Topic

# Bifurcation

About: Bifurcation is a research topic. Over the lifetime, 9676 publications have been published within this topic receiving 135763 citations.

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TL;DR: In this article, a simple homotopy is constructed by the modified Lindstedt-Poincare method, by the solution and the coefficient of linear term are expanded into series of the embedding parameter.

Abstract: A simple homotopy is constructed, by the modified Lindstedt-Poincare method(He,J.H. International Journal of Non-Linear Mechanics , 37, 2002, 309-314 ), the solution and the coefficient of linear term are expanded into series of the embedding parameter. Only one iteration leads to accurate solution.

907 citations

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TL;DR: In this article, the singularities caused by grazing impact are studied using analytical methods, and it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter, a special type of bifurcation occurs.

Abstract: The motion of a single-degree-of-freedom, periodically forced oscillator subjected to a rigid amplitude constraint is considered. Using analytical methods, the singularities caused by grazing impact are studied. It is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter, a special type of bifurcation occurs. The motion after the bifurcation may be non-periodic, and a criterion for this based on orientation and eigenvalues is given.

711 citations

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TL;DR: In this article, a basic tensile bifurcation problem is studied for an incompressible rectangular block constrained to undergo plane deformations, where the sides of the block are traction-free and it is elongated by a uniform, shear-free, relative displacement of its ends.

Abstract: A basic tensile bifurcation problem is studied. Bifurcations from a state of homogeneous inplane tension are investigated for an incompressible rectangular block constrained to undergo plane deformations. The sides of the block are traction-free, and it is elongated by a uniform, shearfree, relative displacement of its ends. For a wide class of incrementally-linear, time-independent materials only two instantaneous moduli enter into the analysis. Symmetric and anti-symmetric bifurcations are examined in each of the characteristic regimes of the governing equations (elliptic, parabolic and hyperbolic). Both diffuse modes and localized shearing modes are considered. Lowest bifurcation stresses are computed for essentially the entire range of possible combinations of material properties and geometry. A number of limiting cases are studied in detail, including those for slender and stubby specimens and for specimens which are rigid in shear. Applications to elastic and elastic/plastic solids are discussed.

620 citations

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TL;DR: A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described, with the emphasis on algorithms based on pseudoarclength continuation for ordinary differential equations.

Abstract: A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for ordinary differential equations. Several illustrative examples computed with the AUTO software package are included. This is Part II of the paper that appeared in the preceding issue [Doedel et al., 1991] and that mainly dealt with algebraic problems.

555 citations