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 studied the Hopf bifurcation in the Volterra model and showed that there is a stable periodic solution in the small, but for parameters such that a long time delay is required to make the equilibrium point locally unstable there is no such stable solution.
Abstract: The model studied is the Volterra prey-predator model with prey population limited to lie below the carrying capacity, and with time delay treated as in paper I by adding an extra ordinary differential equation. This makes it possible to examine periodic solutions in the small by means of the theory of the Hopf bifurcation. It is found that for parameters such that short time delay makes the equilibrium point locally unstable there is a stable periodic solution in the small, but for parameters such that a long time delay is required to make the equilibrium point locally unstable there is no such stable solution.

50 citations

Journal ArticleDOI
TL;DR: Dynamical behaviors of the new system are analyzed, both theoretically and numerically, including some basic dynamical properties, stability and Hopf bifurcation, which shows that both methods justified the same results.

50 citations

Journal ArticleDOI
TL;DR: In this article, the authors use the geometrical characterization given by the change from an unstable to a stable focus through a center for a basic (piecewise) linear system to find two mechanisms for the destabilizing of the basic stationary solution and for the generation of bifurcating periodic orbits.
Abstract: Hopf bifurcation for smooth systems is characterized by a crossing of a pair of complex conjugate eigenvalues of the linearized problem through the imaginary axis. Since this approach is not at hand for non-smooth systems, we use the geometrical characterization given by the change from an unstable to a stable focus through a centre for a basic (piecewise) linear system. In that way we find two mechanisms for the destabilizing of the basic stationary solution and for the generation of bifurcating periodic orbits: a generation switch of the stability properties or the influence of the unstable subsystem measured by the time of duration spent in the subsystem. The switch between stable and unstable subsystems seems to be a general source of destabilization observed in several mechanical systems. We expect that the features analysed for planar systems will help us to understand higher-dimensional systems as well.

50 citations

Journal ArticleDOI
TL;DR: In this paper, the singularity-induced bifurcation (SIB) was shown to arise in parameter dependent differential-algebraic equations (DAEs) of the form x/spl dot/=f and 0=g, and which occurs when an equilibrium path of the DAE crosses the singular surface defined by g=0 and det g/sub y/=0.
Abstract: It has been shown recently that there is a new type of codimension one bifurcation, called the singularity-induced bifurcation (SIB), arising in parameter dependent differential-algebraic equations (DAEs) of the form x/spl dot/=f and 0=g, and which occurs generically when an equilibrium path of the DAE crosses the singular surface defined by g=0 and det g/sub y/=0. The SIB refers to a stability change of the DAE owing to some eigenvalue of a related linearization diverging to infinity when the Jacobian g/sub y/ is singular. In this article an improved version (Theorem 1.1) of the SIB theorem with its simple proof is given, based on a decomposition theorem (Theorem 2.1) of parameter dependent polynomials.

50 citations

Journal ArticleDOI
TL;DR: This paper theoretically demonstrate and numerically evidence that the emergence of geometric power series in the vicinity of simple bifurcation points is a generic behavior, and proposes to use this hallmark as a bIfurcation indicator to locate and compute very efficiently any simple b ifurcation point.

50 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