Universality in globally coupled maps and flows

Abstract

We show that universality in chaotic elements can be lifted to that in complex systems. We construct a globally coupled Flow lattice (GCFL), an analog of GCML of Maps. We find that Duffing GCFL shows the same behavior with GCML; population ratio between synchronizing clusters acts as a bifurcation parameter. Lorenz GCFL exhibits interesting two quasi-clusters in an opposite phase motion. Each of them looks like Will o' the wisp; they dance around in opposite phase.

Model of globally coupled Duffing flows

“…We have verified that our findings are essentially independent of the choice of the potential shape. The doublewell potential Duffing model also follows the symmetry (5), and similar to our single-well Duffing GCFL, the double-well Duffing GCFL also exhibits two-clustered attractor under the symmetry(11) .…”

“…(ii) In GCML, each of the two clusters evolves in its onedimensional orbit and two orbits of the clusters approximately agree with each other (precisely if ϑ = 0.5) modulo a shift of one step. In the Duffing GCFL, on the other hand, the cluster attractor is two dimensional under the T ⊗ P π symmetry (11). Both (i) and (ii) together realize quite similar ϑ-bifurcation diagrams for GCML and GCFL.…”

“…6. Given that the dynamics of the N flows is reduced by synchronization to that of two clusters, that is, assuming the two-clustered configuration, it can be seen that the symmetry (6) of the single Duffing oscillator is lifted to (11). We show below that if (11) holds at time t, then it also holds at time t + ∆t.…”

“…This difference at the element level is reflected in the final structure of the attractors of coupled models and we will discuss this in detail. Some results in this paper were reported at AROB 13 [11]. This paper presents our completed work with a deepened understanding of GCML-GCFL correspondence (especially relation (11) and ( 14)).…”

“…In the event formation probability below, one observes that type B is clearly more frequently produced. (b) is deduced via (5) from the same 10 4 data, but orbits at odd multiples of 2π are converted by the relation (11). (c) is our final GCFL ϑ-bifurcation tree constructed from a total of 6.4 × 10 5 events; both even and odd steps data are combined).…”