• Korean Journal of Mathematics
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• 제9권1호
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• pp.29-43
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• 2001

In this paper we study a Hopf bifurcation in a parabolic multiple free boundary problem. We are dealing with the following problem.

• Journal of Mechanical Science and Technology
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• 제14권1호
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• pp.48-56
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• 2000

The bifurcation problem of thin walled structures in contact with rigid surfaces is formulated by adopting the multiple scales asymptotic technique. The general theory developed in this paper is very useful for the bifurcation analysis of waviness instabilities in the sheet metal forming. The formulation is presented in a full Lagrangian formulation. Through this general formulation, the bifurcation functional is derived within an error of O($(E^4)$) (E: shell's thickness parameter). This functional can be used in numerical solutions to sheet metal forming instability problem.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1191-1205
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• 2008

We consider a delayed predator prey model. The local stability and Hopf bifurcation results are stated taking the time delay as a control parameter. We apply multiple scale analysis to analyze the effects of additive white noises near the Hopf bifurcation point at the positive interior equilibrium state. The governing equations for the amplitude of oscillations on a slow time scale are derived. We identify the process of amplitude of oscillations and derive its transient properties. We show that oscillations, which would decay in the deterministic system whenever time delay lies below its critical value, persists for long time under the validity of multiple scale analysis.

• 대한기계학회논문집B
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• 제21권2호
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• pp.252-263
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• 1997

Steady-state two-dimensional natural convective heat transfer in horizontal cylindrical annuli was studied by solving the governing equations based on the primitive variables. Emphasis was put on the occurrence of the multiple solutions at a given set of parameter values, and on the determination of the bifurcation points at which those multiple solutions begin to branch out. The multicellular flow pattern from the results of melting process in an isothermally heated horizontal cylinder for high Rayleigh numbers, was used as initial guesses for the field variables. This was succeeded in new bifurcation point to tetracellular solutions for an identical set of parameter variables of previous works. The close examination of flow pattern transition around bifurcation point was also conducted. It was found that the mechanisms of flow transition are different depending on the critical Rayleigh number of bifurcation point.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제12권3호
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• pp.179-191
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• 2005

In this paper we consider a Dirichlet problem in the unit disk. We show that the equation has a unique or multiple solutions according to the range of the parameter. Moreover, we prove that the equation admits a nonradial bifurcation at each branch of radial solutions.

• Journal of Mechanical Science and Technology
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• 제14권1호
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• pp.39-47
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• 2000

Bifurcation intability modes of axially compressed circular cylindrical shell are investigated in the limit of zero thickness (i.e., h (thickness) ${\rightarrow}$ 0) analytically, adopting the general stability theory developed by Triantafyllidis and Kwon (1987) and Kwon (1992). The primary state of the shell is obtained in a closed form using the asymptotic technique, and then the straight-forward bifurcation analysis is followed according to the general stability theory to obtain the bifurcation modes in the limit of zero thickness in a full analytical manner. Hence, the closed form bifurcation solution is obtained. Finally, the result is compared with the classical one.

• 대한기계학회논문집B
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• 제28권6호
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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.

• 대한수학회보
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• 제53권3호
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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.

• Structural Engineering and Mechanics
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• 제40권3호
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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.

• Structural Engineering and Mechanics
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• 제16권2호
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• pp.141-152
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• 2003

Characteristics of solutions of softening plasticity are discussed in this article. The localized and non-localized solutions are obtained for a three-bar truss and their stability is evaluated with the aid of the second-order work. Beyond the bifurcation point, the single stable loading path splits into several post-bifurcation paths and the second-order work exhibits several competing minima. Among the multiple post-bifurcation equilibrium states, the localized solutions correspond to the minimum points of the second-order work, while the non-localized solutions correspond to the saddles and local maximum points. To determine the real post-bifurcation path, it is proposed that the structure should follow the path corresponding to the absolute minimum point of the second-order work. The proposal is further proved equivalent to Bazant's path criterion derived on a thermodynamics basis.