartesian coordinates

In this section we simulate the most commonly found situation: the oscillators are represented in Cartesian coordinates, coupling is dissipative, and $r_{s}(t)=s_{2}(t),~{}s\in~{}\{x,\,y,\,z\}$, which means that the control action for the first equation is $m_{x}(t)=k_{x}[x_{2}(t)-x_{1}(t)]$ and the error signal $e(t)=x_{2}(t)-x_{1}(t)$ is in the state variable  $x$, not in the phase. Analogous definitions are used for the other equations. Therefore there is no CA-MV mismatch. The control action is added to one equation at a time. Also, PS will be assumed as the goal.

For the Poincaré system, the performance of this scheme is shown in Figure 2 with 5% frequency mismatch between ${\cal O}_{1}$ and the master oscillator. For gains slightly over 0.1 high quality PS is attained and for lower values of the gain no PS was found. In particular for $k_{x}=0.02$ and $k_{y}=0$, PS is not achieved ($R=0.21\pm 0.09$); also for $k_{x}=0$ and $k_{y}=0.02$ the results are just as poor: $R=0.29\pm 0.05$.

For the van der Pol oscillator we proceed in a similar fashion, that is the control actions have the form $m_{s}(t)=k_{s}[s_{2}(t)-s_{1}(t)],~{}s~{}\in~{}\{x,\,y\}$, and are added directly to the respective equation in ${\cal O}_{1}$, with $k_{s}=0.1$. The results are summarized in Figure 3 for 10% mismatch in all parameters. Synchronization is poor when the control action is added to the second equation ($s=y$), and is only reasonable at large values of $\mu$ for $s=x$.

For the Rössler oscillator the control action used was $m_{s}(t)=k_{s}[s_{2}(t)-s_{1}(t)],~{}s\in\{x,\,y,\,z\}$, with $k_{s}=0.4$. PS was achieved quite consistently for $s=x$ and $s=y$, but more rarely for $s=z$ and only for small frequency mismatches, as summarized in Figure 4. It was quite unexpected to find PS with $z$ as the coupling variable, as it is known that for this variable no CS has been reported using dissipative coupling.

In addition to the phase coherence $R$, the synchronization regions on the $\omega-k_{s}$ plane for the Rössler oscillator were estimated using the same control actions. Specific values of $\omega$ and $k_{s}$ determine a point on that plane. The synchronization of each point on a grid of values was verified by testing the condition $|\phi_{1}-\phi_{2}|<\pi$ directly, where $\phi_{i}=\tan^{-1}(y_{i}/x_{i})$ for $i\in\{1,2\}$. The results for the best case  is shown in Figure 5 for a grid of pairs obtained varying $k_{s}\in[0,\,0.5]$ and $\omega\in[0.85,\,1.15]$ for ${\cal O}_{1}$ and fixed $\omega=1$ for the master oscillator. The percentage of pairs $(\omega,\,k_{s})$ for which synchronization was detected with respect of all the pairs in the grid was computed and called the $\alpha_{s}$ index $s~{}\in~{}\{x,\,y,\,z\}$. For the Rössler system $\alpha_{z}\ll\alpha_{y}\approx\alpha_{x}$ (Table 1).

4.2 Cylindrical coordinates

Because PS is intended here we choose $c(t)=\phi_{1}(t)$, $r_{\phi}(t)=\phi_{2}(t)$, where $\phi_{2}(t)$ is the phase from another similar (master) oscillator. Connecting ${\cal S}$ to position 2 (Figure 1), $e(t)=\phi_{2}(t)-\phi_{1}(t)$ and using dissipative coupling the control action is $m_{\phi}(t)=k_{\phi}[\phi_{2}(t)-\phi_{1}(t)]$, which is a function of phase  error.

For the Poincaré system (9), as before, the parameters for ${\cal O}_{1}$ are $(\omega,\,p,\,\lambda)=(1.05,\,1,\,0.5)$ and for the master oscillator the parameters are $(1,\,1,\,0.5)$. In this case high quality PS ($R=1$) is attained for a controller gain as low as $k_{\phi}=0.02$. This clearly outperforms the scheme of Section 4.1.

For the van der Pol system (11), with $k_{\phi}=0.5$ PS is attained with very high quality for $\mu<1$ and even for larger values of this parameter, although with slightly decaying quality as $\mu$ is increased.

For the Rössler system (13), $\phi_{2}(t)$ was generated using the same equations with $(a,\,b,\,c,\,\omega)=(0.398,\,2.0,\,4.0,\,1)$ and the parameters for ${\cal O}_{1}$ are $(a,\,b,\,c,\,\omega)=(0.398,\,2.0,\,4.0,\,\omega)$ – notice the frequency mismatch – and even with a low gain ($k_{\phi}=0.05$), $|e(t)|=|\phi_{2}(t)-\phi_{1}(t)|<\pi$ is achieved. For this case $R=0.92\pm 0.018$ was obtained over 10 Monte Carlo runs, indicating good PS. That value of gain is close to the threshold under which no PS is attained and is much lower than for the case in which Cartesian coordinates are used with error in a state variable. Taking $k_{\phi}=0.25$ and varying the frequency $\omega$, PS is attained as shown in Figure 6. Comparing this figure to Figure 4 it is seen that although variance is much higher (probably due to longer transients) there is no clear deterioration with the increase of $\omega$ within the investigated range. The maximum $R$ attained (red dash-dot line) is always very close to 0.985.

The synchronization region was also calculated for the Rössler system described in cylindrical coordinates, with $m_{\phi}(t)=k_{\phi}[\phi_{2}(t)-\phi_{1}(t)]$ for which $\alpha_{\phi}=98\%$ (Table 1).

4.3 Control action and manipulated variable mismatch

In this section $r_{\phi}(t)=\phi_{2}(t)$ as in Sec. 4.2. The important difference here is that the main oscillator, ${\cal O}_{1}$, is simulated in Cartesian coordinates and this hampers our ability to add the control action $m_{\phi}(t)$ directly to the equation that determines the phase dynamics, thus resulting in CA-MV mismatch.

In what follows the control action $m_{\phi}(t)=k_{\phi}[\phi_{2}(t)-\phi_{1}(t)]$ is added to each dynamical equation in Cartesian coordinates, one at a time. For the three investigated systems, the phase was computed as $\phi_{1}=\tan^{-1}(y_{1}/x_{1})$ for ${\cal O}_{1}$, and analogously for $\phi_{2}$.

The Poincaré oscillator can be phase synchronized by adding the control action $m_{\phi}(t)$ directly either to the first or to the second equation with basically the same performance, as seen in Figure 7. From a comparison of these results with those in Figure 2, the deleterious effect of the CA-MV mismatch becomes evident.

For the van der Pol system with $m_{\phi}(t)=0.1[\phi_{2}(t)-\phi_{1}(t)]$, the performance is very poor in general and PS is only attained for a limited range of values of $\mu$ and only when $m_{\phi}(t)$ is added to the first equation. For this system CA-MV mismatch is strongly deleterious.

For the Rössler system with $r_{\phi}(t)=\phi_{2}(t)$ rather surprisingly, PS is achieved adding the control action $m_{\phi}(t)=0.4[\phi_{2}(t)-\phi_{1}(t)]$ to either equation (Figure 8). For $1.0\leq\omega\leq 1.02$ the best  – out of the 10 Monte Carlo runs – performance of phase coupling  is similar regardless to which of the three equations $m_{\phi}(t)$ is added, although adding the control action to the second equation is especially good at a higher values of frequency mismatch. Due to the CA-MV mismatch, the general performance is poorer than for the case shown in Figure 2.

4.4 Observability coefficients

In order to verify if observability plays any role at all in PS, here we compute the observability coefficients for the investigated oscillators according to (Letellier et al.,, 2005). Here we report results for all the systems in both coordinate systems. The only situation previously considered in the literature is that for the Rössler system in Cartesian coordinates, and for that system we only quote published results.

Observability matrices for the Poincaré oscillator are

[TABLE]

where it is clearly seen that both matrices are singular. This confirms expectation that this system is nonobservable if we only measure $\rho$ ($\delta_{\rho}=0$) or if we only measure $\phi$ ($\delta_{\phi}=0$), since $\rho$ cannot be found from $\phi$ and vice versa. A very different situation happens in Cartesian coordinates, where the observability matrices for the Poincaré oscillator are

[TABLE]

which yield the coefficients shown in Table 1. The observability matrices for the van der Pol oscillator in cylindrical coordinates are

[TABLE]

where $\rho_{22}=2\rho\sin\phi\cos\phi[\beta\mu\rho(\sin^{2}\phi-\cos^{2}\phi)+1]$, $\phi_{22}=(\beta\mu\rho^{2}\cos^{2}\phi-1)(\sin^{2}\phi-\cos^{2}\phi)+2(\beta\mu\rho^{2}\cos^{2}\phi)\sin^{2}\phi$. The observability coefficients (see Table 1) show a clear decrease in observability from the phase for the more nonlinear situation $(\mu=10)$. For Cartesian coordinates the matrices are

[TABLE]

It is noteworthy that {\cal O}_{x}(\mbox{\boldmathx}) for the van der Pol oscillator is constant and nonsingular. This means that there is a global diffeomorphism connecting the original space $(x,\,y)$ and $(x,\,\dot{x})$. From an observability point of view, this is both very good and quite rare. For the more nonlinear case $\delta_{y}$ is greatly reduced.

For the Rössler oscillator the observability coefficients are provided in Table 1. The observability matrices for the cylindrical coordinates are cumbersome and therefore are not shown. The observability for Cartesian coordinates can be found in the literature. Table 1 also shows the percentage of synchronization conditions for the Poincaré, van der Pol and Rössler oscillators, calculated for 61 equally-spaced values of the frequency in the range $\omega\in[0.85,\,1.15]$ for the slave oscillator ${\cal O}_{1}$ and with $\omega=1$ fixed for the master oscillator. The gain $k_{s}$ was varied over 51 equally-spaced values in the ranges provided in the caption of Table 1. Hence for each case 3111 different situations were tested for phase synchronization.

In using observability coefficients it must be remembered that coefficients from one system should not be compared to those of other systems, as they do not have a clear absolute interpretation. However they are useful in ranking variables within a system and also to gain some insight. So, the observability of the dynamics from the phase is typically very low ($\delta_{\phi}\ll 1$) for all systems with the exception of the van der Pol system with low nonlinearity. This concurs with intuition and says that it is virtually impossible to figure out the full state of the system if only the phase is measured. Also, typically, PS is achieved with greater ease and quality (e.g. $\alpha_{\phi}\gg\alpha_{x},\,\alpha_{y},\,\alpha_{z},\,\alpha_{\rho}$) when $m_{\phi}(t)$ is used, especially when there is no CA-MV mismatch (i.e. using $m_{\phi}$ in cylindrical coordinates – see corresponding values of $\alpha_{\phi}$ in Table 1) or when the CA-MV mismatch is handled with the procedure described in Sec. 3 (i.e. using $m_{x}$ and $m_{y}$ as defined in (3) – see Table 2). So it seems fair to conclude based on these numerical results that observability does not  play a significant role in PS, rather the choice of controlled variable, control action and manipulated variable are far more fundamental.

Be it as it may, we still need to address the following question: is the relation between synchronization and observability, at least for some systems as for instance Rössler, a coincidence? It is believed that the answer has two aspects. First, as discussed before, for complete synchronization it seems reasonable that observability should play some role, so there is no coincidence there. Second, even in the case of PS in some cases, variables that provide good observability of the system dynamics also are good signals for coupling. We conjecture that in such cases the good coupling is a consequence of the phase information conveyed by the variable and is not directly related with observability proper. In other words, such variables are probably good proxy variables for the phase and, if so, need not provide good observability of the dynamics.

4.5 Solving the CA-MV mismatch problem

Figure 9 shows the result of using the control action developed in Sec. 3 (see Eq. 3). Notice that no control action is added to the third equation. Comparing to previous results described for the Rössler system, it is clear that the procedure proposed in Sec. 3 was effective in solving the CA-MV mismatch problem.

To further test the procedure developed in Sec. 3, synchronization regions for the three oscillators investigated were obtained numerically. The results are summarized in Table 2.

An interesting aspect revealed by such results is that $\alpha_{x}$ and $\alpha_{y}$ for the Rössler are smaller than for the other oscillators. Because $m_{x}$ and $m_{y}$ defined in Sec. 3 are multiplied by the state variables $-y$ and $x$, respectively, the control action has the effect of shifting the parameter $\omega$ of the corresponding equation. In the case of the Rössler system in certain regions (see Figure 10) the slave oscillator enters the funnel regime. In such cases the lack of PS is due to the fact that a non-phase-coherent oscillator tries to follow the phase-coherent master. Also, the assumption made in Sec. 3 that the phase should be computable using the arctangent function is no longer valid.

5 Discussion and Conclusions

This paper has presented a feedback control framework that is useful in analyzing master-slave synchronization schemes. In particular, the proposed framework was used to investigate PS. It has been argued that in the case of PS  the key factor for successful synchronization is not the quality of observability provided by a state variable, but rather depends if the that variable is a good proxy for the phase in the case of CA-MV mismatch. When such a mismatch is overcome, for instance following the procedure proposed in Sec. 3, PS is quite simply obtained, at least for the investigated systems. This view seems to underlie the success of phase-locked-loop schemes where the phase detector plays a key role: receiving state variables and computing a phase  error rather than an error in the said variables.

For all investigated oscillators, feeding back phase had very good PS results, measured by the coherence index and synchronization regions, but the observability of the oscillator dynamics seen from the phase are very low in every case (see Table 1). This provides consistent counter-examples where variables with lowest observability coefficients are best coupling variables.

An intriguing matter investigated in the paper is this: how can it be explained that for some systems control actions based on the error of state variables instead of phase error works well? In terms of diffusive coupling this question can be rephrased as: why, say, $m_{x}(t)=k_{x}[x_{2}(t)-x_{1}(t)]$ is sometimes very effective in attaining PS, when from a control point of view the best would be to use $m_{\phi}(t)=k_{\phi}[\phi_{2}(t)-\phi_{1}(t)]$? In this paper it has been conjectured that what matters is how much of “phase information” is there in $x$. In other words, if $x$ is a good proxy variable for $\phi$ then $m_{x}(t)$ will be a proxy for $m_{\phi}(t)$ and should also be effective. There is a clear need for a way to judge the quality of proxy variables for the phase.

Another point that has been raised based on the proposed framework has been named the control action and manipulated variable (CA-MV) mismatch. This situation happens whenever the control action affects a manipulated variable that does not directly determine the control error. For example, defining a phase error and computing the control action based on this is an ideal situation from a control point of view if the aim is PS. Once the control action is available it should affect a manipulated variable directly related to the phase dynamics, or frequency. However if it is added to some equation in Cartesian coordinates, we face CA-MV mismatch, which is a nonideal situation. Still, it has been shown that quality PS was attained in such a situation for the Rössler system for the $x$ and $y$ variables. We conjecture that part of this success is due to the fact that $x$ and $y$ are good proxy variables for the phase. However this does not explain all the details found and other aspects of the problem should be further investigated.

To illustrate this last point, consider Figure 8 where the same  control action $m_{\phi}(t)$ was added to each equation of the Rössler system. Hence the coupling variable (phase) is probably not responsible for the differences in performance, which are probably a consequence of the dynamics associated to each $x$, $y$ and $z$ variables. And in this respect we see that $y$ is clearly very good with $x$ and $z$ being poor. These variables perhaps suffer more intensely with the CA-MV mismatch.

To avoid  the CA-MV mismatch we represent the Rössler system in Cartesian coordinates and use $m_{s}(t)=k_{s}[s_{2}(t)-s_{1}(t)],~{}s\in\{x,\,y,\,z\}$ (Fig. 4). First, we must remember that now there is no CA-MV mismatch, so there is a good chance that the performance using $x$ and $z$ improve somewhat because these variables are sensitive to the CA-MV mismatch. The performance in terms of $z$ has not changed significantly so any possible improvement due to the lack of CA-MV mismatch has been negatively counter-balanced by the fact that $z$ is a poor proxy for the phase. The performance using $y$ deteriorated slightly and that of $x$ improved greatly. We then conjecture that $y$ is a good proxy for the phase otherwise performance would have deteriorated for the lack of phase information. As for the $x$ variable, it must be a very good proxy for the phase, superior to $y$. As mentioned before, this ranking of $x,\,y$ and $z$ in terms of being proxy variables for the phase coincides with the results in Table 2.

To solve  the CA-MV mismatch in the case of the Rössler system, the procedure in Sec. 3 was seen to be effective. Here the phase-based control action is “mapped” to Cartesian coordinates. Phase synchronization performance is then much improved. This procedure is believed to be applicable to systems for which the phase can be estimated as the arctangent of the quotient of two state variables.

In the present investigation only systems for which the phase is well defined were considered. This choice was made in order to be able to focus on other aspects of phase synchronization. However, it is important to be able to extend the present results to systems for which the phase is not readily defined. A more general definition of phase and a quantifier of phase information in a variable are needed. These definitions are currently under investigation.

Acknowledgements

LAA gratefully acknowledges financial support from CNPq. LF is grateful to IFMG Campus Betim for an academic leave. The authors thank Christophe Letellier for commenting on an early draft of this paper.