Hopf normal form with $S_N$ symmetry and reduction to systems of nonlinearly coupled phase oscillators
Coupled oscillator models where oscillators are identical and symmetrically coupled to all others with full permutation symmetry are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, (the strength of coupling) and (an unfolding parameter for the Hopf bifurcation). For small enough there is an attractor that is the product of stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small . Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with symmetry. For fixed and taking the limit , we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of . Using a normalization that maintains nontrivial interactions in the limit , we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.
Coupled oscillator models are used in a wide variety of applications. They appear in neuroscience for studying neuronal oscillation patterns in the brain (see for example [2, 11, 14, 27]); in chemistry (see for example [25, 26]) and in physics (see for example [1, 30]). A powerful method for understanding the dynamics of coupled oscillators comes from noting that limit cycle oscillators give rise to a normally hyperbolic invariant torus that persists for weak enough coupling . In such cases we can describe the asymptotic dynamics in terms of just phases. A specific example of a coupled identical phase oscillator system with global (all-to-all) coupling is that of Kuramoto 
with fixed natural frequency , coupling strength . Although the original work of Kuramoto considered , a more general “Kuramoto-Sakaguchi” coupling (phase interaction) function  is
For the system (1.1,1.2), the only attractors are full synchrony or full asynchrony, depending on the value of the parameter , while in the special case the system is integrable. Many papers have studied the dynamics of this and related systems; see for example [1, 24]. For this permutation symmetric case of identical oscillators, the system above is known to behave in ways that are not generic, even accounting for symmetries. There can be a large number of integrals of the motion  and degenerate bifurcation behaviour on varying . As pointed out in , for weak linear coupling of nonlinear systems near Hopf bifurcation, one expects to have a coupling function that is smooth and -periodic [3, 9]. That is, the generic situation is that all will be non-zero in the Fourier expansion
where the will decay with at a rate that will depend on the smoothness of .
However, recent work by Rosenblum, Pikovsky and co-workers has highlighted that more complex interactions may be present in coupled oscillator systems, and that this may lead to new emergent phenomena such as self-organized quasiperiodicity  on including an additional damped equation, or on including direct dependence of a phase shift on an order parameter [8, 22].
The current paper considers generic nonlinear, but fully permutation symmetric, weak coupling of identical Hopf bifurcations. We show, by examining a generic normal form for equivariant Hopf bifurcation and unfolding parameter , that the system has an attracting invariant torus for . On this torus, the flow can be approximated by (1.1,1.2) at lowest order, but over a longer timescale it can be better approximated by a system of the form
The frequency is a symmetric function of the phases that is close to the frequency at Hopf bifurcation of the uncoupled system, and we have coupling via
where and depend on . More precisely, they are determined by
for some constant coefficients and . A more precise statement that includes the suppressed higher order terms is given in Theorem 3.2 and Corollary 3.3. Most of the discussion, apart from Section 5.3, will assume is fixed, but we assume the given normalization in (1.4) of the sums by , or to ensure non-trivial coupling in the thermodynamic limit .
with , and constants. As (1.4) shows, to the next order we only need to consider interaction terms of up to four phases. Each of the smooth periodic functions for involves only one Fourier mode, except for which involve two, and is a symmetric function of the phases. Note that (1.4) has normal form symmetry
for any , in addition to the permutation symmetries .
The structure of the paper is as follows. In Section 2 we give an outline of the normal form theory for equivariant Hopf bifurcation on where acts naturally by permutation of coordinates. This action decomposes into two irreducible subspaces of complex type, one of dimension one (corresponding to bifurcation to in-phase oscillation) and one of dimension (corresponding to bifurcation to anti-phase oscillation). We include two bifurcation parameters, determining the Hopf bifurcation representing the strength of coupling, in regimes where both are small.
Section 3 considers a set of coupled systems undergoing a generic supercritical Hopf bifurcation in the case of weak coupling . The main result is Theorem 3.2 which is proved in Section 4 by performing an explicit reduction of the normal form to an invariant -torus represented by coupled phase oscillators. Section 5 briefly considers a numerical example of the reduction as well as discussions of the consequences of the new interaction terms on fully synchronous and two-cluster states. The new terms introduce a particular (quadratic) nonlinearity to the equations for the phases of two cluster states, and in Theorem 5.1 we detail a particular new phenomenon. Finally, in Section 6 we discuss some implications of this on the dynamics of all-to-all coupled oscillators near Hopf bifurcation, and we relate to other work in the literature that considers more general nonlinear coupling between oscillators.
2 Equivariant Hopf bifurcation with symmetry
Suppose we have identical and identically interacting smooth () dynamical systems on (), generated by the following coupled ordinary differential equations:
The “coupling parameter” is such that the system completely decouples for . We also assume that each system undergoes a Hopf bifurcation of when a “Hopf parameter” passes through zero for .
Without loss of generality we assume that the uncoupled system for given by
has a linearly stable fixed point at for that undergoes Hopf bifurcation at . Without loss of generality, we can assume that has a complex pair of eigenvalues
where and all other eigenvalues of satisfy .
We also assume that, without loss of generality, is an equilibrium for in some neighbourhood of . As we will be interested in attracting behaviour near bifurcation we assume that the bifurcation is supercritical, i.e. it gives rise to a small amplitude stable limit cycle for .
Note that the Jacobian of (2.1) at will have the form
where represents the Jacobian with respect to the th argument.
We assume that the coupling respects the fact that the uncoupled systems can be permuted arbitrarily, i.e. that the system is equivariant under the action of on by permutation
for any and .
Although Hopf bifurcation in the absence of symmetry can generically be reduced to a two dimensional centre manifold, this is not the case here - the action of the symmetry group means that for the centre manifold at generic bifurcation will generically be either dimensional or dimensional. In the uncoupled case the extra structure means that the centre manifold will be dimensional.
3 Hopf normal form for a weakly coupled system
Using equivariant bifurcation theory  it is possible to write the system of ODEs (2.1) on a centre manifold where in the case the centre manifold in each coordinate is parametrized by . This system on the centre manifold is
where and we note the right hand sides can be chosen to be , with arbitrarily large, in a neighbourhood of the bifurcation. The conditions for Hopf bifurcation mean that for (3.1) we have and has a pair of purely imaginary eigenvalues that pass transversely through the imaginary axis with non-zero speed on changing . The action of on where acts by permutation of coordinates
where and so is symmetric under all permutations of the last arguments that fix the first.
Poincaré-Birkhoff normal form theory means that to all polynomial orders we can assume there is a normal form symmetry given by the action of on
The symmetries (3.2), (3.3) restrict the possible terms that can appear in the normal form; we can characterise these by finding the possible equivariants, one order at a time. This can be expressed in the following form which can be recovered from , where denotes , denotes and denotes .
Proof: For details, see [10, Section 2.1.2].
We summarise so far: if system (2.1) is such that (a) the system decouples for and (b) for each system has a generic Hopf bifurcation at , , then near the dynamics can be written on a centre manifold of dimension as (3.1). We now state the main result of our paper:
Consider system (3.1) with -symmetry (for fixed ) such that the uncoupled systems () undergo a generic supercritical Hopf bifurcation on passing through . There exists and such that for any and the system (3.1) has an attracting -smooth invariant -dimensional torus for arbitrarily large .
Moreover, on this invariant torus, the phases of the flow can be expressed as a coupled oscillator system
for fixed in the limit , where is independent of and
The constants and are generically non-zero. The error term satisfies
uniformly in the phases . The truncation of (3.6) by removing and terms is valid over time intervals where in the limit . In particular, for any , this approximation involves up to four interacting phases.
The proof of this Theorem is given in the next section. We remark that if we set in the theorem above, this gives the Kuramoto-Sakaguchi system as a truncation but with errors , meaning the timescale of validity of the Kuramoto-Sakaguchi system approximation will typically only be . We discuss the implications on timescales of validity of the approximation more precisely in the following corollary which is obtained by integrating the error term in the truncation.
Consider the system and hypotheses as in Theorem 3.2. Then for any and any such that there is an attracting -torus, there is a timescale such that for any solution of (3.6), there is a solution of the truncated equation with
over . If we truncate further to only Kuramoto-Sakaguchi terms by setting , then this will be possible only over a shorter timescale .
and the equations for the other are obtained by permutation of the indices; there is an error term but this is beyond all (polynomial) orders.
Since we are assuming there is a Hopf bifurcation of (3.1) for (the uncoupled system) on varying through , it follows that
and we write . The tail represents the higher order terms in the normal form for the uncoupled system: we can assume and write . The hypothesis that the Hopf bifurcation is generic and supercritical implies
We seek solutions of (4.2) of the form
for some , and constant . Substituting this into (4.2), we require
From (4.2), note that
From this, it is clear that for small enough and there is a stable periodic orbit at fixed such that , with angular frequency and arbitrary but fixed phase .
More precisely, solving , we note (recalling ) that
Now consider the dynamics of the full (but still uncoupled) system. For and any there is a stable invariant torus given by
parametrized by the phases . This invariant torus is foliated by neutrally stable periodic orbits with period and so for each , the torus is normally hyperbolic. By Fenichel’s theorem  there is an (depending on ) such that for the invariant torus persists and is -smooth for arbitrarily large . Note that reducing will restrict the : we will need for the approximation to be valid.
We now aim to find the approximating system (3.6) on this invariant torus for . We follow a method similar to [29, Section 7.3], using coordinate changes and a slow time to “blow up” the weak hyperbolic dynamics. Including all terms up to cubic order (except for the linear term which can be assumed to be contained in by a suitable change in parameters), using Theorem 3.1 we have
where the error term is . The complex normal form coefficients can be expressed using real quantities and (or and ) such that
for . We seek solutions of the following form:
Writing in real and imaginary parts and expanding for small we have
If we define
then, from (4.4),
This implies that (4.7) can be expressed as
Recalling from (4.4) that , , , and so we have
Similarly, we have
In particular, for there are finite limits
Applying similar expansions for the remaining terms in and taking real parts of (4.10) gives
where and , , etc are the normalized sums. The equivalent equation for is obtained by taking imaginary parts of (4.7):