1 Introduction

Synchronization plays a critical role in various fields such as biology [1,2,3], chemistry [4,5,6], neuroscience [7,8,9], and social networks [10,11,12], where it is responsible for enabling coherent and coordinated behavior among complex systems [13]. The process of synchronization occurs when two or more oscillatory systems interact and adjust their rhythms, so they become more similar over time, eventually becoming identical. In general, there is no unified method or theory concerning the measurement of synchronization between two oscillations. Depending on the context, there are different ways of measuring synchronization including cross-correlation analysis [14, 15], coherence [16, 17], calculating the phase locking value (PLV) [18, 19], mutual information (transfer entropy) [20, 21], and Granger causality [22, 23], with each method having its own advantages and limitations depending on the research question and the type of oscillations being studied.

A Poincaré sphere is a geometric representation of the polarization states of a classical electromagnetic wave, such as light [24,25,26]. The sphere provides a useful way to visualize and understand the behavior of polarized light as it interacts with different optical elements [27]. In the context of quantum mechanics, the Poincaré sphere is referred to as the Bloch sphere and is used for the visualization of the pure quantum state of quantum mechanical systems [28]. When it comes to synchronization, the Poincaré sphere can be used to visualize the similarities between the oscillations of two systems. To the best of our knowledge, such a visualization has only been done a few number of times, for example, for the analysis of weakly coupled double pendulum [29,30,31], for depicting the trajectories of chaotic systems [32], and for analyzing the synchronization of nanomechanical arrays [33]. However, there is no instructive methodology for using the Poincaré sphere as a general tool for measuring synchronization.

In this work, our goal is to demonstrate the usefulness of the Poincaré sphere as a generalized measure of synchronization. Besides its common application to polarization states, we show its superiority in measuring the similarities between oscillations. Here, we introduce the notion of total synchronization, which considers amplitude, frequency, and phase synchronization. Using the Poincaré sphere, we show that all these different types of synchronization can be visualized and measured at once. Furthermore, we discuss the advantages of visualizing oscillatory trajectories on the Poincaré sphere as opposed to a more typical two-dimensional representation. In particular, we show that projecting different onto oscillations onto two-dimensional representations can lead to completely different trajectories, even though projecting them onto Poincaré sphere leads to the same one. This implies that the similarities between different oscillations are sometimes only visible on the Poincaré sphere. Finally, we discuss how this new synchronization measurement can be applied to dynamical systems on the example of linear and nonlinear oscillators. Contrary to the simpler two-dimensional projections, we show that the Poincaré sphere allows for a more general parametrization, which is ideal for dealing with nonlinear oscillators. Furthermore, we also provide two concrete application examples, namely visualizing phase jitter in noisy oscillations [34, 35] and quantizing solutions of oscillator-based Ising machines [36,37,38]. The latter are tightly related to the Kuramoto model [39].

The remainder of the paper is organized as follows. Section II reviews measures of polarization and introduces the polarization ellipse as well as the Poincaré sphere. Section III discusses the application of the polarization ellipse and the Poincaré sphere as general measures of synchronization. This concept is then applied to the synchronization analysis of linear and nonlinear oscillators in Sec. IV. Section V discusses the usefulness of the Poincaré sphere on two different application examples. Finally, Section VI concludes this work and provides a brief outlook on future research in this context.

2 Measures of polarization

2.1 Polarization

The propagation of electromagnetic waves through a charge-free medium with permeability \(\mu \), and permittivity \(\varepsilon \) is described by the wave equations

$$\begin{aligned} \triangle {\varvec{E}}&= \frac{1}{c^2} \frac{\partial ^2 {\varvec{E}}}{\partial t^2}\,, \quad \triangle {\varvec{B}} = \frac{1}{c^2} \frac{\partial ^2 {\varvec{B}}}{\partial t^2}\,, \end{aligned}$$
(1)

where \(\triangle \) denotes the Laplace operator, while \(c=1/\sqrt{\mu \varepsilon }\) denotes the propagation velocity of the electrical and magnetic field, \({\varvec{E}}\) and \({\varvec{B}}\), respectively. It suffices to limit our considerations to the steady-state solution of the electrical field

$$\begin{aligned} {\varvec{E}}(t, {\varvec{r}})&= \Re \left\{ {\varvec{E}}_+ {\text {e}}^{{\textrm{j}} [\varOmega t - {{\varvec{k}}}^{\textrm{T}} {\varvec{r}}]} \right\} \,, \end{aligned}$$
(2)

where \(\Re \left\{ \cdot \right\} \) denotes the real part operator, \(\varOmega \) is a radian frequency, \({\varvec{k}}=\left[ \begin{array}{c} k_x, k_y, k_z \end{array}\right] \) is the real wave number vector, and \({\varvec{r}}=\left[ \begin{array}{c} x, y, z \end{array}\right] \) is the position vector. This is because \({\varvec{B}}\) can be determined from \({\varvec{E}}\) by a 90\(^{\circ }\) rotation and multiplication with a constant [40]. Since the divergence \({\varvec{\nabla }}^{\textrm{T}}\) is a real operator, we find that the vector \({\varvec{E}}_+\) is orthogonal to the wave number vector:

$$\begin{aligned} {\varvec{\nabla }}^{\textrm{T}} {\varvec{E}} = 0 \quad \Leftrightarrow \quad \quad {\varvec{\nabla }}^{\textrm{T}} {\varvec{E}}_+ = 0 \quad \Leftrightarrow \quad \quad {{\varvec{k}}}^{\textrm{T}} {\varvec{E}}_+ = 0 \,. \end{aligned}$$

For the sake of simplicity, let us choose our coordinate system such that the wave propagates in z direction. As a consequence, the wave number vector is pointing to this direction, \({\varvec{k}}=k{\varvec{e}}_z\), and the electric field \({\varvec{E}}(t,z)\) lies for any given z solely in the (x, y)-plane, such that we only need to focus on these coordinates.

For our upcoming investigations it suffices to look at spatially-independent signals. In retrospect to our short discussion on electromagnetic waves, this means that we only consider one point in space \(z_{0}\), such that \({\varvec{E}}\) and \({\varvec{B}}\) can generally be both described by the vector signal

$$\begin{aligned} {\varvec{u}}(t) = \Re \left\{ {\varvec{a}} {\text {e}}^{{\textrm{j}}\varOmega t} \right\} \,, \quad {\varvec{a}} = \left[ \begin{array}{c} a_x {\text {e}}^{{\textrm{j}}\beta _x}\\ a_y {\text {e}}^{{\textrm{j}}\beta _y} \end{array}\right] \,, \end{aligned}$$
(3)

where \({\varvec{a}}\) designates the Jones vector having real phases \(\beta _x\) and \(\beta _y\) [41]. To clarify, a possible representation of (2) in terms of (3) reads:

$$\begin{aligned} {\varvec{u}}(t) = {\varvec{E}}(t,z_{0}) \quad \text {with} \quad {\varvec{a}} = {\varvec{E}}_{+} e^{-{\textrm{j}} k_{z}z_{0}} \,. \end{aligned}$$

2.2 Polarization ellipse

Starting from the general vector signal defined in (3), we assume that the amplitudes \(a_x\) and \(a_y\) are nonnegative and at least one of the amplitudes is positive. Since it is not possible to measure a total phase, we can use \(\beta _x\) as a reference and introduce the phase difference

$$\begin{aligned} \xi = \beta _x - \beta _y\,. \end{aligned}$$
(4)

By evaluating the real part of \({\varvec{u}}(t)\), we get the parametric representation of the polarization ellipse

$$\begin{aligned} {\varvec{u}}(t) = \left[ \begin{array}{c} u_x(t)\\ u_y(t) \end{array}\right] = \left[ \begin{array}{l} a_x \cos (\varOmega t + \beta _x)\\ a_y \cos (\varOmega t + \beta _x - \xi ) \end{array}\right] \,. \end{aligned}$$
(5)

For \(a_y=0\) the y-component of \({\varvec{u}}(t)\) is vanishing and the polarization is referred to horizontal polarization, whereas for \(a_x=0\) the polarization is vertical. Having positive amplitudes but a phase difference \(\xi \) of integer multiple of \(\pi \) leads to a line through the origin. This indicates a linear polarization, which becomes diagonal or anti-diagonal for equal amplitudes. In order to realize that in the remaining cases the vector signal \({\varvec{u}}(t)\) indeed constitutes an ellipse, \({\varvec{u}}(t)\) will be reformulated. Initially, we normalize this vector to its positive amplitudes and exploit some trigonometric addition theorems:

$$\begin{aligned} {\varvec{v}} = \left[ \begin{array}{cc} 1 &{} 0\\ \cos (\xi ) &{} \sin (\xi ) \end{array}\right] \left[ \begin{array}{c} \cos (\varOmega t + \beta _x)\\ \sin (\varOmega t + \beta _x) \end{array}\right] \,. \end{aligned}$$

Evidently, the vector on the right-hand side is a unit vector, rotating in dependence of time. Thus, the squared norm of the vector \({\varvec{v}}\) leads to the equation

$$\begin{aligned} {{\varvec{v}}}^{\textrm{T}} \left[ \begin{array}{cc} 1 &{} -\cos (\xi )\\ -\cos (\xi ) &{} 1 \end{array}\right] {\varvec{v}} = \sin ^2(\xi )\,, \end{aligned}$$
(6)

which is a quadratic form of a \(\pi /4\) rotated ellipse. A back substitution to \({\varvec{u}}\) results in the quadratic form of the polarization ellipse

$$\begin{aligned} {{\varvec{u}}}^{\textrm{T}} \left[ \begin{array}{cc} \frac{1}{a_x^2} &{} -\frac{\cos (\xi )}{a_xa_y}\\[1ex] -\frac{\cos (\xi )}{a_xa_y} &{} \frac{1}{a_y^2} \end{array}\right] {\varvec{u}} = \sin ^2(\xi )\,. \end{aligned}$$
(7)

Another way to characterize a polarization is to map the parameters of this ellipse onto a sphere as it is presented in the next subsection.

2.3 Poincaré sphere

Starting from the phase difference and the axes lengths of the polarization ellipse the Stokes parameters [40] are defined by the equations

$$\begin{aligned} s_0&= a_x^2 + a_y^2\,, \end{aligned}$$
(8a)
$$\begin{aligned} s_1&= a_x^2 - a_y^2\,,\end{aligned}$$
(8b)
$$\begin{aligned} s_2&= +2 a_x a_y \cos (\xi ) \,,\end{aligned}$$
(8c)
$$\begin{aligned} s_3&= -2 a_x a_y \sin (\xi )\,, \end{aligned}$$
(8d)

from which in turn the polarization ellipse parameters can easily be determined:

$$\begin{aligned} a_x^2 = \frac{s_0+s_1}{2}\,, \quad a_y^2 = \frac{s_0-s_1}{2}\,, \quad \tan (\xi ) = -\frac{s_3}{s_2}\,. \end{aligned}$$
(9)

Due to the fact that the Stokes parameters are not independent,

$$\begin{aligned} s_0^2 = s_1^2 + s_2^2 + s_3^2\,, \end{aligned}$$
(10)

we normalize these parameters to \(s_0\) and combine them in the vector

$$\begin{aligned} {\varvec{p}} = \left[ \begin{array}{c} p_1\\ p_2\\ p_3 \end{array}\right] = \frac{1}{s_0} \left[ \begin{array}{c} s_1\\ s_2\\ s_3 \end{array}\right] = \left[ \begin{array}{c} \sin (\vartheta ) \cos (\varphi )\\ \sin (\vartheta ) \sin (\varphi )\\ \cos (\vartheta ) \end{array}\right] \,. \end{aligned}$$
(11)

The latter equality indicates that this vector lies on a unit sphere and has the spherical angles \(\vartheta \) and \(\varphi \), see Fig. 2 (right). This sphere is called Poincaré sphere and in accordance to this we designate \({\varvec{p}}\) as the Poincaré vector. In terms of Stokes parameters the quadratic form of the polarization ellipse (7) reads

$$\begin{aligned} {{\varvec{u}}}^{\textrm{T}} \left[ \begin{array}{cc} s_0-s_1 &{} -s_2\\ -s_2 &{} s_0+s_1 \end{array}\right] {\varvec{u}} = \frac{s_3^2}{2}\,. \end{aligned}$$
(12)

An eigenvalue decomposition shows, that the polarization ellipse is rotated by the angle \(\phi \) and has the major axes \(b_x\) and \(b_y\):

$$\begin{aligned} \begin{aligned} b_x&= \sqrt{s_0} \cos (\theta )\,, \quad 2\theta = \frac{\pi }{2} - \vartheta \,,\\ b_y&= \sqrt{s_0} \sin (\theta )\,, \quad 2\phi = \varphi \,. \end{aligned} \end{aligned}$$
(13)

This ellipse in correspondence to the afore-mentioned example of a Stokes vector on a Poincaré sphere is depicted in Fig. 1.

Fig. 1
figure 1

The polarization ellipse according to the Poincaré vector of Fig. 2 (right), with \(\phi =\pi /3\), \(\theta =\pi /6\)

Full size image

For \(\theta =0\) the polarization is linear, for \(\theta =\pi /4\) it is circular, whereas a linear and circular polarized wave is unpolarized. Some special cases of polarization ellipses and its corresponding polarization vectors on the Poincaré sphere are illustrated in Fig. 2, namely horizontal (H), vertical (V), diagonal (D), anti-diagonal (A), right-circular (R), and left-circular (L) polarization. As can be taken from this figure Stokes vectors in the equatorial plane belong to linear polarization whereas Stokes vectors collinear to the north–south axis belong to circular polarization.

Fig. 2
figure 2

Special cases of polarization ellipses and its corresponding polarization vectors on the Poincaré sphere

Full size image

3 Synchronization measurement

3.1 Synchronization ellipse

As a starting point to transfer polarization measures to measurements of synchronization let us consider two oscillations labeled with x and y:

$$\begin{aligned} u_x(t)&= {\text {e}}^{\sigma _x t+\alpha _x} \cos (\varOmega _x t+\beta _x)\,, \end{aligned}$$
(14a)
$$\begin{aligned} u_y(t)&= {\text {e}}^{\sigma _y t+\alpha _y} \cos (\varOmega _y t+\beta _y)\,. \end{aligned}$$
(14b)

In order to measure how synchronized these signals are it seems to be natural to use the relative parameters

$$\begin{aligned} \begin{aligned} \sigma&= \sigma _x-\sigma _y\,, \quad \varOmega = \varOmega _x-\varOmega _y\,,\\ \alpha&= \alpha _x - \alpha _y\,, \quad \beta = \beta _x - \beta _y \end{aligned} \end{aligned}$$
(15)

as a measurement. Since we cannot measure an absolute phase, we will tacitly use \(\beta _x\) as a reference. Moreover, the phase difference \(\beta \) has to be understood modulus \(2\pi \), that is, we restrict our considerations to the interval \(-\pi <\beta \le \pi \). With regard to the relative parameters the oscillations are obviously totally synchronized if all relative parameters vanish. The benefit of the relative parameters is a concretization of the notion of synchronicity by introducing the term \((\sigma , \varOmega , \alpha , \beta )\)-synchronicity of two oscillations. For instance, two oscillations having the same radian frequency (\(\varOmega =0\)) or the same envelope (\(\sigma =0\)) are said to be frequency or envelope synchronized, respectively.

For a visualization of synchronization as in case of polarization we write the signals \(u_x(t)\) and \(u_y(t)\) as the elements of the vector

$$\begin{aligned} {\varvec{u}}(t) = \left[ \begin{array}{l} a_x(t) \cos (\varOmega _x t + \beta _x)\\ a_y(t) \cos (\varOmega _x t + \beta _x - \xi (t)) \end{array}\right] \,. \end{aligned}$$
(16)

Since this expression is similar to the formulation of the polarization ellipse in equation (5), it will be called synchronization ellipse here having positive amplitudes and phase difference but they all vary with time:

$$\begin{aligned} a_x(t)&= {\text {e}}^{\sigma _x t+\alpha _x}\,, \end{aligned}$$
(17a)
$$\begin{aligned} a_y(t)&= {\text {e}}^{\sigma _y t+\alpha _y}\,,\end{aligned}$$
(17b)
$$\begin{aligned} \xi (t)&= \varOmega t + \beta \,. \end{aligned}$$
(17c)

This definition of a synchronization ellipse is admittedly a formal extension of the existing one. Although we can derive a parametric representation of this ellipse

$$\begin{aligned} \frac{u_y(t)}{a_y(t)} = \cos \left( \frac{\varOmega _y}{\varOmega _x} \left[ \arccos \left( \frac{u_x(t)}{a_x(t)} \right) - \beta _x \right] + \beta _y \right) \,, \end{aligned}$$
(18)

one hardly recovers the contour because of the time-variance.

There are some special but important cases, where this contour can be recovered. For example, if the oscillations are frequency synchronized (\(\varOmega _y=\varOmega _x\)), the ellipse has the representations

$$\begin{aligned} {\varvec{u}}(t)&= a_y(t) \left[ \begin{array}{l} {\text {e}}^{\sigma t+\alpha } \cos (\varOmega _x t + \beta _x)\\ \cos (\varOmega _x t + \beta _x - \beta ) \end{array}\right] \,, \end{aligned}$$
(19a)
$$\begin{aligned} \frac{u_y(t)}{a_y(t)}&= \cos \left( \arccos \left( \frac{u_x(t)}{a_x(t)} \right) - \beta \right) \,. \end{aligned}$$
(19b)

The latter equation can be simplified by using a trigonometric addition theorem:

$$\begin{aligned} \frac{u_y(t)}{a_y(t)} = \cos (\beta ) \frac{u_x(t)}{a_x(t)} + \sin (\beta ) \sqrt{1-\left[ \frac{u_x(t)}{a_x(t)} \right] ^2}\,. \end{aligned}$$
(19c)

On the other hand, if we have envelope synchronization (\(\sigma =0\)), the formulation of the synchronization ellipse cannot substantially be simplified. But there is an instructive case if the ratio of the oscillation frequencies is a multiple integer, e.g., \(\varOmega _y=n\varOmega _x\), \(n\in \mathbb {N}\). Without loss of generality, we set the reference phase \(\beta _x\) to zero to reduce the formulation effort. This yields the parametric representation

$$\begin{aligned} \frac{u_y(t)}{a_y(t)} = \cos \left( n \arccos \left( \frac{u_x(t)}{a_x(t)} \right) - \beta \right) \,. \end{aligned}$$
(20a)

Applying the same trigonometric addition theorem as before, this expression becomes

$$\begin{aligned} \frac{u_y(t)}{a_y(t)} = \cos (\beta ) \textrm{T}_n\left( \frac{u_x(t)}{a_x(t)} \right) + \sin (\beta ) \textrm{L}_n\left( \frac{u_x(t)}{a_x(t)} \right) \,, \end{aligned}$$
(20b)

where \(\textrm{T}_n\) denotes an n-th order Chebyshev polynomials of first kind and

$$\begin{aligned} \textrm{L}_n\left( \frac{u_x(t)}{a_x(t)} \right) = \sin \left( n \arccos \left( \frac{u_x(t)}{a_x(t)} \right) \right) \end{aligned}$$
(21)

represents Lissajous figures. Although there is a coincidence of (20b) with the parametric representation of (19c), one might not overlook that the ratio \(a_x(t)/a_y(t)={\text {e}}^\alpha \) is constant here.

Fig. 3
figure 3

Time-variant synchronization ellipses of the equation (22), where the labels of the columns correspondents to the labeling of the subequation-environment. The first four columns are dedicated to frequency synchronization (\(\varOmega =0\)), where the ellipses start at the initial instant \(t=0\) at one of the synchronization points (\(\alpha =0\)) and tend asymptotically toward H or V if \(\sigma \) is positive or negative, respectively. The last column shows envelope synchronization according to (22e), where the synchronization ellipse is a typical Chebyshev polynomial (above) or a Lissajous figure (below)

Full size image

Next, some of the synchronization ellipses are visualized, where it is advised to set \(\alpha =0\). For the first four cases we examine two oscillations being frequency but not envelope synchronized. With this choice of \(\alpha \) we start either at a diagonal (\(\beta \in \{0,\pi \}\)) or circular (\(\beta =\pm \pi /2\)) basic synchronization type and arrive in dependence to the sign of \(\sigma \) asymptotically at the horizontal or vertical synchronization:

$$\begin{aligned}&\bullet \ \beta =0\hspace{2.4ex}: \quad {\varvec{u}}(t) = a_y(t)\, \left[ \begin{array}{c} {\text {e}}^{\sigma t} \\ 1 \end{array}\right] \cos (\varOmega _x t + \beta _x)\,, \end{aligned}$$
(22a)
$$\begin{aligned}&\bullet \ \beta =\pi \hspace{2.2ex}: \quad {\varvec{u}}(t) = a_y(t)\, \left[ \begin{array}{c} {\text {e}}^{\sigma t} \\ -1 \end{array}\right] \cos (\varOmega _x t + \beta _x)\,, \end{aligned}$$
(22b)
$$\begin{aligned}&\bullet \ \beta =-\frac{\pi }{2}: \quad {\varvec{u}}(t) = a_y(t)\, \left[ \begin{array}{r} {\text {e}}^{\sigma t} \cos (\varOmega _x t + \beta _x)\\ -\sin (\varOmega _x t + \beta _x) \end{array}\right] \,, \end{aligned}$$
(22c)
$$\begin{aligned}&\bullet \ \beta =+\frac{\pi }{2}: \quad {\varvec{u}}(t) = a_y(t)\, \left[ \begin{array}{r} {\text {e}}^{\sigma t} \cos (\varOmega _x t + \beta _x)\\ \sin (\varOmega _x t + \beta _x) \end{array}\right] \,. \end{aligned}$$
(22d)

Vice versa, in a final step we focus on two oscillations being envelope but not frequency synchronized. Now, if \(\beta \) is zero or \(\pi \), we get Chebyshev polynomials of the first kind and for \(\beta =\pm \pi /2\) Lissajous figures:

$$\begin{aligned} \begin{aligned}&\bullet \ \beta \in \{0,\pi \} : \quad \frac{u_y(t)}{a_y(t)} = \pm \textrm{T}_n\left( \frac{u_x(t)}{a_x(t)} \right) \,,\\&\bullet \ \beta =\pm \frac{\pi }{2}\hspace{1.8ex}: \quad \frac{u_y(t)}{a_y(t)} = \pm \textrm{L}_n\left( \frac{u_x(t)}{a_x(t)} \right) \,. \end{aligned} \end{aligned}$$
(22e)

The particular synchronization ellipses of these equations are depicted in Fig. 3. An evaluation of synchronization solely based on such ellipses is questionable because they rely on absolute oscillation parameters which causes ambiguous results as it will be shown in the next section.

3.2 Synchronization sphere

Pursuant to time-variant synchronization ellipse parameters the Stokes parameters are also time-variant but different from the ellipse parameters the (normalized) Stokes parameters depend on relative quantities only:

$$\begin{aligned} {\varvec{p}}(t) = \frac{1}{\cosh (\sigma t+\alpha )} \left[ \begin{array}{r} \sinh (\sigma t+\alpha )\\ \cos (\varOmega t+\beta )\\ -\sin (\varOmega t+\beta ) \end{array}\right] \,. \end{aligned}$$
(23)

For this situation the normalization of the Stokes parameters takes an important role, because a vector containing the elements \(s_1(t)\), \(s_2(t)\), and \(s_3(t)\) would lie on a sphere with a time-dependent radius \(s_0(t)\), whereas the Poincaré vector \({\varvec{p}}(t)\) lies on a unit sphere. Irrespective of this, both vectors have the same spherical angles \(\vartheta (t)\) and \(\varphi (t)\), which are implicitly given by

$$\begin{aligned} \cos (\vartheta (t))&= - \frac{\sin \left( \varOmega t + \beta \right) }{\cosh \left( \sigma t + \alpha \right) }\,, \end{aligned}$$
(24a)
$$\begin{aligned} \tan (\varphi (t))&= \frac{\cos \left( \varOmega t + \beta \right) }{\sinh \left( \sigma t + \alpha \right) }\,. \end{aligned}$$
(24b)

There are a variety of special cases, where the trajectory can reside on the Poincaré sphere.

At first we should mention that for frequency and envelope synchronized oscillations (\(\varOmega =\sigma =0\)) the trajectory is time-invariant and degenerates to a point on the Poincaré sphere

$$\begin{aligned} {\varvec{p}}(t) = {\varvec{p}}_0 = \frac{1}{\cosh (\alpha )} \left[ \begin{array}{r} \sinh (\alpha )\\ \cos (\beta )\\ -\sin (\beta ) \end{array}\right] \,. \end{aligned}$$
(25)

This is the same solution as in Sect. 2.3 for constant amplitudes because the oscillations are synchron with respect to its envelope and frequency. Evidently, the two oscillations are totally synchronized if we have a constant diagonal synchronization, where \(\alpha \) and \(\beta \) are additionally vanishing.

Secondly, the trajectory defined in equation (23) describes a spherical helix through \({\varvec{p}}_\infty \)—also known as loxodrome or rhumb line [42]. This spherical helix degenerates for \(\sigma =0\) to a circle in the plane \(p_1(t)=\tanh (\alpha )\) with radius \(1/\cosh (\alpha )\):

$$\begin{aligned} {\varvec{p}}(t) = \frac{1}{\cosh (\alpha )} \left[ \begin{array}{c} \sinh (\alpha )\\ \cos (\varOmega t+\beta )\\ -\sin (\varOmega t+\beta ) \end{array}\right] \,. \end{aligned}$$
(26)

For \(\sigma \ne 0\) all trajectories on the Poincaré sphere are starting from \({\varvec{p}}(0)={\varvec{p}}_0\) and tend asymptotically toward

$$\begin{aligned} \lim \limits _{t\rightarrow \infty } {\varvec{p}}(t) = {\varvec{p}}_\infty = \textrm{sgn}(\sigma ) \left[ \begin{array}{c} 1\\ 0\\ 0 \end{array}\right] \,, \end{aligned}$$
(27)

where \(\textrm{sgn}(\sigma )\) denotes the signum of \(\sigma \). Qualitatively discussed, \(\sigma \) defines the movement of the trajectory in \(p_1\)-direction: \(\sigma =0\) means no movement in that direction, while \(\sigma >0\) and \(\sigma <0\) results in a movement toward H and V, respectively. The parameter \(\alpha \) specifies the initial point of the trajectory in \(p_1\)-direction and thus \(\textrm{sgn}(p_1(0))=\textrm{sgn}(\alpha )\). Similarly, the relative radian frequency \(\varOmega \) prescribes the rotational direction of the trajectory around the \(p_1\)-axis: For \(\varOmega =0\) there is no rotation, whereas the sign of \(\varOmega \) defines the direction of rotation. Apart from typical angle ambiguities \(\beta \) determines the initial angle in the \((p_2,p_3)\)-plane.

Fig. 4
figure 4

Special cases of trajectories on the Poincaré sphere, where the columns corresponds to the labeling of the equations in (30): Above the trajectories on the sphere are shown and below the same trajectories but in the associated plane. For (a+b) we have frequency synchronization (\(\varOmega =0\)). The trajectories start at the initial instant \(t=0\) at one of the synchronization points (\(\alpha =0\)) tend asymptotically toward H or V if \(\sigma \) is positive or negative, respectively. According to the value of \(\beta \) the trajectories in a start from D, A and vice versa in b from R or L. For c we have modulus synchronization (\(\sigma =0, \alpha =0\)), where the trajectory in dependency of the sign of the relative radian frequency rotates in clockwise or counter-clockwise direction in the \((p_2,p_3)\)-plane

Full size image

At last we examine some cases, where the trajectory passes characteristic synchronization states. To this end, we state that the third coordinate vanishes and the trajectory thus stays in the \((p_1,p_2)\)-plane if \(\xi (t)\) is an integer multiple of \(\pi \). If \(\xi (t)\) is an odd integer multiple of \(\pi /2\) then the second coordinate vanishes and the trajectory lies in the \((p_1,p_3)\)-plane. In contrast to this, the first coordinate vanishes and the trajectory is located in the \((p_2,p_3)\)-plane if \(\sigma t + \alpha =0\) holds. A pairwise combination of these cases leads to the following requirements for the base types of synchronization:

$$\begin{aligned}&\bullet (D) \quad \alpha = -\sigma t\,, \quad \beta = -\varOmega t\,, \end{aligned}$$
(28a)
$$\begin{aligned}&\bullet (A) \quad \alpha = -\sigma t\,, \quad \beta = -\varOmega t+\pi \,,\end{aligned}$$
(28b)
$$\begin{aligned}&\bullet (R) \quad \alpha = -\sigma t\,, \quad \beta = -\varOmega t-\pi /2\,,\end{aligned}$$
(28c)
$$\begin{aligned}&\bullet (L) \quad \alpha = -\sigma t\,, \quad \beta = -\varOmega t+\pi /2\,,\end{aligned}$$
(28d)
$$\begin{aligned}&\bullet (H) \quad t \rightarrow \infty \,, \quad \sigma > 0\,,\end{aligned}$$
(28e)
$$\begin{aligned}&\bullet \ (V) \quad t \rightarrow \infty \,, \quad \sigma < 0\,. \end{aligned}$$
(28f)

It should be emphasized that the instant t in the first four requirements is arbitrary. To get a better understanding of the trajectories on the Poincaré sphere let us examine some special cases. An interesting case is frequency synchronicity of the oscillations, where the relative radian frequency \(\varOmega \) vanishes and the equation (23) of the Poincaré vector becomes

$$\begin{aligned} {\varvec{p}}(t) = \frac{1}{\cosh (\sigma t+\alpha )} \left[ \begin{array}{c} \sinh (\sigma t+\alpha )\\ \cos (\beta )\\ -\sin (\beta ) \end{array}\right] \,. \end{aligned}$$
(29)

Hence, the trajectories of frequency synchronized oscillations remain in in a plane defined by the Poincaré vectors \({\varvec{p}}_0\) and  \({\varvec{p}}_\infty \). To be more specific, these trajectories passing diagonal or anti-diagonal synchronization stay solely in the \((p_1,p_2)\)-plane, whereas their trajectories are only located in the \((p_1,p_3)\)-plane if they pass right- or left-circular synchronization, cf. (28a-d).

In order to illustrate some of those trajectories let the particular trajectory start at any of the first four base types of synchronization. At first we assume frequency synchronization (\(\varOmega =0\)) but not envelope synchronization (\(\sigma \ne 0\)). In all this requires \(\alpha \) to be zero and \(\beta \) to be a multiple of \(\pi /2\). Depending on the sign of \(\sigma \) the trajectories asymptotically tends toward horizontal or vertical synchronization. Starting from diagonal or anti-diagonal synchronization this leads to

$$\begin{aligned} \begin{aligned}&\bullet \ (D\rightarrow H) \quad \alpha = 0\,, \quad \beta = 0\,, \quad \sigma> 0\,,\\&\bullet \ (D\rightarrow V) \quad \alpha = 0\,, \quad \beta = 0\,, \quad \sigma< 0\,,\\&\bullet \ (A\rightarrow H) \quad \alpha = 0\,, \quad \beta = \pi \,, \quad \sigma > 0\,,\\&\bullet \ (A\rightarrow V) \quad \alpha = 0\,, \quad \beta = \pi \,, \quad \sigma < 0\,, \end{aligned} \end{aligned}$$
(30a)

which is depicted in the first column of Fig. 4. The column in the middle of this figure shows the trajectories if they start from a right- or left-circular synchronization:

$$\begin{aligned} \begin{aligned}&\bullet \ (L\rightarrow H) \quad \alpha = 0\,, \quad \beta = +\pi /2\,, \sigma> 0\,,\\&\bullet \ (L\rightarrow V) \quad \alpha = 0\,, \quad \beta = +\pi /2\,, \sigma< 0\,,\\&\bullet \ (R\rightarrow H) \quad \alpha = 0\,, \quad \beta = -\pi /2\,, \sigma > 0\,,\\&\bullet \ (R\rightarrow V) \quad \alpha = 0\,, \quad \beta = -\pi /2\,, \sigma < 0\,. \end{aligned} \end{aligned}$$
(30b)

The last column in Fig. 4 comprises trajectories of a modulus synchronization (\(\sigma =0\)). Because of \(\alpha =0\) the Poincaré vector describes a unit circle in the \((p_2,p_3)\)-plane:

$$\begin{aligned} {\varvec{p}}(t) = \left[ \begin{array}{c} 0\\ \cos (\varOmega t+\beta )\\ -\sin (\varOmega t+\beta ) \end{array}\right] \,. \end{aligned}$$
(30c)

As it is shown, the rotational direction depends on the sign of the relative radian frequency \(\varOmega \). Since we do not see in this graphical representation where the trajectory starts and how fast it rotates, the phase \(\beta \) as well as the magnitude of \(\varOmega \) cannot be retrieved.

Fig. 5
figure 5

Example of different oscillations and its synchronization ellipses a–c leading to the same trajectory on the synchronization sphere (d). The labeling is according to equation (31), where the absolute parameters of the oscillations are given in (31a)–(31c), while the identical relative parameters can be taken from (31d)

Full size image

Now, the ambiguity problem of synchronization ellipses is shown by an example. For this purpose, we choose the following three different parameter sets. At first, we have a constant signal and a damped oscillation:

$$\begin{aligned} \sigma _x&= 0\,,\quad \varOmega _x = 0\,, \alpha _x = 0\,,\quad \beta _x = 0\,,\nonumber \\ \sigma _y&= -1/2\,,\quad \varOmega _y = 2\pi \,, \alpha _y = 0\,, \beta _y = 0\,, \end{aligned}$$
(31a)

Secondly, we choose two exponentially decaying oscillations having apart from a sign the same radian frequency:

$$\begin{aligned} \sigma _x&= -1/2\,, \varOmega _x = -\pi \,,\quad \alpha _x = 1\,, \beta _x = 0\,,\nonumber \\ \sigma _y&= -1\,, \varOmega _y = \pi \,,\quad \alpha _y = 1\,, \beta _y = 0\,. \end{aligned}$$
(31b)

The last example comprises two damped oscillations, which differ in their exponential decay and in their radian frequency:

$$\begin{aligned} \sigma _x&= -1/2\,, \varOmega _x = 2\pi \,, \alpha _x = 1\,, \beta _x = 0\,,\nonumber \\ \sigma _y&= -1\,, \varOmega _y = 4\pi \,, \alpha _y = 1\,, \beta _y= 0\,. \end{aligned}$$
(31c)

The resulting oscillations and synchronization ellipses are shown in the first three columns of Fig. 5. The commonness of the particular oscillations is hard to see and the synchronization ellipses are not helpful. They have a totally different contour and this despite the fact that the relative oscillation parameters are all the same:

$$\begin{aligned} \sigma = 1/2\,, \varOmega = -2\pi \,, \alpha = 0\,, \beta = 0\,. \end{aligned}$$
(31d)

In contrast to this, the trajectories on the synchronization sphere, which are depicted in the last column of Fig. 5, are identical in all cases and indicate the right relations. Since \(\alpha \) and \(\beta \) are zero, both oscillations are initially totally synchronized and the trajectory must start from D. Moreover, that \(\sigma \) is positive and the \(\varOmega \) is negative leads to a spherical helix, which is positively rotating around the \(p_1\)-axis and tends asymptotically toward H. To confirm this, the reader is referred to the basic trajectory movements as shown in Fig. 4: The green trajectory (\(\beta =0, \sigma >0\)) in column (a) superposed with the blue trajectory (\(\varOmega <0\)) in column (c) yields the trajectory in Fig. 5.

Up to now, the trajectories on the synchronization sphere can only be created by the knowledge of the relative oscillating parameters. How to overcome this problem by utilizing oscillator signals only is addressed in the next section.

4 Synchronization of oscillators

4.1 State-space model of a linear oscillator

Let us briefly recapitulate the state-space model of an oscillator by considering the scalar, complex, linear, time-invariant differential equation

$$\begin{aligned} \dot{w}_x(t) = \lambda _x w_x(t)\,,\, w_x(0) = {\text {e}}^{\alpha _x+{\textrm{j}}\beta _x}\,. \end{aligned}$$
(32)

Here, \(\alpha _x\) and \(\beta _x\) denote real numbers, whereas the eigenvalue \(\lambda _x = \sigma _x+{\textrm{j}}\varOmega _x\) is a complex number with \(\sigma _x\le 0\) and \(\varOmega _x>0\). The corresponding solution

$$\begin{aligned} w_x(t) = {\text {e}}^{\lambda _x t+\alpha _x+{\textrm{j}}\beta _x} \end{aligned}$$
(33)

separated into its real and imaginary part provides the real-valued solutions of a damped oscillator the associated real differential equation system

$$\begin{aligned} \left[ \begin{array}{c} \dot{u}_x(t)\\ \dot{v}_x(t) \end{array}\right]&= \left[ \begin{array}{rr} \sigma _x &{} -\varOmega _x\\ \varOmega _x &{} \sigma _x \end{array}\right] \left[ \begin{array}{c} u_x(t)\\ v_x(t) \end{array}\right] , \\ \left[ \begin{array}{c} u_x(0)\\ v_x(0) \end{array}\right]&= {\text {e}}^\alpha _x \left[ \begin{array}{c} \cos (\beta _x)\\ \sin (\beta _x) \end{array}\right] . \end{aligned}$$

It describes a damped oscillator and has the solution

$$\begin{aligned} \left[ \begin{array}{c} u_x(t)\\ v_x(t) \end{array}\right]&= {\text {e}}^{\sigma _x t+\alpha _x} \left[ \begin{array}{rr} \cos (\varOmega _x t) &{} -\sin (\varOmega _x t)\\ \sin (\varOmega _x t) &{} \cos (\varOmega _x t) \end{array}\right] \left[ \begin{array}{c} \cos (\beta _x)\\ \sin (\beta _x) \end{array}\right] \,, \end{aligned}$$

which is equivalent to the real and imaginary part of the complex solution in (33):

$$\begin{aligned} u_x(t)&= \Re \{ w_x(t) \} = {\text {e}}^{\sigma _x t+\alpha _x} \cos (\varOmega _x t+\beta _x)\,, \end{aligned}$$
(34a)
$$\begin{aligned} v_x(t)&= \Im \{ w_x(t) \} = {\text {e}}^{\sigma _x t+\alpha _x} \sin (\varOmega _x t+\beta _x)\,. \end{aligned}$$
(34b)

The parameters of the damped oscillator can be determined from the state at the initial instant and at a later instant \(T_0\)

$$\begin{aligned} w_x(0) = {\text {e}}^{\alpha _x+{\textrm{j}}\beta _x} \quad \text {and} \quad w_x(T_0) = {\text {e}}^{\lambda _x T_0} w_x(0)\,. \end{aligned}$$
(35)

By taking the complex logarithm and separating the real and imaginary parts, the desired parameters

$$\begin{aligned} \begin{aligned} \alpha _x&= \ln \left( |w_x(0) |\right) \,, \quad \sigma _x = \frac{\ln \left( |w_x(T_0)|\right) - \alpha _x}{T_0} \,,\\ \beta _x&= \textrm{arc}\left\{ w_x(0)\right\} \,,\quad \varOmega _x = \frac{\textrm{arc}\left\{ w_x(T_0)\right\} - \beta _x}{T_0} \end{aligned} \end{aligned}$$
(36)

become consecutively accessible.

4.2 Synchronization measurement of two linear oscillators

In this section, we will inspect the synchronization signals of two damped oscillators as given in (34), which are labeled with x and y. Unlike to Sect. 3 we do not have direct access to the parameters. Fortunately, this is not necessary because of (17) the ellipse parameter are directly accessible from the oscillator signals

$$\begin{aligned} a_x(t)&= \sqrt{u_x^2(t) + v_x^2(t)}\,, \end{aligned}$$
(37a)
$$\begin{aligned} a_y(t)&= \sqrt{u_y^2(t) + v_y^2(t)}\,, \end{aligned}$$
(37b)
$$\begin{aligned} \tan (\xi (t))&= \frac{v_x(t) u_y(t) - u_x(t) v_y(t)}{u_x(t) u_y(t) + v_x(t) v_y(t)}\,, \end{aligned}$$
(37c)

such that we are able to plot the trajectory on the synchronization sphere. The initial synchronization vector \({\varvec{p}}_0\) allows for a direct computation of \(\alpha \) and \(\beta \):

$$\begin{aligned} \tanh (\alpha ) = p_1(0) \,, \quad \tan (\beta ) \quad = -\frac{p_3(0)}{p_2(0)}\,. \end{aligned}$$
(38)

The determination of the relative parameters \(\sigma \) and \(\varOmega \) is more difficulty, since the trajectory on the synchronization sphere is a parametric representation where the timing cannot be retrieved. But there are some instants, which we can identify from observable events. For example, the instants \(t_\nu \), with \(\varOmega t_\nu +\beta =\nu \pi /2\), \(\nu \in \mathbb {Z}\) can be identified because the trajectory intersects the \((p_1,p_2)\)-plane or the \((p_1,p_3)\)-plane. These characteristic points,

$$\begin{aligned} p_2(t_\nu )&= 0 \ \Leftrightarrow \ p_3(t_\nu ) = \frac{\pm 1}{\cosh (\sigma t_\nu +\alpha )}\,, \quad \nu \ \quad \text {odd}\,, \end{aligned}$$
(39a)
$$\begin{aligned} p_3(t_\nu )&= 0 \ \Leftrightarrow \ p_2(t_\nu ) = \frac{\pm 1}{\cosh (\sigma t_\nu +\alpha )}\,, \quad \nu \ \quad \text {even}\,, \end{aligned}$$
(39b)

facilitate to compute the remaining parameters \(\sigma \) and \(\varOmega \). This way, it is possible to determine all synchronization parameters from the trajectory on the synchronization sphere but admittedly the computations are somewhat cumbersome. Considerably better is to utilize the equation (37) by transforming the quotient \(a_x(t)/a_y(t)\) and \(\tan (\xi (t))\). In this manner we get two linear equations for the synchronization parameters,

$$\begin{aligned}&\frac{1}{2} \ln \left( \frac{u_x^2(t) + v_x^2(t)}{u_y^2(t) + v_y^2(t)} \right) = \sigma t + \alpha \,, \end{aligned}$$
(40a)
$$\begin{aligned}&\arctan \left( \frac{v_x(t) u_y(t) - u_x(t) v_y(t)}{u_x(t) u_y(t) + v_x(t) v_y(t)} \right) = \varOmega t + \beta \,, \end{aligned}$$
(40b)

where the last equation holds modulo \(2\pi \) by taking the signs of the numerator and denominator into account. If we additionally rewrite these equations in dependence of the states \(w_x(t)\) and \(w_y(t)\), we get

$$\begin{aligned} \sigma t + \alpha&= \ln \left( \frac{|w_x(t)|}{|w_y(t) |} \right) \,, \end{aligned}$$
(41a)
$$\begin{aligned} \varOmega t + \beta&= \textrm{arc}\left\{ w_x(t) w_y^*(t)\right\} \,. \end{aligned}$$
(41b)
Fig. 6
figure 6

Top left: Signal with phase and amplitude jitter (blue) and reference signal (red). Bottom left: Interpretation of the marked points. Right: Poincaré sphere with the relative trajectory corresponding to the signals of the left plot. (Color figure online)

Full size image

4.3 Extension to nonlinear oscillators

The synchronization measurement in the last section is indeed an evaluation of two states in polar coordinates with respect to their magnitude and phase relations. This observation is important since the choice of the states is not unique and in addition it enables to conceive the particular oscillators as a nonlinear system of second order. Each of them have a (complex) state

$$\begin{aligned} w_x(t) = {\text {e}}^{\eta _x(t) + {\textrm{j}} \xi _x(t)}\,, \quad w_y(t) = {\text {e}}^{\eta _y(t) + {\textrm{j}} \xi _y(t)}\,. \end{aligned}$$
(42)

The ratio of these states

$$\begin{aligned} w(t) = \frac{w_x(t)}{w_y(t)} = {\text {e}}^{\eta (t) + {\textrm{j}} \xi (t)} \end{aligned}$$
(43)

serves as a measure of synchronization, where \(\eta (t)\) denotes an exponential and \(\xi (t)\) a phase deviation:

$$\begin{aligned} \eta (t)&= \eta _x(t) - \eta _y(t) = \ln \left( w(t) \right) \,, \end{aligned}$$
(44a)
$$\begin{aligned} \xi (t)&= \xi _x(t) - \xi _y(t) = \textrm{arc}\left\{ w(t) \right\} \,. \end{aligned}$$
(44b)

Thus, in general, the ellipse parameters of (37) take the form

$$\begin{aligned} a_x(t)&= \vert w_x(t)\vert \,, \end{aligned}$$
(45a)
$$\begin{aligned} a_y(t)&= \vert w_y(t)\vert \,, \end{aligned}$$
(45b)
$$\begin{aligned} \xi (t)&= \textrm{arc}\left\{ w_x(t) w_y^*(t) \right\} \end{aligned}$$
(45c)

and allow to draw the trajectory on the synchronization sphere:

$$\begin{aligned} p_1(t)&= \frac{|w(t) |^2+1}{\vert w(t)\vert ^2-1}\,, \end{aligned}$$
(46a)
$$\begin{aligned} p_2(t)&= \frac{2\cos (\xi (t))}{\vert w(t)\vert +\vert w(t)\vert ^{-1}}\,, \end{aligned}$$
(46b)
$$\begin{aligned} p_3(t)&= \frac{2\sin (\xi (t))}{\vert w(t)\vert +\vert w(t)\vert ^{-1}}\,. \end{aligned}$$
(46c)

Plugging \(\vert w(t)\vert ={\text {e}}^{\eta (t)}\) into these equations leads to a generalization of equation (23):

$$\begin{aligned} {\varvec{p}}(t) = \frac{1}{\cosh (\eta (t))} \left[ \begin{array}{r} \sinh (\eta (t))\\ \cos (\xi (t))\\ -\sin (\xi (t)) \end{array}\right] \,. \end{aligned}$$
(47)

Vice versa, equation (23) can be seen as a first order Taylor approximation with respect to the arguments. To the best of our knowledge, this type of generalization is only applicable to the synchronization sphere. Thus, we see the superiority of the synchronization sphere of the synchronization ellipse, as only the former allows for visualizing the trajectories of nonlinear oscillators.

5 Application examples

5.1 Preliminaries

Up to this point, we have only shown how synchronization measurements can be calculated in case the complex states are known. However, in practical scenarios, we usually only have access to real measurements, which are not directly suitable for the suggested measures at first glance. Thus, we would like to know briefly elaborate on a simple method for obtaining a complex signal to which our methods can be applied.

Assume we are given a real signal w(t) stemming from a device measurement. Let \(w_{+}(t)\) be the corresponding analytical signal, which is defined as:

$$\begin{aligned} w_{+}(t) = w(t) + {\textrm{j}}\mathcal {H}\{w(t)\} = e^{\eta (t) + {\textrm{j}} \xi (t)}\,, \end{aligned}$$
(48)

where \(\mathcal {H}\{\cdot \}\) denotes the Hilbert transformation. Thus, the necessary parameters read:

$$\begin{aligned} \eta (t) = \ln \left\{ |w_{+}(t) |\right\} \,, \quad \xi (t) = \textrm{arc}\left\{ w_{+}(t)\right\} . \end{aligned}$$
(49)

5.2 Phase jitter

Phase jitter is a phenomenon that can be observed in weakly coupled oscillators, where the phase of one of the oscillators keeps oscillating in a small interval around the phase of another oscillator [34]. Usually, jitter is caused by the presence of a moderate amount of noise, which leads to power exchange between the oscillators and hence a minor perturbation. Phase jitter can sometimes be difficult to detect, especially when amplitude variations are additionally present, usually due to the heterogeneity of the oscillators. However, on the Poincaré sphere, this phenomenon is clearly visible, as it leads to a simple trajectory. An example of this phenomenon is depicted in Fig. 6. Here, the top left plot depicts a signal with phase jitter (blue) and a self-defined reference signal (red). Although the signals are very similar, it is very hard to tell how they are exactly related. Using the projection given in (46) after making use of (49), we obtain the illustration depicted on the right side of Fig. 6. On the Poincaré sphere, we obtain a periodic trajectory, where we (chronologically) marked four special time instants. Recalling that the point corresponding to diagonal polarization (D) corresponds to a perfect synchronization, we obtain an interpretation for every one of these points, which is given in the table on the bottom left part of Fig. 6.

The trajectory on the Poincaré sphere can also be well explained by using a second-order approximation of (47), which results in:

$$\begin{aligned} {\varvec{p}}(t) \approx \frac{1}{1+\eta ^2(t)/2} \left[ \begin{array}{c} \eta (t) \\ 1-\xi ^2(t)/2 \\ -\xi (t) \end{array}\right] \end{aligned}$$
(50)

This shows that small amplitude differences predominantly influence the \(p_{1}\), while small phase deviations predominantly influence \(p_{3}\). Also, we see that amplitude deviations lead to larger \(p_{1}\) coordinates, i.e., a latitudinal movement along the surface of the sphere, while phase deviations lead to larger \(p_{3}\) and smaller \(p_{2}\) coordinates, i.e., a longitudinal movement along the surface of the sphere.

5.3 The synchronization sphere as a quantization tool for oscillator-based Ising machines

Fig. 7
figure 7

Left: Practical synchronization on the example of 5 oscillators of an Ising machine. The upper plot shows three oscillator outputs that are considered to be in-phase. The lower plot depicts the output of the two other oscillators, which are in anti-phase w.r.t. the oscillations in the top plot. Right: Projection of all oscillations on the right side using \(w_1\) as a reference. (Color figure online)

Full size image

While total synchronization is achievable in theory, only weaker forms of synchronization occur under practical conditions. In particular, we would like to highlight the case where two oscillations are synchronized with respect to their frequencies, however, due to parasitic effects or parameter mismatches, they have a constant but small phase and/or amplitude deviation. In this case, it can be useful to define tolerance values for the amplitude/phase deviations. If the amplitude/phase deviation of the two signals is less than the tolerance values, we say that the signals are practically synchronized.

Phase synchronization plays an important role in novel computing technologies. As a running example, we make use of the oscillator-based Ising machine [36,37,38]. These machines are optimization devices consisting of coupled oscillators, which encode their solutions into the phases of their oscillators. Specifically, every oscillator represents a binary variable, similar to a binary bit, and exhibits a bistable phase behavior in which the device asymptotically exhibits either a phase shift of 0 or \(\pi \) with respect to some reference oscillation. As such, the phase of every oscillator is of paramount importance and a good quantization method is required to be able to decode the solution of the problem. Here, we speak of quantization, because technical Ising machines usually only exhibit practical synchronization.

To demonstrate the usefulness of the Poincaré sphere, we simulated the solution of a Max-Cut problem with 5 oscillators using the methodology presented in [38]. The Max-Cut problem asks us to partition the graph into two subgraphs such that the number of edges connecting the two subgraphs is maximized. Now, once we solved the problem, we mapped the resulting trajectories onto the Poincaré sphere using the corresponding analytical signals. The oscillations are presented on the left of Fig. 7 and their projection onto the Poincaré sphere is presented on the right side of the same figure. As can be seen, the oscillations are synchronized in frequency but neither in their phase nor their amplitude. From a practical point of view, however, the amplitude and phase differences are quite marginal and hence negligible, i.e., the oscillators are practically synchronized. A good way to detect whether oscillations are practically synchronized is to define a tolerance (\(\alpha ,\beta \))-ellipse on the surface of the sphere. Here, we make use of the fact that the oscillator’s amplitude and phase parameters can be well approximated with

$$\begin{aligned} \eta _{\mu }(t)&= \sigma _{\mu }(t) t + \alpha _{\mu } \,, \quad \lim _{t \rightarrow \infty } \sigma _{\mu }(t) = 0 \,, \end{aligned}$$
(51a)
$$\begin{aligned} \dot{\xi }_{\mu }(t)&= \varOmega _{\mu }(t) \,, \quad \lim _{t \rightarrow \infty } \varOmega _{\mu }(t) = \varOmega _{0} \,, \end{aligned}$$
(51b)

where \(\mu = 1,\dots ,5\). These definitions follow from the observation that the amplitude of every oscillator converges to some constant value, while their frequencies eventually synchronize to \(\varOmega _{0}\). Asymptotically, every oscillator only exhibits some constant amplitude deviation \(\alpha _{\mu }\) or phase deviation \(\beta _{\mu }\). According to (51), we asymptotically have \(\varOmega =0\) and \(\sigma =0\) for all oscillators. Inserting this into (23) yields:

$$\begin{aligned} {\varvec{p}}_{\infty } = \frac{1}{\cosh (\alpha )} \left[ \begin{array}{r} \sinh (\alpha )\\ \cos (\beta )\\ -\sin (\beta ) \end{array}\right] \approx \frac{1}{1+\alpha ^2/2} \left[ \begin{array}{c} \alpha \\ 1-\beta ^2/2 \\ -\beta \end{array}\right] \,, \end{aligned}$$

where the latter expression is a second-order approximation similar to (50). Again, we see that small deviations in phase and amplitude translate to longitudinal and latitudinal movements (along the surface of the sphere). Therefore, we can define a small region on the surface of the sphere, an ellipse with the semi-major axis \(\alpha \) and semi-minor axis \(\beta \), in which all practically synchronized trajectories must asymptotically reside.

Going back to the trajectories of the oscillator-based Ising machine (left of Fig. 7), we see that all trajectories generally exhibit an amplitude deviation, are asymptotically frequency synchronized, but also have a constant phase deviation of either \(\beta _{\mu }\) or \(\beta _{\mu }+\pi \), where \(\beta _{\mu } \approx 0\). Thus, we define two tolerance ellipses centered around \((\textrm{D})\) and \((\textrm{A})\), the first being for trajectories with a phase deviation of \(\beta _{\mu }\) and the second being for the trajectories with a phase deviation of \(\beta _{\mu }+\pi \). As can be seen on the right side of Fig. 7, all trajectories eventually converge to one of the two tolerance ellipses, which allows us to then quantize the solutions by simply checking in which ellipse the end points of these trajectories reside. The solution of the Max-Cut problem is therefore given by the graphs partitions \(\{1,3,5\}\) and \(\{2,4\}\), cf. [38].

6 Conclusion and outlook

In this work, we have shown how the Poincaré sphere can be used as a powerful tool for the measurement of synchronization. Here, we started by reviewing the measures of polarization, the most important ones being the polarization ellipse and the Poincaré sphere. Next, we discussed two different synchronization measurements, the synchronization ellipse, which is inspired by the polarization ellipse, and the synchronization sphere, which is based on the Poincaré sphere. Moreover, we extended the notion of synchronization to the notion of total synchronization, which includes amplitude, phase, and frequency synchronization. Also, we discussed and visualized different types of synchronization on the synchronization ellipse and synchronization sphere. Here, we pointed out the advantage of the synchronization sphere over the synchronization ellipse, as some types of synchronizations can only be clearly seen when using the spherical representation. Finally, we discussed the synchronization of linear and nonlinear oscillators, where we also provided two application examples demonstrating the usefulness of the Poincaré sphere. The first one discussed the identification of phase jitter in oscillatory systems. The second example discussed using the Poincaré sphere as a quantization tool in the context of oscillator-based Ising machines.

Since synchronization is commonly tightly related to stability properties of dynamical systems, future research should look into making use of the Poincaré as a visualization tool for the dynamical behavior of such systems. Considering the undeniable impact of Henri Poincaré in the context of nonlinear dynamics, it may, for example, be possible to draw a relationship between the Poincaré map and sphere. Poincaré maps are the standard tool for analyzing the stability of periodic orbits. The spherical representation could be a visual extension of the standard tool, which allows gaining more insights on the stability properties of these orbits. In fact, it may even be possible to depict certain attractor regions onto the surface of the sphere, similar to the tolerance regions discussed on the example of an Ising machine.