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 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.

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.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

2009 ◽  
Vol 19 (11) ◽  
pp. 3733-3751 ◽  
Author(s):  
SUQI MA ◽  
ZHAOSHENG FENG ◽  
QISHAI LU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Value Of Time ◽  
Asymptotically Stable ◽  
Universal Unfolding ◽  
Double Hopf Bifurcation ◽  
Bifurcation Points ◽  
The Rose

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

Seismic Response Analysis on a Chaotic System

2013 ◽  
Vol 639-640 ◽  
pp. 911-916
Author(s):  
Cui Xiang Liang
Keyword(s):  
Seismic Response ◽  
Chaotic System ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Response Analysis ◽  
Bifurcation Diagrams ◽  
Seismic Response Analysis ◽  
Nonlinear Seismic Response ◽  
Hyperbolic Equilibrium ◽  
Hyperbolic Equilibrium Point

This paper is concerned with the dynamical behavior of a chaotic system which is a model for seismic response of structures. The local bifurcation of the non-hyperbolic equilibrium point of the chaotic system is investigated by using center manifold method. The transcritical bifurcation is analyzed in detail. Based on numerical simulations, spectrums of maximal Lyapunov exponent and the bifurcation diagrams are presented for the dynamic analysis. The method proposed can be used as a reference of nonlinear seismic response analysis.

scholarly journals Regularization of the Boundary-Saddle-Node Bifurcation

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Xia Liu
Keyword(s):  
Vector Field ◽  
Center Manifold ◽  
Continuous System ◽  
Critical Value ◽  
Continuous Systems ◽  
Local Behavior ◽  
Saddle Node Bifurcation ◽  
Node Point ◽  
Perturbation Problem ◽  
Saddle Node

In this paper we treat a particular class of planar Filippov systems which consist of two smooth systems that are separated by a discontinuity boundary. In such systems one vector field undergoes a saddle-node bifurcation while the other vector field is transversal to the boundary. The boundary-saddle-node (BSN) bifurcation occurs at a critical value when the saddle-node point is located on the discontinuity boundary. We derive a local topological normal form for the BSN bifurcation and study its local dynamics by applying the classical Filippov’s convex method and a novel regularization approach. In fact, by the regularization approach a given Filippov system is approximated by a piecewise-smooth continuous system. Moreover, the regularization process produces a singular perturbation problem where the original discontinuous set becomes a center manifold. Thus, the regularization enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation.

Diffusive Interaction Leading to Dephasing of Coupled Neural Oscillators

1997 ◽  
Vol 07 (04) ◽  
pp. 869-876 ◽  
Author(s):  
Seung Kee Han ◽  
Christian Kurrer ◽  
Yoshiki Kuramoto
Keyword(s):  
Asymptotic Behavior ◽  
Phase Velocity ◽  
Limit Cycle ◽  
Coupled Oscillators ◽  
Control Parameter ◽  
Critical Value ◽  
Homoclinic Bifurcation ◽  
Neural Oscillators ◽  
Diffusive Coupling ◽  
Synchronization Of Oscillators

It is usually believed that strong diffusive coupling in one of the dynamical variables is well-suited for imposing synchronization of oscillators. But it was recently shown that weak diffusive coupling, counter-intuitively, can lead to dephasing of coupled neural oscillators. In this paper, we investigate how diffusively coupled oscillators become dephasing. For this we study a system of coupled neural oscillators on a limit cycle generated through a homoclinic bifurcation. We examine the asymptotic behavior of diffusive coupling as the control parameter approaches the critical value for which the homoclinic bifurcation occurs. In this study, we show that the gradient of phase velocity near the limit cycle is essential in generating dephasing through diffusive interaction.

scholarly journals Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Changzhi Li ◽  
Dhanagopal Ramachandran ◽  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Yongjian Liu
Keyword(s):  
Chaotic Maps ◽  
Period Doubling ◽  
Tipping Points ◽  
Bifurcation Parameter ◽  
Bifurcation Diagrams ◽  
Bifurcation Points ◽  
Slowing Down ◽  
Sine Map ◽  
Autocorrelation Method ◽  
Period Doubling Bifurcations

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

scholarly journals Computational Analysis and Bifurcation of Regular and Chaotic Ca2+ Oscillations

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3324
Author(s):  
Xinxin Qie ◽  
Quanbao Ji
Keyword(s):  
Numerical Simulation ◽  
Center Manifold ◽  
Computational Analysis ◽  
System Model ◽  
Phase Portraits ◽  
Bifurcation Parameter ◽  
Bifurcation Diagrams ◽  
Saddle Node Bifurcation ◽  
The Stability ◽  
Torus Bifurcation

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc.

Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions

1997 ◽  
Vol 07 (07) ◽  
pp. 1691-1698 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Hopf Bifurcation ◽  
Geometric Structure ◽  
Trivial Solution ◽  
Mixed Mode ◽  
Mode Interaction ◽  
Hopf Bifurcations ◽  
Bifurcation Points ◽  
Mode Interactions ◽  
The Stability ◽  
Secondary Bifurcations

In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.

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