scholarly journals Bifurcation analysis for degenerate problems with mixed regime and absorption

2020 ◽  
pp. 2050017
Author(s):  
Ahmed Alsaedi ◽  
Vicenţiu D. Rădulescu ◽  
Bashir Ahmad
Keyword(s):  
Differential Operator ◽  
Bifurcation Analysis ◽  
Transonic Flow ◽  
Combined Effects ◽  
Bifurcation Problem ◽  
Nontrivial Solutions ◽  
Degenerate Problems ◽  
Mixed Regime ◽  
Degenerate Operator ◽  
Bifurcation Behavior

We are concerned with the study of a bifurcation problem driven by a degenerate operator of Baouendi–Grushin type. Due to its degenerate structure, this differential operator has a mixed regime. Studying the combined effects generated by the absorption and the reaction terms, we establish the bifurcation behavior in two cases. First, if the absorption nonlinearity is dominating, then the problem admits solutions only for high perturbations of the reaction. In the case when the reaction dominates the absorption term, we prove that the problem admits nontrivial solutions for all the values of the parameter. The analysis developed in this paper is associated with patterns describing transonic flow restricted to subsonic regions.

scholarly journals The bifurcation analysis and optimal feedback mechanism for an SIS epidemic model on networks

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Lijuan Chen ◽  
Shouying Huang ◽  
Fengde Chen ◽  
Mingjian Fu
Keyword(s):  
Optimal Control ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Feedback Mechanism ◽  
Optimal Control Strategy ◽  
Optimal Feedback ◽  
Sis Epidemic Model ◽  
Vital Effect ◽  
Bifurcation Behavior ◽  
Sis Epidemic

AbstractIt is well known that the feedback mechanism or the individual’s intuitive response to the epidemic can have a vital effect on the disease’s spreading. In this paper, we investigate the bifurcation behavior and the optimal feedback mechanism for an SIS epidemic model on heterogeneous networks. Firstly, we present the bifurcation analysis when the basic reproduction number is equal to unity. The direction of bifurcation is also determined. Secondly, different from the constant coefficient in the existing literature, we incorporate a time-varying feedback mechanism coefficient. This is more reasonable since the initiative response of people is constantly changing during different process of disease prevalence. We analyze the optimal feedback mechanism for the SIS epidemic network model by applying the optimal control theory. The existence and uniqueness of the optimal control strategy are obtained. Finally, a numerical example is presented to verify the efficiency of the obtained results. How the topology of the network affects the optimal feedback mechanism is also discussed.

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.

DESIGN-ORIENTED BIFURCATION ANALYSIS OF POWER ELECTRONICS SYSTEMS

2011 ◽  
Vol 21 (06) ◽  
pp. 1523-1537 ◽  
Author(s):  
CHI K. TSE ◽  
MING LI
Keyword(s):  
Hopf Bifurcation ◽  
Power Electronics ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Current Status ◽  
Main Difficulty ◽  
Design Problems ◽  
Practical Design ◽  
Underlying Causes ◽  
Bifurcation Behavior

Bifurcation analysis has been applied to many power electronics circuits. Literature abounds with results regarding the various ways in which such circuits lose stability under variation of some selected parameters, e.g. via period-doubling bifurcation, Hopf bifurcation, border collision, etc. The current status of research in the identification of bifurcation behavior in power electronics has reached a stage where the salient types of bifurcation behavior, their underlying causes and the theoretical parameters affecting them have been well understood. Currently, the emphasis of research in this field has gradually shifted toward applications that are of direct relevance to practical design of power electronics. One direction is to apply some of the available research results in bifurcation behavior to the design of practical power electronics circuits. The main difficulty is that the abstract mathematical presentations of the available results are not directly applicable to practical design problems. In this paper we will discuss how research efforts may be directed to bridge this gap.

Multiple Solutions for Nonlinear Periodic Problems

2013 ◽  
Vol 56 (2) ◽  
pp. 366-377 ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Periodic Problem ◽  
Point Theory ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Periodic Problems ◽  
Superlinear Growth ◽  
Special Case ◽  
Nonlinear Nonhomogeneous Differential Operator

Abstract We consider a nonlinear periodic problem driven by a nonlinear nonhomogeneous differential operator and a Carathéodory reaction term f (t; x) that exhibits a (p – 1)-superlinear growth in x 2 R near 1 and near zero. A special case of the differential operator is the scalar p-Laplacian. Using a combination of variational methods based on the critical point theory with Morse theory (critical groups), we show that the problem has three nontrivial solutions, two of which have constant sign (one positive, the other negative).

scholarly journals Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Youpei Zhang
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Smooth Solution ◽  
Singular Term ◽  
Variable Method ◽  
Combined Effects ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Nonhomogeneous Differential Operator

Abstract We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation $f(z,x,y)$ f ( z , x , y ) which is only locally defined in $x \in {\mathbb {R}} $ x ∈ R . Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.

scholarly journals Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

2020 ◽  
Vol 10 (1) ◽  
pp. 636-658
Author(s):  
Fuliang Wang ◽  
Die Hu ◽  
Mingqi Xiang
Keyword(s):  
Sobolev Inequality ◽  
Nehari Manifold ◽  
Combined Effects ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Singular Nonlinearity ◽  
Existence And Multiplicity ◽  
Singular Nonlinearities ◽  
The Nehari Manifold

Abstract The aim of this paper is to study the existence and multiplicity of solutions for a class of fractional Kirchho problems involving Choquard type nonlinearity and singular nonlinearity. Under suitable assumptions, two nonnegative and nontrivial solutions are obtained by using the Nehari manifold approach combined with the Hardy-Littlehood-Sobolev inequality.

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 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.

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