• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.667-680
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• 2016

We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the $S^1$-invariant functions and the $S^1$-invariant linear subspaces.

• Korean Journal of Mathematics
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• v.20 no.3
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• pp.333-341
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• 2012

We investigate the number of solutions for the superlinear elliptic bifurcation problem with Dirichlet boundary condition. We get a theorem which shows the existence of at least $k$ weak solutions for the superlinear elliptic bifurcation problem with boundary value condition. We obtain this result by using the critical point theory induced from invariant linear subspace and the invariant functional.

• Journal of applied mathematics & informatics
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• v.14 no.1_2
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• pp.319-328
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• 2004

In this paper we investigate the qualitative behaviour of numerical approximation to a class delay differential equation. We consider the numerical solution of the delay differential equations undergoing a Hopf bifurcation. We prove the numerical approximation of delay differential equation had a Hopf bifurcation point if the true solution does.

• Journal of Theoretical and Applied Mechanics
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• v.1 no.1
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• pp.29-67
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• 1995

A procedure is formulated, in this paper, to compute the bifurcation modes born by the stability change of normal modes, and to compute the forced responses associated with bifurcation modes in inertially and elastically coupled nonlinear oscillators. It is assumed that a saddle-loop is formed in Poincare map at the stability chage of normal modes. In order to test the validity of procedure, it is applied to one-to-one internal resonant systems in which the solutions are guaranteed within the order of a small perturbation parameter. The procedure is also applied to the exact system in which normal modes are written in exact form and the stability of normal modes can be exactly determined. In this system the stability change of normal modes occurs several times so that various types of bifurcation modes are created. A method is described to identify a fixed point on Poincare map as one of bifurcation modes. The limitations and advantage of proposed procedure are discussed.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.28 no.6
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• pp.697-704
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• 2004

Bifurcation behavior of ignition and extinction of catalytic reaction is theoretically investigated in a stagnation-point flow. Considering that reaction takes place only on the catalytic surface, where conductive heat losses are allowed to occur, activation energy asymptotics with a overall one-step Arrhenius-type catalytic reaction is employed. For the cases with and without the limiting reactant consumption, the analysis provides explicit expressions, which indicate the possibility of multiple steady-state solution branches. The difference between the solutions with and without reactant consumption is in the existence of an upper solution branch, and the neglect of reactant consumption is inappropriate for determining extinction conditions. For larger values of reactant consumption, the solution response is all monotone, suggesting that multiple solutions are not possible. It is shown that bifurcation Damkohler numbers increase (decrease) with increasing of conductive heat loss (gain) on the catalytic surface, which means that smaller (larger) values of the strain rate allow the surface reaction to tolerate larger heat losses (gains). Lewis number of the limiting reactant can also significantly affect bifurcation behavior in a similar way to the effect of heat loss.

• Journal of Electrical Engineering and Technology
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• v.8 no.2
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• pp.206-214
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• 2013

This paper proposes a new method for monitoring local voltage stability using the saddle node bifurcation set or loadability boundary in two dimensional power parameter space. The method includes three main steps. First step is to determine the critical buses and the second step is building the static voltage stability boundary or the saddle node bifurcation set. Final step is monitoring the voltage stability through the distance from current operating point to the boundary. Critical buses are defined through the right eigenvector by direct method. The boundary of the static voltage stability region is a quadratic curve that can be obtained by the proposed method that is combining a variation of standard direct method and Thevenin equivalent model of electric power system. And finally the distance is computed through the Euclid norm of normal vector of the boundary at the closest saddle node bifurcation point. The advantage of the proposed method is that it gets the advantages of both methods, the accuracy of the direct method and simple of Thevenin Equivalent model. Thus, the proposed method holds some promises in terms of performing the real-time voltage stability monitoring of power system. Test results of New England 39 bus system are presented to show the effectiveness of the proposed method.

• Structural Engineering and Mechanics
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• v.5 no.5
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• pp.577-586
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• 1997

In nonlinear finite element stability analysis of structures, the foremost necessary procedure is the computation to precisely locate a singular equilibrium point, at which the instability occurs. The present study describes global and local procedures for the computation of stability points including bifurcation points and limit points. The starting point, at which the procedure will be initiated, may be close to or arbitrarily far away from the target point. It may also be an equilibrium point or non-equilibrium point. Apart from the usual equilibrium path, bypass and homotopy path are proposed as the global path to the stability point. A local iterative method is necessary, when it is inspected that the computed path point is sufficiently close to the stability point.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1347-1372
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• 2020

We study bifurcation for the following fractional Schrödinger equation $$\{\left.\begin{eqnarray}(-{\Delta})^su+V(x)u&=&{\lambda}f(u)&&{\text{in}}\;{\Omega}\\u&>&0&&{\text{in}}\;{\Omega}\\u&=&0&&{\hspace{32}}{\text{in}}\;{\mathbb{R}}^n{\backslash}{\Omega}\end{eqnarray}\right$$ where 0 < s < 1, n > 2s, Ω is a bounded smooth domain of ℝⁿ, (-∆)^s is the fractional Laplacian of order s, V is the potential energy satisfying suitable assumptions and λ is a positive real parameter. The nonlinear term f is a positive nondecreasing convex function, asymptotically linear that is $\lim_{t{\rightarrow}+{\infty}}\;{\frac{f(t)}{t}}=a{\in}(0,+{\infty})$. We discuss the existence, uniqueness and stability of a positive solution and we also prove the existence of critical value and the uniqueness of extremal solutions. We take into account the types of Bifurcation problem for a class of quasilinear fractional Schrödinger equations, we also establish the asymptotic behavior of the solution around the bifurcation point.

• Structural Engineering and Mechanics
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• v.40 no.3
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• pp.335-346
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• 2011

This paper deals with the bifurcations of non-semi-simple eigenvalues at critical point of Hopf bifurcation to understand the dynamic behavior of the system. By using the Puiseux expansion, the expression of the bifurcation of non-semi-simple eigenvalues and the corresponding topological structure in the parameter space are obtained. The zero-order approximate solutions in the vicinity of the critical points at which the multiple Hopf bifurcation may occur are developed. A numerical example, the flutter problem of an airfoil in simplified model, is given to illustrate the application of the proposed method.

• Advances in nano research
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• v.13 no.4
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• pp.379-391
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• 2022

One factor that can heighten the risk of the rapture intracranial aneurysm (IA) is bifurcations, which can cause the IA to evaluate. This study presents the effect of geometric of intracranial vascular on the bifurcation analysis of the aneurysm occurrence. The aneurysm mechanism is mathematically modeled based on the nano pipe structures under the thermal stresses, and the impact of the aneurysm geometric on the stability and bifurcation points is analyzed. Because of the dimension of these structures, the classical theories could not predict their behavior perfectly, so the nonclassical and nonlocal theories are required for the mechanical modeling of the aneurysm. The presented results show that the bifurcation point of the aneurysm mechanism is dependent on the environment temperature, and the temperature change plays an essential role in the stability of these structures.