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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01 Jan 1973
TL;DR: In this paper, the mathematical problems of hydrodynamic stability and topological degree theory and applications are discussed. But the real world is not considered. And there is no solution to the nonlinear elliptic boundary value problems of second order.
Abstract: Nonlinear elliptic boundary value problems of second order.- Functional analysis.- Bifurcation at a simple eigenvalue.- Bifurcation of periodic solutions.- The mathematical problems of hydrodynamic stability.- Topological degree theory and applications.- The real world.
605 citations
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
TL;DR: In this article, the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action was studied and it was shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease would disappear if the starting position lies outside this region.
524 citations
TL;DR: It is analytically shown that transients from a state of incoherence firing can be immediate and the stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived.
Abstract: An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized "locked" solutions is analyzed.
470 citations
TL;DR: A predator-prey system with nonmonotonic functional response is considered and global qualitative and bifurcation analyses are combined to determine the global dynamics of the model.
Abstract: A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation a...
458 citations