Resolution Dependency at the Accumulation Point

where we define Γ(N, k) = (D−+D+−D−+) D−+ (3.28) as the information codimension. Note that here we admit the possible dependence of the relevant information dimensions onτ, which implicitly defines the underlying measures.

Thus for systems whose joint information dimension is not just the sum of the marginal ones Persistent Mutual Information should scale logarithmically with probability resolution. This is in contrast to ‘simple’ regimes of dissipative systems like logistic map studied above. When the attractor consists of a finite number of points as is the case for period-p cycle, information dimensions of all the support spaces are the same (which also happen to be zero). For any τ, which due to our definition of PPMI does not have to be finite, higher sample sizes only ensure PMI converges to log(p) in a manner specific to the estimator used.

This can be interpreted in terms of ensembles. Flat initial distribution sampled with N

points can be considered as being equivalent to starting with N closed systems. By the optimal ‘lack of information’ argument we then assume that the distribution out of which the systems were picked was flat. PMI then corresponds to the average information about the future that would be obtained should one of the systems be examined. For non-fractal attractors increasing the number of samples becomes, after some N∗ _{e.g.> ppointless, in} the sense that this average value would not change. PMI dependency on sample size of this form would manifest itself in the average information about the possible future state increasing without end at a logarithmic rate.

3.2.1 Resolution Dependency at the Accumulation Point

We now compute the PMI at the period-doubling accumulation pointrc. Figure 3.9 shows the result for a variety of resolution ranges that we control by varying k (computationally faster than increasing N, it at the same time lowers the errors).

2 4 6 8 10 12 14 2 4 6 8 10 12 Resolution PMI N = 100 N = 200 N = 500 N = 1000 N = 2000 N = 5000 N = 10000 N = 20000 N = 50000 N = 100000 N = 200000 N = 500000

Figure 3.9: Kraskov-Grassberger estimate of PMI as a function of the resolution Ψ(N)−Ψ(k) at the accumulation pointrcof the logistic map, with added noise of order 10−12. Settling time and time gapτ are all 104_{; nearest neighbour index 1≤k≤20.}

mation dimension D, D = D− = D+ = D−+. Therefore the information codimension Γ,

which controls the slope, is simply unity.

We do indeed see that the trends follow the slope of unity, but then begin to decline. This is unexpected in that an apparent decline can be interpreted as the start of convergence towards some finite PMI value. By definition at rc the periodicity is infinite and thus PMI should not stop increasing.

A possible reason is that the (floored) finite precision value with which we approximaterc will necessarily give a finite periodicity. Yet when this is tried for the very low approxima- tions torc, associated with periodicities visible on the scales given above, PMI converges in an abrupt manner, very different to the one observed in figure 3.9. Also herercis given to 94 decimal places, which by trial and error we know to give the period (whether regular or chaotic) higher than the range of observed ordinate values.

Neither is it the case that our resolution limits the ‘visible’ periodic dependencies, since then PMI would settle with resolution in the same sudden manner as described above.

In order to understand what factors effect the change in slope we vary several pa- rameters. Figure 3.10 shows results for higher values of ts, and τ. Plots are seen to follow the expected slope for longer, and from examining further variations we conclude that the

2 4 6 8 10 12 1 2 3 4 5 6 7 8 9 10 11 12 Resolution PMI

Figure 3.10: Same details and legend as for figure 3.9, but settling time and time gapτ are all 109_.

main cause of this is τ.

The reason why increasing τ leads us to resolve higher periodicities lies in the specific way we collect data. The methodology section above justified using a single trajectory and tak- ing consecutive time steps as independent initial positions on the attractor. We thus have at most (N +τ) sequential datapoints, of which we collect, again, at most 2N.

The way the trajectory arranges itself on the attractor is related to its fractal nature. Ev- ery second point of the trajectory will be in some portion of the state space. Every fourth point will come back closer. Every eighth point will be even closer. Thus to detect higher periodicities a longer and longer trajectory is needed. As a result the maximum resolvable periodic will be a function of (N +τ). The further the ‘past’ and ‘future’ are separated, the better will be the resolution of the underlying attractor. In order to see the plots begin to deviate from the expect slope a higher resolution range is needed for a higher τ.

It is also interesting to see the step-wise manner in which the plots increase for the low end of the resolution scale. To some extent this is equivalent to the oversampling part of the plots when PMI was computed using the binning strategies. Here increasing the resolution only changes the PMI when the effective neighbourhood size is small enough to only resolve the higher periodicities. The fact that the jump appears discontinuous indicates that there is a spatial gap between points that are near to each other every pth _{step and those that}

are near to each other every (p+ 1)th _{step. Especially in the first figure 3.9 it is possible} to see that the jumps correspond to I(τ) = log(p), as expected from the period-doubling behaviour.

3.3

Permanently Persistent Mutual Information

PPMI, or permanently persistent mutual information, is defined as

I(∞) = lim

τ→∞I(τ). (3.29)

It represents the information that does not get eroded away but is ultimately preserved across time. We consider PPMI in the context of measures of (strong) emergence. This section relates the observed PMI for the logistic map to PPMI, and concludes with the corresponding analysis of the tent map as another 1D example.

For the majority of the logistic map regimes I(τ, ts,resolution) decayed with τ to some constant asymptotic value. The speed of this is varied but was generally much slower at values of r corresponding to chaotic bands of high periodicity. Also, unless the settling time was high enough PMI would display (otherwise transient) peaks after period-doublings. Convergence speed also varied between chaotic and regular regimes - being almost instan- taneous in the latter. We conclude that for most regimes the limit defined in the equation above does in fact exist, though when the underlying measures are fractal the definition above needs to be supplemented by some (finite) resolution at which the infinite τ limit is taken. We make the same assumption for the tent map.