^{1,a)}and Ernest Barreto

^{1,b)}

### Abstract

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos,attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.

In nature and in many practical applications, it is not uncommon to observe the emergence of coherent macroscopic behavior in large populations of interacting rhythmic units despite noise and the presence of heterogeneity in the population. For systems of globally coupled phase oscillators, collective synchrony and simple macroscopic oscillations have been extensively studied.

^{1–6}Large networks of more complicated (e.g., chaotic) oscillators can also exhibit these behaviors

^{7}but can also display a collective chaotic state.

^{8}It is not clear, however, if heterogeneous networks constructed of

*simple*phase oscillators that are not independently capable of producing chaos can exhibit complex macroscopic behavior (such as chaos) in the thermodynamic limit of large system size. In the current work, we use a recently developed mean-field analysis method and demonstrate that chaos can exist in the macroscopic mean field for a heterogeneous network of globally coupled phase oscillators with a bimodal frequency distribution and time-periodic coupling. We propose a qualitative mechanism for how this arises, and we identify period-doubling scenarios, attractor crises, and Smale horseshoe dynamics.

I. INTRODUCTION

II. FORMULATION

A. Our Model

B. OA reduction

C. No Chaos in the Classic Kuramoto Model

D. The Bimodal Kuramoto Model with Time-varying Coupling

III. MACROSCOPIC DYNAMICS FROM A TIME-VARYING BIMODAL NETWORK

A. The Incoherent State and Periodic and Quasiperiodic Solutions

B. Period Doubling Cascades, Chaos, and a Crisis

IV. DISCUSSION

### Key Topics

- Bifurcations
- 32.0
- Oscillators
- 32.0
- Chaos
- 31.0
- Mean field theory
- 30.0
- Attractors
- 29.0

## Figures

Bimodal distribution of natural frequencies, *g*(ω), as a sum of two Lorentzians.

Bimodal distribution of natural frequencies, *g*(ω), as a sum of two Lorentzians.

(Color online) Phase diagram for the static bimodal Kuramoto network with coupling *K* = *K* _{0}. The incoherent state is stable in the upper region, and black curves denote bifurcations that lead to coherent collective states (TC = transcritical, HB = Hopf, SN = saddle-node, HC = homoclinic, SNIPER = saddle-node-infinite-period). In the time-varying system, *K*(*t*) = *K* _{0} + *A* sin (2π*t*/τ), and the parameters sweep along the short diagonal green lines. The lettered red points indicate cases of interest for which (ω_{0},Δ) are as follows: (a) (2.0,1.5); (b) (0.7,1.35); (c) (2.5,0.8); (d) (1.29,0.8); (e) (1.4,0.65). We set *K* _{0} = 4 and we have *A* = 0.3 for cases (a)-(c) and *A* = 1 for case (d).

(Color online) Phase diagram for the static bimodal Kuramoto network with coupling *K* = *K* _{0}. The incoherent state is stable in the upper region, and black curves denote bifurcations that lead to coherent collective states (TC = transcritical, HB = Hopf, SN = saddle-node, HC = homoclinic, SNIPER = saddle-node-infinite-period). In the time-varying system, *K*(*t*) = *K* _{0} + *A* sin (2π*t*/τ), and the parameters sweep along the short diagonal green lines. The lettered red points indicate cases of interest for which (ω_{0},Δ) are as follows: (a) (2.0,1.5); (b) (0.7,1.35); (c) (2.5,0.8); (d) (1.29,0.8); (e) (1.4,0.65). We set *K* _{0} = 4 and we have *A* = 0.3 for cases (a)-(c) and *A* = 1 for case (d).

Behavior of the reduced (left) and full (right) time-varying equations for cases (a)-(c) with *K* _{0} = 4, τ = 5, and *A* = 0.3. A: Case (a), the persistence of the incoherent state. The fluctuations in the full system are due to finite-system-size effects. B: Case (b), a stable fixed point in the static system becomes a limit cycle (libration) in the weakly time-varying system. C: case (c), a limit cycle in the static system becomes a quasi-periodic state in the weakly time-varying system.

Behavior of the reduced (left) and full (right) time-varying equations for cases (a)-(c) with *K* _{0} = 4, τ = 5, and *A* = 0.3. A: Case (a), the persistence of the incoherent state. The fluctuations in the full system are due to finite-system-size effects. B: Case (b), a stable fixed point in the static system becomes a limit cycle (libration) in the weakly time-varying system. C: case (c), a limit cycle in the static system becomes a quasi-periodic state in the weakly time-varying system.

(Color online) (A) Bifurcation diagram showing the static asymptotic structures past which we sweep. Solid lines are stable equilibria; dashed lines are unstable equilibria; lines with symbols represent the maxima (circles) and minima (squares) of limit cycles. (B) State space diagram showing the limit cycle and the equilibria (circles; solid for stable and open for unstable) for K = 3.94, when these coexist. *X* = ρ cos ψ, *Y* = ρ sin ψ, and in both panels, Δ = 0.8 and ω_{0} = 1.29.

(Color online) (A) Bifurcation diagram showing the static asymptotic structures past which we sweep. Solid lines are stable equilibria; dashed lines are unstable equilibria; lines with symbols represent the maxima (circles) and minima (squares) of limit cycles. (B) State space diagram showing the limit cycle and the equilibria (circles; solid for stable and open for unstable) for K = 3.94, when these coexist. *X* = ρ cos ψ, *Y* = ρ sin ψ, and in both panels, Δ = 0.8 and ω_{0} = 1.29.

Period doubling and chaos in the time-varying network for case (d). The panels show *Y* = ρ sin ψ versus *X* = ρ cos ψ for A = 0.55 (A), 0.56 (B), 0.61 (C), and 0.65 (D), with ω = 1.29, Δ = 0.8, and *K* _{0} = 4.

Period doubling and chaos in the time-varying network for case (d). The panels show *Y* = ρ sin ψ versus *X* = ρ cos ψ for A = 0.55 (A), 0.56 (B), 0.61 (C), and 0.65 (D), with ω = 1.29, Δ = 0.8, and *K* _{0} = 4.

(Color online) (A) Bifurcation diagram, obtained using a Poincaré surface of section, showing the magnitude ρ of the macroscopic mean field versus the amplitude *A* of coupling variation, in case (d). (B) Plot of the two largest Lyapunov exponents versus *A*. Both panels were computed using the reduced system, Eq. (15).

(Color online) (A) Bifurcation diagram, obtained using a Poincaré surface of section, showing the magnitude ρ of the macroscopic mean field versus the amplitude *A* of coupling variation, in case (d). (B) Plot of the two largest Lyapunov exponents versus *A*. Both panels were computed using the reduced system, Eq. (15).

(Color) The chaotic attractor of the macroscopic mean-field slightly above and below the interior crisis value at *A* _{ c } = 0.641. For these diagrams, a stroboscopic surface of section, (ρ((*n* + 1)τ), ψ((*n* + 1)τ)) vs. (ρ(*n*τ), ψ(*n*τ)), was used. (A) the attractor for . Superimposed on this is a trajectory segment for showing escape via the mediating period-three orbit. (B) the expanded attractor for . The pre-crisis attractor is overlaid in blue.

(Color) The chaotic attractor of the macroscopic mean-field slightly above and below the interior crisis value at *A* _{ c } = 0.641. For these diagrams, a stroboscopic surface of section, (ρ((*n* + 1)τ), ψ((*n* + 1)τ)) vs. (ρ(*n*τ), ψ(*n*τ)), was used. (A) the attractor for . Superimposed on this is a trajectory segment for showing escape via the mediating period-three orbit. (B) the expanded attractor for . The pre-crisis attractor is overlaid in blue.

(Color) A horseshoe in the stroboscopic map of the reduced equation Eq. (15) for *A* = 0.65.

(Color) A horseshoe in the stroboscopic map of the reduced equation Eq. (15) for *A* = 0.65.

The bifurcation diagram for the magnitude ρ of the macroscopic mean-field versus the period τ of the coupling time variation. A Poincaré surface of section is used. *A* = 0.65, ω_{0} = 0.8, and Δ = 1.29.

The bifurcation diagram for the magnitude ρ of the macroscopic mean-field versus the period τ of the coupling time variation. A Poincaré surface of section is used. *A* = 0.65, ω_{0} = 0.8, and Δ = 1.29.

(Color online) Bifurcation diagram and Lyapunov exponents for case (e); compare Fig. 6. ω_{0} = 1.4, Δ = 0.65.

(Color online) Bifurcation diagram and Lyapunov exponents for case (e); compare Fig. 6. ω_{0} = 1.4, Δ = 0.65.

(Color online) Very slow parameter variation with τ = 50.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.1]10.1063/1.3638441.1

(Color online) Very slow parameter variation with τ = 50.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.1]10.1063/1.3638441.1

(Color online) Fast parameter variation with τ = 1.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.2]10.1063/1.3638441.2

(Color online) Fast parameter variation with τ = 1.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.2]10.1063/1.3638441.2

(Color online) Very fast parameter variation with τ = 0.1 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.3]10.1063/1.3638441.3

(Color online) Very fast parameter variation with τ = 0.1 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.3]10.1063/1.3638441.3

(Color online) Chaos: intermediate parameter variation with τ = 5.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.4]10.1063/1.3638441.4

(Color online) Chaos: intermediate parameter variation with τ = 5.0 (enhanced online). [URL: http://dx.doi.org/10.1063/1.3638441.4]10.1063/1.3638441.4

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