4,387 research outputs found
High order amplitude equation for steps on creep curve
We consider a model proposed by one of the authors for a type of plastic
instability found in creep experiments which reproduces a number of
experimentally observed features. The model consists of three coupled
non-linear differential equations describing the evolution of three types of
dislocations. The transition to the instability has been shown to be via Hopf
bifurcation leading to limit cycle solutions with respect to physically
relevant drive parameters. Here we use reductive perturbative method to extract
an amplitude equation of up to seventh order to obtain an approximate analytic
expression for the order parameter. The analysis also enables us to obtain the
bifurcation (phase) diagram of the instability. We find that while
supercritical bifurcation dominates the major part of the instability region,
subcritical bifurcation gradually takes over at one end of the region. These
results are compared with the known experimental results. Approximate analytic
expressions for the limit cycles for different types of bifurcations are shown
to agree with their corresponding numerical solutions of the equations
describing the model. The analysis also shows that high order nonlinearities
are important in the problem. This approach further allows us to map the
theoretical parameters to the experimentally observed macroscopic quantities.Comment: LaTex file and eps figures; Communicated to Phys. Rev.
Turbulence and turbulent pattern formation in a minimal model for active fluids
Active matter systems display a fascinating range of dynamical states,
including stationary patterns and turbulent phases. While the former can be
tackled with methods from the field of pattern formation, the spatio-temporal
disorder of the active turbulence phase calls for a statistical description.
Borrowing techniques from turbulence theory, we here establish a quantitative
description of correlation functions and spectra of a minimal continuum model
for active turbulence. Further exploring the parameter space, we also report on
a surprising type of turbulence-driven pattern formation far beyond linear
onset: the emergence of a dynamic hexagonal vortex lattice state after an
extended turbulent transient, which can only be explained taking into account
turbulent energy transfer across scales.Comment: Supplemental videos available at https://youtu.be/gbf6cRho03w
https://youtu.be/n0qUUhAUJFQ https://youtu.be/LGmamkM012
The Nonlinear Analysis of Perturbation Solution for a Parabolic Differential System
By investigation of perturbation solution for nonlinear reaction-diffusion system, we derive related differential model for perturbations that involves weak nonlinearities up to third order. For a first time, this model is shown to result in derivation of the system for amplitude distribution by means of nonlinear integration on orthogonal basis in spatial region. The obtained time-dependent system (TDS) contains all possible functional relations between the modes of wave train under consideration along with delayed relations, and after numerical simulation it provides some conclusions concerning the natural frequency of the investigated self-organization process in active medium. The related matrix and modulo operations which substantiate the derivation of the TDS are also considered
Coordinated optimization of visual cortical maps : 1. Symmetry-based analysis
In the primary visual cortex of primates and carnivores, functional architecture can be characterized by maps of various stimulus features such as orientation preference (OP), ocular dominance (OD), and spatial frequency. It is a long-standing question in theoretical neuroscience whether the observed maps should be interpreted as optima of a specific energy functional that summarizes the design principles of cortical functional architecture. A rigorous evaluation of this optimization hypothesis is particularly demanded by recent evidence that the functional architecture of orientation columns precisely follows species invariant quantitative laws. Because it would be desirable to infer the form of such an optimization principle from the biological data, the optimization approach to explain cortical functional architecture raises the following questions: i) What are the genuine ground states of candidate energy functionals and how can they be calculated with precision and rigor? ii) How do differences in candidate optimization principles impact on the predicted map structure and conversely what can be learned about a hypothetical underlying optimization principle from observations on map structure? iii) Is there a way to analyze the coordinated organization of cortical maps predicted by optimization principles in general? To answer these questions we developed a general dynamical systems approach to the combined optimization of visual cortical maps of OP and another scalar feature such as OD or spatial frequency preference. From basic symmetry assumptions we obtain a comprehensive phenomenological classification of possible inter-map coupling energies and examine representative examples. We show that each individual coupling energy leads to a different class of OP solutions with different correlations among the maps such that inferences about the optimization principle from map layout appear viable. We systematically assess whether quantitative laws resembling experimental observations can result from the coordinated optimization of orientation columns with other feature maps
The case of the trapped singularities
A case study in bifurcation and stability analysis is presented, in which
reduced dynamical system modelling yields substantial new global and predictive
information about the behaviour of a complex system. The first smooth pathway,
free of pathological and persistent degenerate singularities, is surveyed
through the parameter space of a nonlinear dynamical model for a
gradient-driven, turbulence-shear flow energetics in magnetized fusion plasmas.
Along the route various obstacles and features are identified and treated
appropriately. An organizing centre of low codimension is shown to be robust,
several trapped singularities are found and released, and domains of
hysteresis, threefold stable equilibria, and limit cycles are mapped.
Characterization of this rich dynamical landscape achieves unification of
previous disparate models for plasma confinement transitions, supplies valuable
intelligence on the big issue of shear flow suppression of turbulence, and
suggests targeted experimental design, control and optimization strategies.Comment: 21 pages, 12 figures, 34 postscript figure file
Coupled self-organization: Thermal interaction between two liquid films undergoing long-wavelength instabilities
The effects of thermal coupling between two thin liquid layers, separated by
a gas layer, are discussed. The liquid layers undergo long-wavelength
instabilities driven by gravitational and thermocapillary stresses. To study
the dynamics, both a linear stability analysis and a full numerical solution of
the thin-film equations are performed. The results demonstrate that the
stability properties of the combined system differ substantially from the case
where both layers evolve independently from each other. Most prominently,
oscillatory instabilities, not present in single-liquid layer configurations,
may occur.Comment: 12 pages, 9 figure
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