Scaling of the Critical Function for the Standard Map: Some Numerical Results

The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.Comment: 26 pages, 3 figures, 13 table

Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis

We study the behaviour of the Standard map critical function in a neighbourhood of a fixed resonance, that is the scaling law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances $p/q$, $q \geq 2$, $p \neq 0$ and $p$ and $q$ relatively prime, the scaling law follows a power--law with exponent $1/q$.Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit

Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviors of these two PT-symmetric dynamical models in complex phase space exhibit strong qualitative similarities.Comment: 22 page, 16 figure

On the phylogeny of Mustelidae subfamilies: analysis of seventeen nuclear non-coding loci and mitochondrial complete genomes

<p>Abstract</p> <p>Background</p> <p>Mustelidae, as the largest and most-diverse family of order Carnivora, comprises eight subfamilies. Phylogenetic relationships among these Mustelidae subfamilies remain argumentative subjects in recent years. One of the main reasons is that the mustelids represent a typical example of rapid evolutionary radiation and recent speciation event. Prior investigation has been concentrated on the application of different mitochondrial (mt) sequence and nuclear protein-coding data, herein we employ 17 nuclear non-coding loci (>15 kb), in conjunction with mt complete genome data (>16 kb), to clarify these enigmatic problems.</p> <p>Results</p> <p>The combined nuclear intron and mt genome analyses both robustly support that Taxidiinae diverged first, followed by Melinae. Lutrinae and Mustelinae are grouped together in all analyses with strong supports. The position of Helictidinae, however, is enigmatic because the mt genome analysis places it to the clade uniting Lutrinae and Mustelinae, whereas the nuclear intron analysis favores a novel view supporting a closer relationship of Helictidinae to Martinae. This finding emphasizes a need to add more data and include more taxa to resolve this problem. In addition, the molecular dating provides insights into the time scale of the origin and diversification of the Mustelidae subfamilies. Finally, the phylogenetic performances and limits of nuclear introns and mt genes are discussed in the context of Mustelidae phylogeny.</p> <p>Conclusion</p> <p>Our study not only brings new perspectives on the previously obscured phylogenetic relationships among Mustelidae subfamilies, but also provides another example demonstrating the effectiveness of nuclear non-coding loci for reconstructing evolutionary histories in a group that has undergone rapid bursts of speciation.</p

Multigene phylogeny of the Mustelidae: Resolving relationships, tempo and biogeographic history of a mammalian adaptive radiation

<p>Abstract</p> <p>Background</p> <p>Adaptive radiation, the evolution of ecological and phenotypic diversity from a common ancestor, is a central concept in evolutionary biology and characterizes the evolutionary histories of many groups of organisms. One such group is the Mustelidae, the most species-rich family within the mammalian order Carnivora, encompassing 59 species classified into 22 genera. Extant mustelids display extensive ecomorphological diversity, with different lineages having evolved into an array of adaptive zones, from fossorial badgers to semi-aquatic otters. Mustelids are also widely distributed, with multiple genera found on different continents. As with other groups that have undergone adaptive radiation, resolving the phylogenetic history of mustelids presents a number of challenges because ecomorphological convergence may potentially confound morphologically based phylogenetic inferences, and because adaptive radiations often include one or more periods of rapid cladogenesis that require a large amount of data to resolve.</p> <p>Results</p> <p>We constructed a nearly complete generic-level phylogeny of the Mustelidae using a data matrix comprising 22 gene segments (~12,000 base pairs) analyzed with maximum parsimony, maximum likelihood and Bayesian inference methods. We show that mustelids are consistently resolved with high nodal support into four major clades and three monotypic lineages. Using Bayesian dating techniques, we provide evidence that mustelids underwent two bursts of diversification that coincide with major paleoenvironmental and biotic changes that occurred during the Neogene and correspond with similar bursts of cladogenesis in other vertebrate groups. Biogeographical analyses indicate that most of the extant diversity of mustelids originated in Eurasia and mustelids have colonized Africa, North America and South America on multiple occasions.</p> <p>Conclusion</p> <p>Combined with information from the fossil record, our phylogenetic and dating analyses suggest that mustelid diversification may have been spurred by a combination of faunal turnover events and diversification at lower trophic levels, ultimately caused by climatically driven environmental changes. Our biogeographic analyses show Eurasia as the center of origin of mustelid diversity and that mustelids in Africa, North America and South America have been assembled over time largely via dispersal, which has important implications for understanding the ecology of mustelid communities.</p