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Generating macroscopic chaos in a network of globally coupled phase oscillators
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Image of FIG. 1.
FIG. 1.

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

Image of FIG. 2.
FIG. 2.

(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 (2πt/τ), and the parameters sweep along the short diagonal green lines. The lettered red points indicate cases of interest for which (ω0,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).

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

(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 ω0 = 1.29.

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

(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).

Image of FIG. 7.
FIG. 7.

(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(nτ), ψ(nτ)), 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.

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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, ω0 = 0.8, and Δ = 1.29.

Image of FIG. 10.
FIG. 10.

(Color online) Bifurcation diagram and Lyapunov exponents for case (e); compare Fig. 6. ω0 = 1.4, Δ = 0.65.

Image of FIG. 11.
FIG. 11.

(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

Image of FIG. 12.
FIG. 12.

(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

Image of FIG. 13.
FIG. 13.

(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

Image of FIG. 14.
FIG. 14.

(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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Generating macroscopic chaos in a network of globally coupled phase oscillators