^{1,a)}

### Abstract

We consider a pair of coupled heterogeneous phase oscillator networks and investigate their dynamics in the continuum limit as the intrinsic frequencies of the oscillators are made more and more disparate. The Ott/Antonsen *Ansatz* is used to reduce the system to three ordinary differential equations. We find that most of the interesting dynamics, such as chaotic behaviour, can be understood by analysing a gluing bifurcation of periodic orbits; these orbits can be thought of as “breathing chimeras” in the limit of identical oscillators. We also add Gaussian white noise to the oscillators'dynamics and derive a pair of coupled Fokker-Planck equations describing the dynamics in this case. Comparison with simulations of finite networks of oscillators is used to confirm many of the results.

The synchronisation of coupled oscillator networks is an ongoing topic of interest. Here, we consider a pair of networks of phase oscillators, with all-to-all coupling both within and between networks. When the oscillators are identical, the system is known to support both stationary and “breathing” chimera states, where one population is synchronised while the other is not. We use the degree of heterogeneity of the oscillators as a parameter, and by varying it, we investigate the new dynamics induced by this “frozen” disorder. We find that for sufficiently high disorder, the differential equations governing the dynamics of the macroscopic order parameters of the network show chaotic behaviour. This chaotic behaviour is the result of a gluing bifurcation involving a saddle fixed point in a three-dimensional phase space. We also consider adding Gaussian white noise to the oscillatordynamics and show that this “temporal” disorder can have surprising effects on the network dynamics.

I thank the referees for their useful comments, which helped to improve the manuscript.

I. INTRODUCTION

II. MODEL EQUATIONS

III. RESULTS

A. Small heterogeneity

B. Large heterogeneity

C. Results for a finite network

IV. NOISE

V. CONCLUSION

### Key Topics

- Oscillators
- 50.0
- Bifurcations
- 43.0
- Coupled oscillators
- 22.0
- Chaotic dynamics
- 10.0
- Fokker Planck equation
- 9.0

## Figures

Bifurcation set of Eqs. (7)–(9) for small heterogeneity. Curves of the three local bifurcations are labelled in the figure. Squares: homoclinic bifurcation of and , destroying and . Circles: symmetric homoclinic connection to *S*, gluing and together to form . Stars: symmetric heteroclinic bifurcation of and , destroying .

Bifurcation set of Eqs. (7)–(9) for small heterogeneity. Curves of the three local bifurcations are labelled in the figure. Squares: homoclinic bifurcation of and , destroying and . Circles: symmetric homoclinic connection to *S*, gluing and together to form . Stars: symmetric heteroclinic bifurcation of and , destroying .

Typical examples of the periodic orbits (top, when *A* = 0.35) and (bottom, when *A* = 0.5). The left column shows time series of (solid) and (dashed) and the right panels show orbits in the plane. Other parameters: *D* = 0.0005.

Typical examples of the periodic orbits (top, when *A* = 0.35) and (bottom, when *A* = 0.5). The left column shows time series of (solid) and (dashed) and the right panels show orbits in the plane. Other parameters: *D* = 0.0005.

Circles joined by lines: the gluing bifurcation shown in Fig. 1. Solid lines are contours of equal values of the saddle index for *S*, , where the eigenvalues of the Jacobian at *S* are . Note that the zero contour corresponds to the pitchfork bifurcation curve shown in Fig. 1.

Circles joined by lines: the gluing bifurcation shown in Fig. 1. Solid lines are contours of equal values of the saddle index for *S*, , where the eigenvalues of the Jacobian at *S* are . Note that the zero contour corresponds to the pitchfork bifurcation curve shown in Fig. 1.

Dynamics of the phase oscillator network (1) and (2) for *N* = 300 at (top to bottom) *A* = 0.05, 0.15, 0.35, 0.45, when *D* = 0.0007. Left panelsshow , colour-coded; right panels show and , where and . Parameters: .

Dynamics of the phase oscillator network (1) and (2) for *N* = 300 at (top to bottom) *A* = 0.05, 0.15, 0.35, 0.45, when *D* = 0.0007. Left panelsshow , colour-coded; right panels show and , where and . Parameters: .

Bifurcation set of Eqs. (7)–(9) for . Curves of the three local bifurcations are labelled in the figure. Circles: symmetric homoclinic connection to *S*, gluing and together. Crosses: saddle-node bifurcation of the periodic orbit (and of .) The thin line labelled “1” is the curve on which the saddle index changes from less than 1 (below the curve) to greater than 1 (above the curve).

Bifurcation set of Eqs. (7)–(9) for . Curves of the three local bifurcations are labelled in the figure. Circles: symmetric homoclinic connection to *S*, gluing and together. Crosses: saddle-node bifurcation of the periodic orbit (and of .) The thin line labelled “1” is the curve on which the saddle index changes from less than 1 (below the curve) to greater than 1 (above the curve).

Parameter sweeps of Eqs. (7)–(9) for *A* decreasing (panels a and b) and increasing (c and d), when . Panels a and c show the value of when increases through 0.5 during a long simulation and after transients have died out. Panels b and d show the most positive Lyapunov exponent of the solution.

Parameter sweeps of Eqs. (7)–(9) for *A* decreasing (panels a and b) and increasing (c and d), when . Panels a and c show the value of when increases through 0.5 during a long simulation and after transients have died out. Panels b and d show the most positive Lyapunov exponent of the solution.

Largest Lyapunov exponent for Eqs. (7)–(9) as a function of *D* and *A*. Other curves are: dashed: gluing bifurcation; solid: ; dash-dotted: Hopf; crosses joined by lines: saddle-node bifurcation of periodic orbits. Compare with Fig. 5.

Largest Lyapunov exponent for Eqs. (7)–(9) as a function of *D* and *A*. Other curves are: dashed: gluing bifurcation; solid: ; dash-dotted: Hopf; crosses joined by lines: saddle-node bifurcation of periodic orbits. Compare with Fig. 5.

The largest Lyapunov exponent for the network (1) and (2) with *N* = 100 and *D* = 0.004.

The largest Lyapunov exponent for the network (1) and (2) with *N* = 100 and *D* = 0.004.

(a): Behaviour of the network (1) and (2) at *A* = 0.15 ( is shown colour-coded). (b): (solid blue) and (dashed green) as functions of time for the state in the top panel, where and . (c) and (d): Lyapunov spectrum, i.e., all 2*N* Lyapunov exponents, for the state shown here. [(d) is an enlargement of the rightmost part of (c).] Parameters: *N* = 100 and *D* = 0.004.

(a): Behaviour of the network (1) and (2) at *A* = 0.15 ( is shown colour-coded). (b): (solid blue) and (dashed green) as functions of time for the state in the top panel, where and . (c) and (d): Lyapunov spectrum, i.e., all 2*N* Lyapunov exponents, for the state shown here. [(d) is an enlargement of the rightmost part of (c).] Parameters: *N* = 100 and *D* = 0.004.

(a): Behaviour of the network (1) and (2) at *A* = 0.4 ( is shown colour-coded). (b): (solid blue) and (dashed green) as functions of time for the state in the top panel, where and . (c) and (d): Lyapunov spectrum, i.e., all 2*N* Lyapunov exponents, for the state shown here. [(d) is an enlargement of the rightmost part of (c).] Parameters: *N* = 100 and *D* = 0.004.

(a): Behaviour of the network (1) and (2) at *A* = 0.4 ( is shown colour-coded). (b): (solid blue) and (dashed green) as functions of time for the state in the top panel, where and . (c) and (d): Lyapunov spectrum, i.e., all 2*N* Lyapunov exponents, for the state shown here. [(d) is an enlargement of the rightmost part of (c).] Parameters: *N* = 100 and *D* = 0.004.

Left column: (solid) and (dashed) from numerically solving Eqs. (20) and (21) for (top), (middle), and (bottom). Right column: (solid) and (dashed), where and , for the same noise intensities as in the left column. Parameters: .

Left column: (solid) and (dashed) from numerically solving Eqs. (20) and (21) for (top), (middle), and (bottom). Right column: (solid) and (dashed), where and , for the same noise intensities as in the left column. Parameters: .

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