scholarly journals On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system

2019 ◽  
Vol 24 (7) ◽  
pp. 3077-3088
Author(s):  
Shixing Li ◽  
◽  
Dongming Yan
Keyword(s):  
Steady State ◽  
Steady State Bifurcation

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

Steady-State Bifurcation for a Biological Depletion Model

2016 ◽  
Vol 26 (04) ◽  
pp. 1650066 ◽  
Author(s):  
Yan’e Wang ◽  
Jianhua Wu ◽  
Yunfeng Jia
Keyword(s):  
Steady State ◽  
Bifurcation Theory ◽  
Steady States ◽  
Two Dimensional ◽  
Implicit Function ◽  
Flux Boundary Condition ◽  
One Dimensional ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Theoretical Results

A two-species biological depletion model in a bounded domain is investigated in which one species is a substrate and the other is an activator. Firstly, under the no-flux boundary condition, the asymptotic stability of constant steady-states is discussed. Secondly, by viewing the feed rate of the substrate as a parameter, the steady-state bifurcations from constant steady-states are analyzed both in one-dimensional kernel case and in two-dimensional kernel case. Finally, numerical simulations are presented to illustrate our theoretical results. The main tools adopted here include the stability theory, the bifurcation theory, the techniques of space decomposition and the implicit function theorem.

Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions

2008 ◽  
Vol 3 (2) ◽  
Author(s):  
Ankur Gupta ◽  
Saikat Chakraborty
Keyword(s):  
Steady State ◽  
Pattern Formation ◽  
Three Dimensional ◽  
Diffusion Reaction ◽  
Dynamic Simulations ◽  
Transverse Mixing ◽  
Autocatalytic Reactions ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Parametric Analyses

Interaction between transport and reaction generates a variety of complex spatio-temporal patterns in chemical reactors. These patterned states, which are typically initiated by autocatalytic effects and sustained by differences in diffusion/local mixing rates, often cause undesired effects in the reactor. In this work, we analyze the dynamic evolution of mixing-limited spatial pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors using two-dimensional (2-D) convection-diffusion-reaction (CDR) models that are obtained through rigorous spatial averaging of the three-dimensional (3-D) CDR model using Liapunov-Schmidt technique of bifurcation theory. We use the spatially-averaged 2-D CDR model (and its "regularized" form) to perform steady-state bifurcation analysis that captures the region of multiple solutions, and we analyze the stability of these multiple steady states to transverse perturbations using linear stability analysis. Parametric analyses of the steady-state bifurcation diagrams and stability boundaries show that when transverse mixing is significantly slower than the rate of autocatalytic reaction, mixing-limited patterns emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Our dynamic simulations show the emergence of three different types of spatial patterns namely, Band, Anti-phase and Target, depending on the nature of transverse perturbation. The temporal evolution of these patterns consists of rapid intensification of the concentration-segregation process (especially when transverse mixing is much slower than reaction) followed by slow diffusion-mediated return to symmetry that occurs at time scales much larger than the reactor residence time. Our parametric analysis of the dynamics reveals that while larger Péclet numbers (both axial and transverse) increase the stability and decay time of the patterned states, larger Damköhler numbers lead to faster ignition resulting in the opposite effect.

Bifurcations and Pattern Formation in a Predator–Prey Model

2018 ◽  
Vol 28 (11) ◽  
pp. 1850140 ◽  
Author(s):  
Yongli Cai ◽  
Zhanji Gui ◽  
Xuebing Zhang ◽  
Hongbo Shi ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Global Bifurcation ◽  
Periodic Pattern ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Steady State Bifurcation

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

Sign in / Sign up

Export Citation Format

Share Document