Slow Passage Through a Hopf Bifurcation: From Oscillatory to Steady State Solutions

1993 ◽  
Vol 53 (4) ◽  
pp. 1045-1058 ◽  
Author(s):  
Lisa Holden ◽  
Thomas Erneux
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Slow Passage ◽  
Steady State Solutions

Bifurcation Phenomena in a Lotka–Volterra Model with Cross-Diffusion and Delay Effect

2017 ◽  
Vol 27 (07) ◽  
pp. 1750105 ◽  
Author(s):  
Shuling Yan ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Volterra Model ◽  
Delay Effect ◽  
Cross Diffusion ◽  
Bifurcation Phenomena ◽  
Bifurcation Direction ◽  
Steady State Solutions

This paper focuses on a Lotka–Volterra model with delay and cross-diffusion. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady-state solutions. Furthermore, we obtain some criteria to determine the bifurcation direction and stability of Hopf bifurcating periodic orbits by using Lyapunov–Schmidt reduction.

scholarly journals Hopf bifurcation in a reaction-diffusive-advection two-species competition model with one delay

2021 ◽  
pp. 1-24
Author(s):  
Qiong Meng ◽  
Guirong Liu ◽  
Zhen Jin
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Competition Model ◽  
Dirichlet Boundary Conditions ◽  
Species Competition ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Existence And Multiplicity ◽  
The Stability ◽  
Steady State Solutions

In this paper, we investigate a reaction-diffusive-advection two-species competition model with one delay and Dirichlet boundary conditions. The existence and multiplicity of spatially non-homogeneous steady-state solutions are obtained. The stability of spatially nonhomogeneous steady-state solutions and the existence of Hopf bifurcation with the changes of the time delay are obtained by analyzing the distribution of eigenvalues of the infinitesimal generator associated with the linearized system. By the normal form theory and the center manifold reduction, the stability and bifurcation direction of Hopf bifurcating periodic orbits are derived. Finally, numerical simulations are given to illustrate the theoretical results.

On the magnetic resonance of an alined spin-assembly

1972 ◽  
Vol 330 (1581) ◽  
pp. 265-270 ◽  
Keyword(s):  
Steady State ◽  
Magnetic Resonance ◽  
Equations Of Motion ◽  
Double Resonance ◽  
Relaxation Times ◽  
Spin Orientation ◽  
Optical Double Resonance ◽  
Set Up ◽  
Slow Passage ◽  
Steady State Solutions

The state of a spin-assembly of arbitrary J , undergoing magnetic resonance, is characterized by the multipole components p q k of the instantaneous spin-polarization which describe spin-orientation ( k = 1), spin-alinement ( k = 2), etc. Equations of motion analogous to Bloch’s equations ( k = 1) are set up for the multipole components of different k , introducing terms which describe phenomenologically ( a ) the pumping of the longitudinal multipole components ( q = 0), and ( b ) the independent but anisotropic relaxation of multipole components of different k . Steady-state solutions are obtained. In particular, the slow-passage magnetic resonance functions for the alinement components, which involve three relaxation times, are calculated explicitly. For the particular case of isotropic relaxation, these resonance functions reduce to the form originally derived for optical double resonance for a J = 1 assembly. It is emphasized that the damping constant which is involved is that for alinement.

Patterns in a Modified Leslie–Gower Model with Beddington–DeAngelis Functional Response and Nonlocal Prey Competition

2020 ◽  
Vol 30 (05) ◽  
pp. 2050074 ◽  
Author(s):  
Jianping Gao ◽  
Shangjiang Guo
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Response ◽  
Steady State Solution ◽  
State Solution ◽  
Bifurcation Curve ◽  
Homogeneous Steady State ◽  
Steady State Bifurcation ◽  
Steady State Solutions

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions.

Explicit steady state solutions for a particularM(x)/M/1 queueing system

1977 ◽  
Vol 24 (4) ◽  
pp. 651-659 ◽  
Author(s):  
George L. Jensen ◽  
Albert S. Paulson ◽  
Pasquale Sullo
Keyword(s):  
Steady State ◽  
Queueing System ◽  
Steady State Solutions

Nonlinear Vibrations of Viscoelastic Plane Truss Under Harmonic Excitation

2014 ◽  
Vol 14 (04) ◽  
pp. 1450009 ◽  
Author(s):  
Andrew Yee Tak Leung ◽  
Hong Xiang Yang ◽  
Ping Zhu
Keyword(s):  
Steady State ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Homotopy Continuation ◽  
Response Curves ◽  
System A ◽  
Structural Responses ◽  
Voigt Model ◽  
Two Degree Of Freedom ◽  
Steady State Solutions

This paper is concerned with the steady state bifurcations of a harmonically excited two-member plane truss system. A two-degree-of-freedom Duffing system having nonlinear fractional derivatives is derived to govern the dynamic behaviors of the truss system. Viscoelastic properties are described by the fractional Kelvin–Voigt model based on the Caputo definition. The combined method of harmonic balance and polynomial homotopy continuation is adopted to obtain steady state solutions analytically. A parametric study is conducted with the help of amplitude-response curves. Despite its seeming simplicity, the mechanical system exhibits a wide variety of structural responses. The primary and sub-harmonic resonances and chaos are found in specific regions of system parameters. The dynamic snap-through phenomena are observed when the forcing amplitude exceeds some critical values. Moreover, it has been shown that, suppression of undesirable responses can be achieved via changing of viscosity of the system.

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