spectrum [2.166]. The largest real Lyapunov exponent was positive and characterized the overriding behavior of symmetry changes during valve formation. For all taxa, sum of the positive Lyapunov exponents signified chaotic asymmetric behavior characterizing instability (Table 2.2). The remainder of the real Lyapunov exponents in descending order were very small (large negative values), summed, and signified stability (Table 2.2). For Lyapunov exponents calculated from KS entropy, some random instability was present during valve simulation as well (Table 2.2).
Figure 2.16 Cyclotella meneghiniana symmetry for normal and abnormal valves.
Figure 2.17 Cyclotella meneghiniana symmetry values for normal forming and external vegetative and initial valves.
The change in symmetry was compared with the change in chaotic instability for taxa used in the valve formation simulation. Symmetry increased monotonically in the order of least symmetric to most symmetric: Actinoptychus splendens, Cyclotella meneghiniana, Arachnoidiscus ornatus, Aulacodiscus oregonus, Coscinodiscus sp., Arachnoidiscus ehrenbergii (1), Arachnoidiscus ehrenbergii forming valve, Asterolampra marylandica, Arachnoidiscus ehrenbergii (2), and Actinoptychus senarius (Table 2.2). Rescaling symmetry and chaotic instability values, a comparative bar graph showed that chaotic instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella meneghiniana, Arachnoidiscus ornatus, and Aulacodiscus oregonus than higher symmetric taxa with the exception of Arachnoidiscus ehrenbergii (2) (Figure 2.23). Rescaled random instability was more evident in less symmetric taxa Actinoptychus splendens, Cyclotella meneghiniana, and Aulacodiscus oregonus as well, in contrast to higher symmetric taxa, with the exception of the Arachnoidiscus ehrenbergii forming valve (Figure 2.24). Overall, less symmetry is associated with higher chaotic and random instability (Figures 2.23 and 2.24).
Figure 2.18 Asterolampra marylandica (SanDBay02_03dx1100) as a sequence of 24 cumulative concentric rings simulating valve formation from annulus to valve margin.
2.4 Discussion
From average symmetry per taxon for all valves, triangular and non-regular structured surface circular taxa exhibited less symmetry than highly regularly structured circular and four-edged or greater polygonal taxa (Figure 2.12). By inspection, the result is evident, and the method of using entropy to determine symmetry states has explicit utility. Additionally, although we measured rotational symmetry, other symmetries are present as well (Figure 2.3). Separate tests for each type of symmetry could be made, and quantification of each type of symmetry would need to be instituted to determine the contribution of a particular symmetry throughout centric diatom development.
Figure 2.19 Plots of symmetry vs. 24-valve formation simulation steps from annulus to valve margin for eight centric diatom taxa. Order from least to highest symmetry curve per taxon is read left to right in legend. Symmetry increases exponentially from the annulus to the completed valve.
Figure 2.20 Valve formation steps for eight taxa as a probability density function of entropy values resulting in a gamma distribution.
Figure 2.21 Valve formation steps for eight taxa as a cumulative density function of entropy values.
Figure 2.22 Valve formation steps for eight taxa as a skew-right normal prior probability density function of entropy values.
Table 2.2 Valve formation simulation for eight centric diatom taxa and characterization of instability and stability from Lyapunov exponents. Positive sums indicate chaotic instability, negative sums indicate stability, and Lyapunov exponents from Kolmogorov-Sinai (KS) entropy indicate random instability.
| Taxon* | File name | Chaotic Instability: Sum of + Lyapunov exponents | Stability: Sum of - Lyapunov exponents | Random Instability: Lyapunov exponent from K-S entropy |
| Actinoptychus senarius | motewx1500_2 | 5.0775 | -429.424 | 4399.39 |
| Actinoptychus splendens | MisBy5_1hx2000 | 4.987 | -388.348 | 3079.95 |
| Arachnoidiscus ehrenbergii (forming valve) | halfmax500 | 4.96527 | -390.747 | 3186.1 |
| Arachnoidiscus ehrenbergii (1) | provbay5_12ix400 | 5.36916 | -379.328 | 2359.42 |
| Arachnoidiscus ehrenbergii (2) | ProvBay5_12lx450 | 5.04099 | -398.893 | 4656.9 |
| Arachnoidiscus ornatus | PoiDumox700 | 5.18264 | -454.589 | 2669.33 |
| Asterolampra marylandica |
SanDBay02_03dx1100
|