scispace - formally typeset
Search or ask a question
Topic

Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.


Papers
More filters
Journal ArticleDOI
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

Journal ArticleDOI
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

Journal ArticleDOI
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

Dissertation
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

Journal ArticleDOI
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


Network Information
Related Topics (5)
Nonlinear system
208.1K papers, 4M citations
89% related
Differential equation
88K papers, 2M citations
88% related
Partial differential equation
70.8K papers, 1.6M citations
88% related
Eigenvalues and eigenvectors
51.7K papers, 1.1M citations
85% related
Boundary value problem
145.3K papers, 2.7M citations
84% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023122
2022326
2021187
2020195
2019166
2018220