SINGULARITY THEORY AND FORCED SYMMETRY BREAKING IN EQUATIONS

A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

SINGULARITY THEORY and FORCED SYMMETRY BREAKING IN EQUATIONS

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bifurcation and singularity analysis of a molecular network for the induction of long-term memory.

Withdrawal reflexes of the mollusk Aplysia exhibit sensitization, a simple form of long-term memory (LTM). Sensitization is due, in part, to long-term facilitation (LTF) of sensorimotor neuron synapses. LTF is induced by the modulatory actions of serotonin (5-HT). Pettigrew et al. developed a computational model of the nonlinear intracellular signaling and gene network that underlies the induction of 5-HT-induced LTF. The model simulated empirical observations that repeated applications of 5-HT induce persistent activation of protein kinase A (PKA) and that this persistent activation requires a suprathreshold exposure of 5-HT. This study extends the analysis of the Pettigrew model by applying bifurcation analysis, singularity theory, and numerical simulation. Using singularity theory, classification diagrams of parameter space were constructed, identifying regions with qualitatively different steady-state behaviors. The graphical representation of these regions illustrates the robustness of these regions to changes in model parameters. Because persistent protein kinase A (PKA) activity correlates with Aplysia LTM, the analysis focuses on a positive feedback loop in the model that tends to maintain PKA activity. In this loop, PKA phosphorylates a transcription factor (TF-1), thereby increasing the expression of an ubiquitin hydrolase (Ap-Uch). Ap-Uch then acts to increase PKA activity, closing the loop. This positive feedback loop manifests multiple, coexisting steady states, or multiplicity, which provides a mechanism for a bistable switch in PKA activity. After the removal of 5-HT, the PKA activity either returns to its basal level (reversible switch) or remains at a high level (irreversible switch). Such an irreversible switch might be a mechanism that contributes to the persistence of LTM. The classification diagrams also identify parameters and processes that might be manipulated, perhaps pharmacologically, to enhance the induction of memory. Rational drug design, to affect complex processes such as memory formation, can benefit from this type of analysis.

应用力学进展：祝贺郑哲敏先生八十华诞应用力学报告会

    本书为祝贺郑哲敏先生八十华诞的学术报告会的文集，其中收录邀请报告12篇，定向征文58篇。这些论文涉及爆炸力学、岩土力学、冲击力学、材料力学性能、生物力学、物理力学、海洋工程力学、环境流体力学等几大方面，绝大多数为论文作者科研项目的最新成果。

 

 

NONDEGENERATE UMBILICS, THE PATH FORMULATION and GRADIENT BIFURCATION PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Path formulation for Z(2) circle plus Z(2)-equivariant bifurcation problems

M. Manoel and I. Stewart 0101) classify Z(2) circle plus Z(2)-equivariant bifurcation problems up to codimension 3 and 1 modal parameter, using the classical techniques of singularity theory of Golubistky and Schaeffer [8]. In this paper we classify these same problems using an alternative form: the path formulation (Theorem 6.1). One of the advantages of this method is that the calculates to obtain the normal forms are easier. Furthermore, in our classification we observe the presence of only one modal parameter in the generic core. It differs from the classical classification where the core has 2 modal parameters. We finish this work comparing our classification to the one obtained in [10].

D-n-forced symmetry breaking of O(2)-equivariant problems

We use singularity theory to classify forced symmetry-breaking bifurcation problemsf(z, lambda, mu) = f(1)(z, lambda) + muf(2)(z, lambda, mu) = 0,where f(1) is O(2)-equivariant and f(2) is D-n-equivariant with the orthogonal group actions on z is an element of R-2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

double-struck D signn-forced symmetry breaking of double-struck O sign (2)-equivariant problems

We use singularity theory to classify forced symmetry-breaking bifurcation problems f(z, λ, μ) = f1 (z, λ) + μf2(z, λ, μ) = 0, where f1 is double-struck O sign (2)-equivariant and f2 is double-struck D sign n-equivariant with the orthogonal group actions on z ∈ ℝ2. Forced symmetry breaking occurs when the symmetry of the equation changes when parameters are varied. We explicitly apply our results to the branching of subharmonic solutions in a model periodic perturbation of an autonomous equation and sketch further applications.

Path formulation for multiparameter D3-equivariant bifurcation problems

We implement a singularity theory approach, the path formulation, to classify D3-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a Ba-miniversal unfolding f0 of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of F0 onto its unfolding parameter space. We apply our results to degenerate bifurcation of period-3 subharmonics in reversible systems, in particular in the 1:1-resonance.

O papel da equivalência de contato na Cr-classificação de germes de aplicações Cr-estáveis

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Teoria de singularidades e classificação de problemas de bifurcação Z2-equivariantes de Corank 2

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Problemas de bifurcação de corank2 com dois parâmetros e a formulação por caminhos

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Contorno aparente, envoltórias e equações diferenciais implícitas

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Colored diffraction catastrophes.

On fine scales, caustics produced with white light show vividly colored diffraction fringes. For caustics described by the elementary catastrophes of singularity theory, the colors are characteristic of the type of singularity. We study the diffraction colors of the fold and cusp catastrophes. The colors can be simulated computationally as the superposition of monochromatic patterns for different wavelengths. Far from the caustic, where the luminosity contrast is negligible, the fringe colors persist; an asymptotic theory explains why. Experiments with caustics produced by refraction through irregular bathroom-window glass show good agreement with theory. Colored fringes near the cusp reveal fine lines that are not present in any of the monochromatic components; these lines are explained in terms of partial decoherence between rays with widely differing path differences.