On bifurcation and symmetry of solutions of symmetric nonlinear equations with odd-harmonic forcings

In this work we study existence, bifurcation, and symmetries of small solutions of the nonlinear equation Lx = N(x, p, epsilon) + mu f, which is supposed to be equivariant under the action of a group OHm, and where f is supposed to be OHm-invariant. We assume that L is a linear operator and N(., p, epsilon) is a nonlinear operator, both defined in a Banach space X, with values in a Banach space Z, and p, mu, and epsilon are small real parameters. Under certain conditions we show the existence of symmetric solutions and under additional conditions we prove that these are the only feasible solutions. Some examples of nonlinear ordinary and partial differential equations are analyzed. (C) 1995 Academic Press, Inc.

Bifurcation of limit cycles from a centre in R-4 in resonance 1:N

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

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.

A note on non-degenerate umbilics and the path formulation for bifurcation problems

Path formulation can be used to classify and structure efficiently multiparameter bifurcation problems around fundamental singularities: the cores. The non-degenerate umbilic singularities are the generic cores for four situations in corank 2: the general or gradient problems and the ℤ 2-equivariant (general or gradient) problems. Those categories determine an interesting 'Russian doll' type of structure in the universal unfoldings of the umbilic singularities. One advantage of our approach is that we can handle one, two or more parameters using the same framework (even considering some special parameter structure, for instance, some internal hierarchy). We classify the generic bifurcations that occur in those cases with one or two parameters.

3-Dimensional hopf bifurcation via averaging theory

We consider the Lorenz system ẋ = σ(y - x), ẏ = rx - y - xz and ż = -bz + xy; and the Rössler system ẋ = -(y + z), ẏ = x + ay and ż = b - cz + xz. Here, we study the Hopf bifurcation which takes place at q± = (±√br - b,±√br - b, r - 1), in the Lorenz case, and at s± = (c+√c2-4ab/2, -c+√c2-4ab/2a, c±√c2-4ab/2a) in the Rössler case. As usual this Hopf bifurcation is in the sense that an one-parameter family in ε of limit cycles bifurcates from the singular point when ε = 0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two 2-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other 3- or n-dimensional differential systems.

Some remarks on bifurcation analysis of a nonlinear vibrating system excited by a shape memory alloy material (SMA)

In this paper, the dynamical response of a coupled oscillator is investigated, taking in consideration the nonlinear behavior of a SMA spring coupling the two oscillators. Due to the nonlinear coupling terms, the system exhibits both regular and chaotic motions. The Poincaré sections for different sets of coupling parameters are verified. © 2011 World Scientific Publishing Company.

The dual currency bifurcation of Cuba's economy in the 1990s: causes, consequences and cures

Includes bibliography

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set. © 2012 Elsevier Ltd.

Hopf bifurcation in the full repressilator equations

In this paper, we prove that the full repressilator equations in dimension six undergo a supercritical Hopf bifurcation.

The Hopf bifurcation in the Shimizu-Morioka system

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Characterization of grazing bifurcation in airfoils with control surface freeplay nonlinearity

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

On bifurcation of solutions of the Yamabe problem in product manifolds

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds. (C) 2011 Elsevier Masson SAS. All rights reserved.

Twelve-month clinical outcomes after coronary stenting with the Genous Bio-engineered R Stent in patients with a bifurcation lesion: from the e-HEALING (Healthy Endothelial Accelerated Lining Inhibits Neointimal Growth) registry

Background The e-Healthy Endothelial Accelerated Lining Inhibits Neointimal Growth (e-HEALING) registry was designed to capture clinical data on the use of the endothelial progenitor cell capture stent (ECS) in routine clinical practice. In this analysis, we investigated the 12-month clinical outcomes in patients treated with an ECS for a bifurcation lesion. Methods The worldwide, prospective, nonrandomized e-HEALING registry aimed to enrol 5000 patients treated for coronary artery disease with one or more ECS between October 2005 and October 2007. Clinical follow-up was obtained at 1, 6, and 12 months. The primary endpoint was target vessel failure (TVF), defined as the composite of cardiac death, myocardial infarction, and target vessel revascularization at 12 months. Results A total of 573 patients were treated for at least one bifurcation lesion and were assessed in the current analysis. Baseline characteristics showed a median age of 65 years; 21% were diabetic patients and 36% had unstable angina. A total of 63% of the bifurcation lesions were located in the left artery descending and the mean stent length was 20.7 +/- 12.6 mm. At 12 months, TVF was 12.7% and target lesion revascularization was 7.5%. Definite or probable stent thrombosis occurred in 1.7% of the patients. Moreover, one or more stents per lesion [hazard ratio (HR): 2.79, 95% confidence interval (CI): 1.60-4.86, P < 0.001], predilatation (HR: 0.39, 95% CI: 0.17-0.87, P = 0.023), and lesions located in the right coronary artery (HR: 4.56, 95% CI: 1.07-19.5, P = 0.041) were independent predictors of TVF. Conclusion In the e-HEALING registry, coronary bifurcation stenting with the ECS results in favorable clinical outcomes and low incidences of repeat revascularization and stent thrombosis. Coron Artery Dis 23:201-207 (C) 2012 Wolters Kluwer Health vertical bar Lippincott Williams & Wilkins.