On reversibility of macroscopic and microscopic dynamics in the Kuramoto model
Introduction
The first theoretical results on low-dimensional behavior of the population of identical Kuramoto oscillators with global coupling have been reported in the seminal paper [1] of Watanabe and Strogatz. The low-dimensional dynamics of Kuramoto oscillators is intimately related to certain geometric and group-theoretic objects that play a central role in several fields of Mathematical Physics. Group-theoretic study, performed in [2] unveiled underlying symmetries in the system. The collective dynamics has been explained in terms of conformal mappings and Lie groups, the mathematical concepts that appeared for the first time in studies on coupled oscillators. About the same time, Ott and Antonsen have also used complex analysis and harmonic functions theory in order to derive their result on macroscopic dynamics in the model with non-identical oscillators, see [3]. Recent continuations of this research direction exposed connections between Kuramoto models and hyperbolic geometry of the unit disk, see [4], [5].
We consider the system Here, denote phases of oscillators, is their intrinsic frequency (common for all oscillators, as they are identical). Finally, is a global complex-valued coupling function. It is easy to check that the specific choice turns (1) into the original Kuramoto model (with identical oscillators). For some other forms of the coupling function we can obtain models with phase-shifted, time-delayed, or noisy couplings. The main assumption is that the coupling is global, that is — the same between each pair of oscillators.
By introducing the substitution we represent each oscillator by a point on the unit circle .
We introduce the centroid (complex order parameter) of the population at moment : is a point in the unit disk . Then, is the real order parameter that measures coherence of the population.
Further, introduce the group of all Möbius (linear fractional) transformations of the complex plane that preserve the unit disk. General disk-preserving Möbius transformation can be written as This group is isomorphic to the matrix group . (More precisely, it is isomorphic to the group as the identity matrix and correspond to the same transformation.)
For our purposes, it is more convenient to use another parametrization of elements of this group:
Throughout the present paper we will denote this group of disk-preserving Möbius transformations by and refer to it as Möbius group. Parametrization (4) corresponds to the decomposition , where stands for the unit disk in complex plane.
Consider the population of coupled phase oscillators whose states are represented by points on the unit circle with the corresponding phases satisfying (1). It has been shown in [2] that there exists a one-parametric family of transformations belonging to the group such that with the parameters of satisfying the following system We refer to (5) as the Watanabe–Strogatz system as it has been first derived in their paper [1].
Therefore, the dynamics of all oscillators are completely determined by the global variables and . The system (5) describes evolution on the Möbius group .
We now set up mathematical framework and notations that will be used throughout the present paper. State of each individual oscillator at moment is given by its phase , or, more conveniently, by a point on the unit circle . State of the whole system is given by the distribution of oscillators’ phases. After a suitable normalization, we can represent distribution of phases at the moment by a probability measure on the unit circle . By passing to thermodynamic limit, , we will mainly deal with absolutely continuous measures and denote their probability density functions by .
Möbius group acts naturally on the unit circle , on hyperbolic disk and the space of all probability measures on . Action of Möbius transformation on the individual oscillator is denoted as . On the other hand, evolution of the whole system is given by the action of on the space and denoted by: or, in terms of the corresponding p.d.f.:
Having that the population evolves by the action of , we conclude that the motions are restricted to orbits of this group. Since has the real dimension 3, it follows that its group orbits are 3-dimensional invariant submanifolds in the space . These invariant submanifolds are determined by the initial distribution of oscillators.
However, there is one important special case. If the initial distribution of oscillators is uniform on the unit circle, i.e. , then the dynamics takes place on the 2-dimensional invariant submanifold, that consists of the functions of the form
Functions of the form (6) are well known in Mathematical Physics as Poisson kernels. In fact, actions of Möbius transformations (4) on the uniform measure on the circle yield precisely Poisson kernels. Notice that each Poisson kernel is uniquely determined by a single complex number , such that . This complex number is the mean value (complex order parameter) of the distribution (6). The shape of this distribution depends only on the real parameter , while is a simple rotational parameter. We conclude that the set of all probability measures with p.d.f. of the form (6) constitutes two-dimensional invariant (for the dynamics (1)) submanifold in the space . In the context of coupled oscillators this invariant submanifold has been recently referred to as Ott–Antonsen manifold, see for instance [6]. We will denote this submanifold by .
In the present paper we focus on the variable that appears in (4), (5). This variable is the global phase in the system. We will examine its impact on the dynamics. In order to explain this, assume that the initial distribution of oscillators in (1) is uniform. Then the evolution takes place on the invariant submanifold , i.e. is function of the form (6) for each . Further, assume that the evolution is cyclic, i.e. for some . At the first glance it seems that the evolution is periodic and oscillators perform simple rotations with the period . However, a closer look unveils that the evolution may be much subtler than what can be observed at the level of oscillators’ densities. Indeed, if , then the Möbius transformations (4) that act on the system at moments and are different.
In other words, if we observe macroscopic dynamics, i.e. just densities of oscillators’ phases, we will see simple rotations. However, if we pass to the microscopic level, we may find out that the dynamics of each individual oscillator is more complicated. This is due to the impact of the global phase that appears as a “hidden” variable in the system.
If the initial distribution is not uniform, then the evolution takes place on a certain 3-dimensional submanifold. Then impact of will be clearly visible on macroscopic level. In this case is not hidden variable, as it affects shape of the density function.
In order to access certain physical aspects of the global phase we will study the question of reversibility of the collective dynamics of oscillators. Roughly, reversibility is understood as a possibility of inverting the dynamics and recovering an initial state of the system. The notion of reversibility will be introduced in a more rigorous way in Section 4. In brief, we will address the following questions:
Suppose that the oscillators evolve on time interval by (1) starting from a certain distribution . Can we then recover an initial distribution from the distribution at the moment ? For instance, if we revert the time in (1) will the system eventually evolve from back to ? How various factors (such as noise) may affect this evolution?
In addition, we will examine the role of individual oscillators in collective dynamics. For instance, suppose that the distribution returned into its initial shape; does it necessarily mean that all oscillators returned to their initial positions? May it happen that the distribution returns to initial shape, but individual oscillators changed their positions in a complicated way?
Finally, what is special about evolution on the two-dimensional submanifold ? Is there any physically meaningful difference when compared to evolution on a general three-dimensional submanifold?
In the next section we briefly explain some related concepts in Mathematical Physics that served as an important inspiration for our study. In Section 3 we introduce one concrete example of the 3-dimensional invariant submanifold that will serve as a model example in the sequel. In Section 4 two notions of reversibility in the Kuramoto model are precisely introduced. In Sections 5 Simulations: deterministic systems exhibit reversible dynamics, 6 Simulations: noise-induced irreversibility we present simulation results in deterministic and noisy setups respectively. Finally, Section 7 contains some conclusions and the brief discussion on their significance in a broader context.
Section snippets
Motivation: -coherent states in mathematical physics
We have pointed out in the previous section that the dynamics (1) is restricted to orbits of the group , that is — on invariant submanifolds in . These invariant submanifolds are, in general, 3-dimensional. However, there is an important special case of invariant submanifold that consists of Poisson kernels (6). We have denoted this two-dimensional invariant submanifold by . In this section we briefly emphasize strong analogies with the notion of coherent states in quantum theories
Invariant submanifolds: two model examples
In Introduction we have explained that the distribution of oscillators evolves on an invariant submanifold in . In this section we introduce one particular example of such an invariant submanifold.
The starting point for this example is the family of so-called von Mises distributions. This family of distributions provides one of several important examples of probability distributions on the unit circle in directional statistics, see [18]. Von Mises distributions are defined by the following
Concepts of reversibility
As introduced above, states of the system (1) are probability measures on . We will identify two measures that differ only by rotation, i.e. if are such that , for some , we will say that and correspond to the same state.
Now, fix the initial state of the system . Suppose that does not contain a majority cluster and consider the following dynamical system and We have simply changed the
Simulations: deterministic systems exhibit reversible dynamics
In this section we present some simulations of (1) with the simplest coupling function (2).
This choice of the coupling function puts (1) into familiar form of the classical Kuramoto model: We set for and for . Intrinsic frequencies are set at . This yields repulsive coupling at the time interval and attractive coupling for . Then the order parameter monotonically decreases for and increases for . We choose
Simulations: noise-induced irreversibility
In this section we consider the Kuramoto model with the common multiplicative noise where is realization of the scalar Gaussian noise with intensity , i.e. . Underline that is a common noise for all oscillators, which means that the coupling is noisy, but still global.
The coupling strength is again set at for and at for . Initial conditions for the two simulations are chosen in the same way as in
Conclusion
The present communication is a continuation of previous studies on geometry and low-dimensional dynamics in the simple Kuramoto model with identical oscillators and global coupling. We have exposed some geometric subtleties of the dynamics and their physically relevant implications. The main focus of the present study is on reversibility of the collective motions of Kuramoto oscillators. Reversibility is understood in the sense that changing signs of the coupling function yields an inverse
CRediT authorship contribution statement
Vladimir Jaćimović: Conception and design of study, Analysis and/or interpretation of data, Writing - original draft, Writing - review & editing. Aladin Crnkić: Acquisition of data, Analysis and/or interpretation of data, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
All authors approved the version of the manuscript to be published.
References (31)
- et al.
Constants of motion for superconducting Josephson arrays
Physica D
(1994) - et al.
Collective motions of globally coupled oscillators and some probability distributions on circle
Phys. Lett. A
(2017) - et al.
Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action
Chaos
(2009) - et al.
Low dimensional behavior of large systems of globally coupled oscillators
Chaos
(2008) - et al.
Hyperbolic geometry of Kuramoto oscillator networks
J. Phys. A
(2017) - et al.
Dynamics of the Kuramoto-Sakaguchi oscillator network with asymmetric order parameter
Chaos
(2019) - et al.
Is the Ott-Antonsen manifold attracting?
Phys. Rev. Res.
(2020) Der stetige Übergang von der Mikro-zur Makromechanik
Naturwissenschaften
(1926)Coherent States in Quantum Physics
(2009)Coherent states for arbitrary Lie group
Comm. Math. Phys.
(1972)