11,688 research outputs found
On the Hyperbolicity of Lorenz Renormalization
We consider infinitely renormalizable Lorenz maps with real critical exponent
and combinatorial type which is monotone and satisfies a long return
condition. For these combinatorial types we prove the existence of periodic
points of the renormalization operator, and that each map in the limit set of
renormalization has an associated unstable manifold. An unstable manifold
defines a family of Lorenz maps and we prove that each infinitely
renormalizable combinatorial type (satisfying the above conditions) has a
unique representative within such a family. We also prove that each infinitely
renormalizable map has no wandering intervals and that the closure of the
forward orbits of its critical values is a Cantor attractor of measure zero.Comment: 63 pages; 10 figure
Basins of Attraction for Chimera States
Chimera states---curious symmetry-broken states in systems of identical
coupled oscillators---typically occur only for certain initial conditions. Here
we analyze their basins of attraction in a simple system comprised of two
populations. Using perturbative analysis and numerical simulation we evaluate
asymptotic states and associated destination maps, and demonstrate that basins
form a complex twisting structure in phase space. Understanding the basins'
precise nature may help in the development of control methods to switch between
chimera patterns, with possible technological and neural system applications.Comment: Please see Ancillary files for the 4 supplementary videos including
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The genotype-phenotype relationship in multicellular pattern-generating models - the neglected role of pattern descriptors
Background: A deep understanding of what causes the phenotypic variation arising from biological patterning
processes, cannot be claimed before we are able to recreate this variation by mathematical models capable of
generating genotype-phenotype maps in a causally cohesive way. However, the concept of pattern in a
multicellular context implies that what matters is not the state of every single cell, but certain emergent qualities
of the total cell aggregate. Thus, in order to set up a genotype-phenotype map in such a spatiotemporal pattern
setting one is actually forced to establish new pattern descriptors and derive their relations to parameters of the
original model. A pattern descriptor is a variable that describes and quantifies a certain qualitative feature of the
pattern, for example the degree to which certain macroscopic structures are present. There is today no general
procedure for how to relate a set of patterns and their characteristic features to the functional relationships,
parameter values and initial values of an original pattern-generating model. Here we present a new, generic
approach for explorative analysis of complex patterning models which focuses on the essential pattern features
and their relations to the model parameters. The approach is illustrated on an existing model for Delta-Notch
lateral inhibition over a two-dimensional lattice.
Results: By combining computer simulations according to a succession of statistical experimental designs,
computer graphics, automatic image analysis, human sensory descriptive analysis and multivariate data modelling,
we derive a pattern descriptor model of those macroscopic, emergent aspects of the patterns that we consider
of interest. The pattern descriptor model relates the values of the new, dedicated pattern descriptors to the
parameter values of the original model, for example by predicting the parameter values leading to particular
patterns, and provides insights that would have been hard to obtain by traditional methods.
Conclusion: The results suggest that our approach may qualify as a general procedure for how to discover and
relate relevant features and characteristics of emergent patterns to the functional relationships, parameter values
and initial values of an underlying pattern-generating mathematical model
The multipliers of periodic points in one-dimensional dynamics
It will be shown that the smooth conjugacy class of an unimodal map which
does not have a periodic attractor neither a Cantor attractor is determined by
the multipliers of the periodic orbits. This generalizes a result by M.Shub and
D.Sullivan for smooth expanding maps of the circle
The effect of functional roles on group efficiency
The usefulness of ‘roles’ as a pedagogical approach to support small group performance can be often read, however, their effect is rarely empirically assessed. Roles promote cohesion and responsibility and decrease so-called ‘process losses’ caused by coordination demands. In addition, roles can increase awareness of intra-group interaction. In this article, the effect of functional roles on group performance, efficiency and collaboration during computer-supported collaborative learning (CSCL) was investigated with questionnaires and quantitative content analysis of e-mail communication. A comparison of thirty-three questionnaire observations, distributed over ten groups in two research conditions: role (n = 5, N = 14) and non-role (n = 5, N = 19), revealed no main effect for performance (grade). A latent variable was interpreted as ‘perceived group efficiency’ (PGE). Multilevel modelling (MLM) yielded a positive marginal effect of PGE. Groups in the role condition appear to be more aware of their efficiency, compared to groups in the ‘non-role’ condition, regardless whether the group performs well or poor. Content analysis reveals that students in the role condition contribute more ‘task content’ focussed statements. This is, however, not as hypothesised due to the premise that roles decrease coordination and thus increase content focused statements; in fact, roles appear to stimulate coordination and simultaneously the amount of ‘task content’ focussed statements increases
Invariant measures for Cherry flows
We investigate the invariant probability measures for Cherry flows, i.e.
flows on the two-torus which have a saddle, a source, and no other fixed
points, closed orbits or homoclinic orbits. In the case when the saddle is
dissipative or conservative we show that the only invariant probability
measures are the Dirac measures at the two fixed points, and the Dirac measure
at the saddle is the physical measure. In the other case we prove that there
exists also an invariant probability measure supported on the quasi-minimal
set, we discuss some situations when this other invariant measure is the
physical measure, and conjecture that this is always the case. The main
techniques used are the study of the integrability of the return time with
respect to the invariant measure of the return map to a closed transversal to
the flow, and the study of the close returns near the saddle.Comment: 12 pages; updated versio
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