scholarly journals Bifurcation Analysis of a Coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-Type Model

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Lei Shi
Keyword(s):  
Asymptotic Behavior ◽  
Bifurcation Analysis ◽  
Trivial Solution ◽  
Type Model ◽  
Bifurcation Parameter ◽  
Local Behavior ◽  
Nontrivial Solutions ◽  
Ginzburg Landau ◽  
The Stability ◽  
Bifurcation And Stability

We study the bifurcation and stability of trivial stationary solution(0,0)of coupled Kuramoto-Sivashinsky- and Ginzburg-Landau-type equations (KS-GL) on a bounded domain(0,L)with Neumann's boundary conditions. The asymptotic behavior of the trivial solution of the equations is considered. With the lengthLof the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches is studied, and the stability of the bifurcated solutions is analyzed as well.

BIFURCATION ANALYSIS OF THE SWIFT–HOHENBERG EQUATION WITH QUINTIC NONLINEARITY

2009 ◽  
Vol 19 (09) ◽  
pp. 2927-2937 ◽  
Author(s):  
QINGKUN XIAO ◽  
HONGJUN GAO
Keyword(s):  
Asymptotic Behavior ◽  
Bifurcation Analysis ◽  
Trivial Solution ◽  
Center Manifold ◽  
Local Behavior ◽  
Bifurcation Diagrams ◽  
Nontrivial Solutions ◽  
One Dimensional ◽  
Quintic Nonlinearity ◽  
Dimensional Domain

This paper is concerned with the asymptotic behavior of the solutions u(x,t) of the Swift–Hohenberg equation with quintic nonlinearity on a one-dimensional domain (0, L). With α and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches are also studied. Global bounds for the solutions u(x,t) are established and then the global attractor is investigated. Finally, with the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are closer, and their stabilities are analyzed.

BIFURCATION ANALYSIS OF AN AMPLITUDE EQUATION

2013 ◽  
Vol 23 (05) ◽  
pp. 1350081
Author(s):  
LEI SHI ◽  
HONGJUN GAO
Keyword(s):  
Neumann Boundary ◽  
Stationary Solutions ◽  
Amplitude Equation ◽  
Parabolic Partial Differential Equations ◽  
Bifurcation Parameter ◽  
Local Behavior ◽  
Nontrivial Solutions ◽  
Amplitude Equations ◽  
The Stability ◽  
Bifurcation And Stability

We study the bifurcation and stability of constant stationary solutions (u0, v0) of a particular system of parabolic partial differential equations as amplitude equations on a bounded domain (0, L) with Neumann boundary conditions. In this paper, the asymptotic behavior of the solutions (u0, v0) of the amplitude equations are considered. With the length L of the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches are studied. We also analyze the stability of the bifurcated solutions.

Bifurcation in the Swift–Hohenberg Equation

2015 ◽  
Vol 11 (3) ◽  
Author(s):  
Qingkun Xiao ◽  
Hongjun Gao
Keyword(s):  
Asymptotic Behavior ◽  
Trivial Solution ◽  
Control Parameter ◽  
Center Manifold ◽  
Cylindrical Domain ◽  
Local Behavior ◽  
Bifurcation Diagrams ◽  
Nontrivial Solutions ◽  
Bifurcation Points ◽  
Quintic Polynomial

This paper is concerned with the asymptotic behavior of the solutions u(x, t) of the Swift–Hohenberg equation with quintic polynomial on the cylindrical domain Q=(0,L)×R+. With the control parameter α in the Swift–Hohenberg equation and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches is also investigated. With the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are close, and their stabilities are analyzed.

BIFURCATION ANALYSIS OF THE 1D AND 2D GENERALIZED SWIFT–HOHENBERG EQUATION

2010 ◽  
Vol 20 (03) ◽  
pp. 619-643 ◽  
Author(s):  
HONGJUN GAO ◽  
QINGKUN XIAO
Keyword(s):  
Boundary Conditions ◽  
Bifurcation Analysis ◽  
Trivial Solution ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Spatial Dimension ◽  
Nontrivial Solutions ◽  
Asymptotic Expressions ◽  
Spatial Dimensions ◽  
The Stability

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed.

scholarly journals Assessing the Impacts of Competition and Dispersal on a Multiple Interactions Type Model

2021 ◽  
Vol 29 (3) ◽  
Author(s):  
Murtala Bello Aliyu ◽  
Mohd Hafiz Mohd ◽  
Mohd Salmi Md. Noorani
Keyword(s):  
Bifurcation Analysis ◽  
Species Coexistence ◽  
Real Life ◽  
Period Doubling ◽  
Type Model ◽  
Coexistence Mechanisms ◽  
Multiple Species ◽  
The Stability ◽  
Transcritical Bifurcations ◽  
Multiple Interactions

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems.

On Post-Critical Behavior of a Beam on an Elastic Foundation

2018 ◽  
Vol 18 (06) ◽  
pp. 1850082 ◽  
Author(s):  
Lidija Z. Rehlicki ◽  
Marko B. Janev ◽  
Branislava N. Novaković ◽  
Teodor M. Atanacković
Keyword(s):  
Total Energy ◽  
Elastic Foundation ◽  
Nonlinear Problem ◽  
Bifurcation Analysis ◽  
Nontrivial Solutions ◽  
Buckling Mode ◽  
Nonlinear Equilibrium ◽  
The Stability ◽  
The One ◽  
Two Parameter

In this paper, we analyze the nonlinear equilibrium equation corresponding to the two-parameter bifurcation problem arising in the stability analysis of an elastic simply supported beam on the Winkler type elastic foundation for the case when bimodal buckling occurs. We perform the bifurcation analysis of the nonlinear problem, by using Lyapunov–Schmidt reduction, thus obtaining the number of the nontrivial solutions to the nonlinear problem and qualitatively characterizing the solution patterns. We also give the formulation of the problem and bifurcation analysis from the total energy viewpoint and determine the energy of each bifurcating solution. We assert that the solution with the smallest energy is the one that will be observed in the post-critical state. For specific choice of parameters, the bifurcating solution in the form of the second buckling mode has the smallest total energy. The numerical results illustrating the theory are also provided.

Bifurcation and Stability in Dissipative Media (Plasticity, Friction, Fracture)

1994 ◽  
Vol 47 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Quoc Son Nguyen
Keyword(s):  
Fracture Mechanics ◽  
Bifurcation Analysis ◽  
Recent Literature ◽  
Mechanical Systems ◽  
Stability And Bifurcation Analysis ◽  
Dissipative Media ◽  
The Stability ◽  
Bifurcation And Stability ◽  
Static Behaviour ◽  
Unified Description

This paper addresses stability and bifurcation analysis for common systems of solids in the framework of plasticity, of friction and of fracture mechanics. Although physically different, usual time-independent laws adopted in these domains lead to a certain mathematical similarity concerning the quasi-static behaviour of materials and structures. These mechanical systems can be described practically in the same mathematical manner concerning their quasi-static evolution and in particular concerning the stability of their response. Our objective is to present within this framework a review of principal results of the recent literature on these subjects in relation with some energy-related considerations and with an unified description based upon energy and dissipation analysis.

APPLICATION OF A TWO-DIMENSIONAL HINDMARSH–ROSE TYPE MODEL FOR BIFURCATION ANALYSIS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350055 ◽  
Author(s):  
SHYAN-SHIOU CHEN ◽  
CHANG-YUAN CHENG ◽  
YI-RU LIN
Keyword(s):  
Hopf Bifurcation ◽  
Bifurcation Analysis ◽  
Sufficient Conditions ◽  
Type Model ◽  
Two Dimensional ◽  
Saddle Node Bifurcation ◽  
Trapping Region ◽  
Saddle Node ◽  
The Stability ◽  
Two Equilibria

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

Dynamics in a Van Der Pol Model with Delay

2012 ◽  
Vol 204-208 ◽  
pp. 4586-4589
Author(s):  
Chang Jin Xu ◽  
Pei Luan Li ◽  
Ling Yun Yao
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Parameter ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Van Der Pol Model ◽  
The Stability

In this paper, the dynamics of a van der pol model with delay are considered. It is shown that the asymptotic behavior depends crucially on the time delay parameter. By regarded the delay as a bifurcation parameter, we are particularly interested in the study of the Hopf bifurcation problem. The length of delay which preserves the stability of the equilibrium is calculated. Some numerical simulations for justifying the analytical findings are included.

Hopf Bifurcation Analysis in a Modified Optically Injected Semiconductor Lasers Model

2014 ◽  
Vol 631-632 ◽  
pp. 254-260
Author(s):  
Jiang Ang Zhang ◽  
Wen Ju Du ◽  
Kutorzi Edwin Yao
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Semiconductor Lasers ◽  
Bifurcation Analysis ◽  
Autonomous System ◽  
Dynamic Properties ◽  
Equilibrium Points ◽  
Bifurcation Parameter ◽  
Numerical Example ◽  
The Stability

In this paper, a modified optically injected semiconductor lasers model is studied in detail. More precisely, we study the stability of the equilibrium points and basic dynamic properties of the autonomous system by means of nonlinear dynamics theory. The existence of Hopf bifurcation is investigated by choosing the appropriate bifurcation parameter. Furthermore, formulas for determining the stability and the conditions for generating Hopf bifurcation of the equilibria are derived. Then, a numerical example is given.

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