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Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.[2] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).[3] A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.[4][5][6]

An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.[33]

Sharkovskii's theorem is the basis of the Li and Yorke[35] (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.

Unlike fixed-point attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set, which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and the fractal dimension can be calculated for them.

Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961.[13] Lorenz and his collaborator Ellen Fetter[75] were using a simple digital computer, a Royal McBee LGP-30, to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.[76] Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions.

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Second, we analyzed rate constancy among lineages using Tajima's relative rate test [37]. When the concatenated sequences were considered, results showed that the null hypothesis of rate constancy was rejected in almost all pairs of contrasts (P < 0.01). It is noteworthy that the F-genome evolved at a faster rate and the G-genome evolved at a slower rate than other genomes (Additional data file 4). To explore the potential impact of rate heterogeneity on tree reconstruction, we adopted the RY-coding strategy that discards fast-evolving transitions and consequently makes phylogenetic reconstructions less susceptible to uneven occurrence of multiple hits among lineages [34, 38]. The tree obtained from the re-coded data set was topologically identical to that shown in Figure 1 (Table 2). To further test the potential long-branch attraction effect of the fast-evolving F-genome, we identified genes that evolved more rapidly in the F-genome than in the A-, B-, and C-genomes. We calculated the ratio of the mean distance between the F-genome and each of A-, B-, and C-genomes to the mean distances among A-, B-, and C-genomes for each gene. We then progressively excluded fast-evolving genes of the F-genome in a decreasing order of the ratios. The topology based on the remaining genes did not change until more than 50 genes were excluded (Additional data file 5). These suggest that rate heterogeneity was not severe enough to cause significant systematic bias.

The following additional data are available with the online version of this paper. Additional data file 1 is a figure showing the relative location on rice chromosomes of the 142 genes sampled in this study. Additional file 2 is a table listing the detailed information on each of 142 loci. Additional file 3 is a table listing the GC content variation among lineages and the result of Chi-square test for the concatenated data set. Additional file 4 is a table summarizing the Tajima's relative rate test for concatenated sequences using Leersia as outgroup, with estimates of the ratio of substitution rate between lineages. Additional file 5 includes figures showing the results of testing the effect of rate bias caused by the fast-evolving genes of the F-genome. Additional file 6 is a table summarizing 14 alternative models used in BI analyses. Additional file 7 is a table indicating the effect of model components on model fit judged by Bayes factor comparisons of competing models. Additional file 8 is a figure showing the consensus networks of a collection of 106 optimal ML trees from the 106 genes with the complete set of seven species, applying thresholds of 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3, respectively. Additional file 9 is a table presenting the results of the ILD test for pairwise comparisons of process partitions. Additional file 10 is a table summarizing the number of genes that failed the ILD test with the target gene at P < 0.01 for total, intron, exon and the third codon sites, respectively, and the P value of the ILD test between the target gene and all the rest of genes. Additional file 11 is a table presenting the topologies of bootstrap 75% majority-rule consensus trees by different methods of analyses for each gene. Additional file 12 is a figure showing saturation analyses in the concatenated datasets of total, intron, exon, and third codon positions, respectively. Additional file 13 is a table summarizing the proportions of topology (or clades) identical to those shown in Figure 1 inferred from randomly sampled genes or sites in 500 replicates. Additional file 14 is a table listing the primers for PCR amplification and the GenBank accession numbers of the sequences of 142 loci sampled in this present study.

Additional data file 8: Consensus networks of a collection of 106 optimal ML trees from the 106 genes with the complete set of seven species, applying thresholds of 0.05, 0.1, 0.15, 0.2, 0.25 and 0.3, respectively. (PDF 25 KB)

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