Fig 1.
(Color online) Bifurcation diagrams of r1, r2 and λ against ω′ = 4ω0/K.
K = 4, β = 0.1, and Δ′ = 4Δ/K = 0.4. Solid (open) data points represent stable (unstable) states. In top panels, red, blue, wine, and dark green symbols are for partial synchronous states ,
,
, and
, respectively. Thick black and dark green lines represent the maximum and minimum values of r1 and r2 for stable standing wave synchronous states. In middle and bottom panels, from left to right, real and imaginary parts of the eigenvalues λ for partial synchronous states from
to
are displayed. Squares, circles, and triangles denote eigenvalues λ1, λ2, and λ3, respectively.
Fig 2.
(Color online) Bifurcation diagrams of r1 and λ against ω′ = 4ω0/K at Δ′ = 4Δ/K = 0.95 (left column) and against Δ′ at ω′ = 1.5 (right column).
Solid (open) data points represent stable (unstable) states. In (a), red, blue, and wine lines are for partial synchronous states ,
, and
, respectively. In (b), wine and dark green lines are for partial synchronous states
and
, respectively. Thick black and dark green lines refer to the standing wave synchronous state. In the panels from (c) to (f), squares, circles, and triangles denote the real and imaginary parts of eigenvalues λ1, λ2, and λ3, respectively. The inset of (d) shows that HB occurs at a lower Δ′ than PB2. Note that the incoherent state changes its stability across the pitchfork bifurcation (PB1).
Fig 3.
(Color online) Bifurcation diagrams on the (Δ′ = 4Δ/K, ω′ = 4ω0/K) plane for (a) β = 0 and (b) β = 0.1.
Line color codes: black and red for two pitchfork bifurcations, PB1 and PB2, respectively; green and pink for two saddle-node bifurcations, SN1 and SN2, respectively; blue for HB (Hopf bifurcation); cyan for HC (homoclinic bifurcation). Acronyms: SNIPER for saddle node infinite period; CP for cusp point of SN1 and SN2; TB for Takens-Bogdanov point. To present the topological structure of the phase space in different phase domains, we plot the phase portraits on the (r1, r2) plane in several insets with the parameters chosen from different phase domains. The dashed arrows pointing to insets refer to the phase domain represented by the insets. In each inset, several phase portraits (wiggly lines) are plotted with arrows representing the evolution from or towards the solutions in Eq (12). In these insets, solid (open) dots represent stable (unstable) partial synchronous states, while the dark yellow curves represent stable standing wave partial synchronous state denoted by L. The solutions in the same color in different insets are the same solution. K = 4.
Fig 4.
(Color online) Phase diagrams on the plane of Δ′ = 4Δ/K and β at in (a),
in (b),
in (c), and
in (d).
K = 4.
Fig 5.
(Color online) Stability diagrams of the incoherent state for asymmetrical bimodal frequency distribution on various parameter planes.
(a) (K, p1) plane at β = 0.9, Δ1 = 0.3, Δ2/Δ1 = 0.02, and ω0 = 0.1; (b) (K, Δ2/Δ1) plane at β = 0.9, Δ1 = 0.3, ω0 = 0.1, and p1 = 0.8; (c) (K, ω0) plane at β = 0.9, Δ1 = 0.3, Δ2/Δ1 = 0.02, and p1 = 0.8; (d) (K, β) plane at Δ1 = 0.3, Δ2/Δ1 = 0.02, ω0 = 0.1, and p1 = 0.8. The shaded regions with red boundary lines, obtained from Eq (15), mark the stable incoherent state. The blue and green lines in (d) are critical curves K = 2Δ1/cosβ for p1 = 1 and K = 2Δ2/cosβ for p1 = 0, respectively.