Individual Chaos
At the left border of the parameter space ( = 0), the system consists of uncoupled cells, thus each cell independently exhibits the chaotic dynamics of the logistic map. Note that for
= 0, the delayα has no effect. Hence the corresponding parameter sets all describe the same coupling mechanism. The MLE starts at 0.9 for = 0, which is the Lyapunov exponent of an isolated logistic map. For increasing, it first gradually declines to a plateau of about 0.5−0.6, then sharply drops to 0, giving the region a distinct boundary. See fig. 4.7a for a typical system state and time series of this regime.
Collective Chaos
In the context of a system of coupled oscillators, the termorder can refer to different concepts, including collective or predictable behavior. The second chaotic regime at the bottom of the parameter space (fig. 4.7b) is noteworthy because the coupling leads to dynamics that are collective but not predictable, indicating that these concepts of order do not always coincide. With increasing coupling strength, the behavior becomes slightly more spatially ordered. The right border with= 1 deserves special attention, see below. The Lyapunov exponent reaches values of 0.2−0.4.
Fixed Point
The logistic map has an unstable fixed point at 0.75, which for the coupled system becomes stable for some control parameters (fig. 4.7c). The synchronization first occurs atα = 23. This is probably related to a result by Masoller and Marti [62], who studied a related system, and observe that a coupling mechanism with mixed delays (in their case randomly chosen) leads to better synchronizability than uniform delays. The transition occurs via a dying period-2 oscillation, which is totally synchronized at small α, while it forms a checkerboard geometry for largeα. This transition will be examined analytically in section 4.2.4.
Periodic
In the phase diagram (fig. 4.6b), the region around the fixed point forms a regime of periodic oscillations, in the sense that the time series of each cell is periodic (fig. 4.7d). When approach- ing the inner border to the fixed point regime, the oscillations decay, while at the outer border the cells lose their coherency. The spatial structure varies in a smooth transition from spatially uniform (total synchronization) to anti-phase clusters forming checkerboard patterns. These two extremes of spatial order will be treated as different regimes.
Total Synchronization
In total synchronization (fig. 4.7e), every cell has exactly the same time series, thus the state of the system is spatially uniform. At the boundary to the collective chaos regime a period doubling cascade is observed. This is the only regime exhibiting periods longer than 2. Note
that formally, total synchronization includes the fixed point, but it is useful to treat it as a separate regime.
Checkerboard
As introduced in section 4.1.4, the symmetry of the lattice geometry allows a splitting into two clusters with only inter-cluster connections. Within the period-2 regime, there is such a splitting into two anti-synchronized clusters (fig. 4.7f). At the border to the non-periodic regimes, the oscillations lose coherency. Since there exist two incompatible phases of the checkerboard structure, there are cases where both phases are present (separated by a line of defects). For odd lattice sizes this configuration can not exist globally, and therefore such defects must exist. Since the checkerboard structure is equally common for even and odd lattice sizes, global defects seem not to have an important effect on its stability.
The Clustered Regimes
The rest of the parameter space is governed by dynamics that are neither chaotic, nor strictly ordered. Since there is a transition between spatially smooth states and local checkerboard patches, I split these cases into two regimes, namely states with positive and negative neigh- bor correlation. This corresponds to a tendency for coherent clusters or checkerboard-clusters (fig. 4.7g and h), respectively.
Amritkar and Jalan [6] study a related system (scale-free network) forα= 1, and observe several regimes. For running from 0 to 1, they find turbulent behavior, phase synchronized clusters, intermittent phase synchronization, and finally a regime of anti-phase clusters. This is in good agreement with the regimes observed here.
The Border at = 1
This section will explain why the extremal parameter = 1 corresponds to an exceptional coupling mechanism. In this case, the coupling equation (eq. 4.2) becomes
xt+1 = 1 4 X x∼x0 ((1−α)f(x0t) +α x0t), (4.4) and the next iterationxt+1 of a cell depends only on the statex0t of the neighbors but not on xt.
Suppose the lattice size is even. To understand the consequences of the extremal coupling, consider the lattice as a checkerboard with each cell labeled as either black or white. The next step of any black cell depends only on the current state of its neighbors, all of which are white. Hence, knowing the initial condition (t = 0) of the white cells is enough to determine the state of all black cells at the odd times, and all white cells at the even times. In other words, this part of the system information forms a subsystem that is uncoupled from the rest of the lattice. The rest of the cells forms a similar subsystem, consisting of the black cells at odd times and the white cells at even times.
For small α, this behavior is clearly reflected in the dynamics. The collective chaos re- gion for small α and large is mainly characterized by locally correlated states. However, in the neighborhood of = 1 there appear fluctuating checkerboard patches, which can be explained as an interleaving of two independent clusters in a checkerboard pattern. In the
Fig. 4.7: Typical lattice states (left) and corresponding time series of a single cell (right) for the main regimes: a) Individual chaos b) Collective chaos c) Fixed point d) Periodic e) Total synchronization f) Checkerboard g) Coherent clusters h) Checkerboard-clusters.
ordered regimes at larger α, there is no apparent change when approaches 1, because both subsystems approach the same ordered state.
In conclusion, in all cases, the two subsystems have similar dynamics. In the chaotic regime, the independence leads to uncorrelated behavior of the two halves, while in the fixed point regime independent systems still lead to a uniform result. One might expect to occasionally see opposing phases in the periodic regime, which are not observed. I conjecture that the starting procedure of the system favors certain phases, leading to a symmetry breaking (which is discussed in section 4.2.5). See also section 4.5.2 for the fact that, dependent on the imple- mentation, numerical roundoff errors can induce a coupling in the order of the computational accuracy (about 10−15 for double precision floating-point arithmetic).