Topic
Bifurcation diagram
About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.
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TL;DR: In this article, the authors focus on the characteristics of these different transitions to MMOs and try to construct a bifurcation diagram, showing that the transition from a simple periodic, period doubled, or chaotic attractor arising from a Feigenbaum route constitutes an interior crisis.
46 citations
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TL;DR: The bIfurcation sequence leading to a snaking bifurcation diagram linking single localized states to "localized patterns" or clusters of localized states is shown and a parameter region where cluster states are inhibited is demonstrated.
Abstract: We report on experimental observations of homoclinic snaking in a vertical-cavity semiconductor optical amplifier. Our observations in a quasi-one-dimensional and two-dimensional configurations agree qualitatively well with what is expected from recent theoretical and numerical studies. In particular, we show the bifurcation sequence leading to a snaking bifurcation diagram linking single localized states to ``localized patterns'' or clusters of localized states and demonstrate a parameter region where cluster states are inhibited.
46 citations
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TL;DR: In this article, the authors apply the multiple-scale method to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point, illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force.
Abstract: The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.
46 citations
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01 Jan 1978
TL;DR: In this paper, the existence and structure of branches of bifurcation is derived and a new and compact proof of the existence of multiple branches is derived, where the dependence of the solution on a naturally occurring parameter is replaced by the dependence on a form of pseudo-arclength.
Abstract: I. Existence and Structure of Bifurcation Branches The problem of bifurcation is formulated as an operator equation
in a Banach space, depending on relevant control parameters, say of
the form G(u,λ) = 0. If dimN(G_u(u_O,λ_O)) = m the method of Lyapunov-Schmidt
reduces the problem to the solution of m algebraic equations.
The possible structure of these equations and the various types of
solution behaviour are discussed. The equations are normally derived
under the assumption that G^O_λeR(G^O_u). It is shown, however, that
if G^O_λeR(G^O_u) then bifurcation still may occur and the local structure
of such branches is determined. A new and compact proof of the
existence of multiple bifurcation is derived. The linearized
stability near simple bifurcation and "normal" limit points is then
indicated. II. Constructive Techniques for the Generation of Solution Branches A method is described in which the dependence of the solution
arc on a naturally occurring parameter is replaced by the dependence
on a form of pseudo-arclength. This results in continuation procedures
through regular and "normal" limit points. In the neighborhood
of bifurcation points, however, the associated linear operator
is nearly singular causing difficulty in the convergence of continuation
methods. A study of the approach to singularity of this
operator yields convergence proofs for an iterative method for determining
the solution arc in the neighborhood of a simple bifurcation
point. As a result of these considerations, a new constructive
proof of bifurcation is determined.
46 citations
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TL;DR: In this paper, the authors considered a three-dimensional delayed differential equation representing a bidirectional associate memory (BAM) neural network with three neurons and two discrete delays and obtained the pitchfork bifurcation curve of the system.
Abstract: In this paper, we consider a three-dimensional delayed differential equation representing a bidirectional associate memory (BAM) neural network with three neurons and two discrete delays. By analyzing the number and stability of equilibria, the pitchfork bifurcation curve of the system is obtained. Furthermore, on the pitchfork bifurcation curve, by using the sum of two delays as the bifurcation parameter, we find that the system can undergo a Hopf bifurcation at the origin and the three-dimensional ordinary differential equation describing the flow on the center manifold is given.
46 citations