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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 the last few years there has been considerable interest in the asymptotic behavior of maps of the interval into itself under iteration as discussed by the authors, and some of the interest has been generated by population biology.
Abstract: In the last few years there has been considerable interest in the asymptotic behavior of maps of the interval into itself under iteration. Some of this interest has come from the theory of dynamical systems (where most authors have studied maps of the circle), and some of the interest has been generated by population biology. In population biology, maps of the unit interval have been used as models for the dynamics of populations with discrete generations. One of the questions of most interest in the theory has been the determination of the limit sets of points for a map f : I -* I. Here I=[0 , 1]. The limit set ofxs t is the set of limit points of the sequence {f~(x)}. The superscript denotes repeated composition. Of particular interest are periodic orbits: points x such that f~(x)= x for some i>0. Even greater interest focuses upon attracting periodic orbits: if

132 citations

Journal ArticleDOI
TL;DR: In this article, a new chemical pattern is discussed, which is a propagationless solitary island in an infinite medium, and its existence and stability for a certain simple model is demonstrated analytically.
Abstract: A new chemical pattern is discussed, which is a propagationless solitary island in an infinite medium We demonstrate analytically its existence and stability for a certain simple model The localization turns out to be a consequence of the rapid diffusion of an inhibiting substance occurring in a potentially excitable system In order to extract thc; important features of the localized pattern, the method of singular perturbation is employed, with the following results: (1) A stable motionless solitary pattern can exist either for a monostablc or bistable system (2) Under suitable conditions such a pattern undergoes the Hop£ bifurcation, leading to a "breathing motion" of the activated droplet The analysis is restricted to the one-dimensional case throughout

131 citations

Journal ArticleDOI
TL;DR: In this paper, the dynamic behavior of nanoscale electrostatic actuators is studied and a two parameter mass-spring model is shown to exhibit a bifurcation from the case excluding an equilibrium point to the case including two equilibrium points as the geometrical dimensions of the device are altered.
Abstract: The dynamic behaviour for nanoscale electrostatic actuators is studied. A two parameter mass-spring model is shown to exhibit a bifurcation from the case excluding an equilibrium point to the case including two equilibrium points as the geometrical dimensions of the device are altered. Stability analysis shows that one is a stable Hopf bifurcation point and the other is an unstable saddle point. In addition, we plot the diagram phases, which have periodic orbits around the Hopf point and a homoclinic orbit passing though the unstable saddle point.

131 citations

Journal ArticleDOI
TL;DR: In this paper, the chaotic dynamics of a micro mechanical resonator with electrostatic forces on both sides is investigated using the Melnikov function, an analytical criterion for homoclinic chaos in the form of an inequality is written in terms of the system parameters.

130 citations

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method for computing the locally closest bifurcation to the nominal parameters of a stable equilibrium in a saddle-node or Hopf bifurbation.
Abstract: Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ0, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ0 to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ0. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ0 of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.

130 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023122
2022326
2021187
2020195
2019166
2018220