• Structural Engineering and Mechanics
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• v.11 no.2
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• pp.199-210
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• 2001

Axially compressed circular cylinders repeat symmetry-breaking bifurcation in the postbuckling region. There exist stable equilibria with all symmetry broken in the buckled configuration, and the minimum postbuckling strength is attained at the deep bottom of closely spaced equilibrium branches. The load level corresponding to such postbuckling stable solutions is usually much lower than the initial buckling load and may serve as a strength limit in shell stability design. The primary concern in the present paper is to compute these possible postbuckling stable solutions at the deep bottom of the postbuckling region. Two computational approaches are used for this purpose. One is the application of individual procedures in computational bifurcation theory. Path-tracing, pinpointing bifurcation points and (local) branch-switching are all applied to follow carefully the postbuckling branches with the decreasing load in order to attain the target at the bottom of the postbuckling region. The buckled shell configuration loses its symmetry stepwise after each (local) branch-switching procedure. The other is to introduce the idea of path jumping (namely, generalized global branch-switching) with static imperfection. The static response of the cylinder under two-parameter loading is computed to enable a direct access to postbuckling equilibria from the prebuckling state. In the numerical example of an elastic perfect circular cylinder, stable postbuckling solutions are computed in these two approaches. It is demonstrated that a direct path jump from the undeformed state to postbuckling stable equilibria is possible for an appropriate choice of static perturbations.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.419-422
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• 2005

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The renormalization-group technique for KAM tori is used to obtain the criteria for large-scale stochasticity in the system.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.140-144
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• 2004

Global bifurcations and chaos in modal interactions of an imperfect circular plate with one-to-one internal resonance are investigated. The case of primary resonance, in which an excitation frequency is near natural frequencies, is considered. The damping force is not included in the analysis. The Melnikov's method for heteroclinic orbits of the autonomous system was used to obtain the criteria for chaotic motion.

• East Asian mathematical journal
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• v.30 no.3
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• pp.335-353
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• 2014

We establish multiple extence of positive solutions for parameterized nonhomogeneous elliptic equations involving critical Sobolev exponent. The approach to the problem is variational method.

• Journal of Korean Society of Steel Construction
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• v.24 no.3
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• pp.337-348
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• 2012

This study investigated the characteristics of bifurcation and the instability due to the initial imperfection of the space truss, which is sensitive to the initial conditions, and the calculated buckling load by the analysis of Eigen-values and the determinant of tangential stiffness. A two-free nodes model, a star dome, and a three-ring dome model were selected as case studies in order to examine the unstable phenomenon due to the sensitivity to Eigen mode, and the influence of the rise-span ratio and the load parameter on the buckling load were analyzed. The sensitivity to the imperfection of the two-free nodes model changed the critical path after reaching the limit point through the bifurcation mode, and the buckling load level was reduced by the increase in the amount of imperfection. The two sensitive buckling patterns for the model can be explained by investigating the displaced position of the free node, and the asymmetric Eigen mode was a major influence on the unstable behavior due to the initial imperfection. The sensitive mode was similar to the in-extensional mechanism basis of the simplified model. Since the rise-span ratio was higher, the effect of local buckling is more prominent than the global buckling in the star dome, and bifurcation on the equilibrium path occurring as the value of the load parameter was higher. Additionally, the buckling load levels of the star dome and the three-ring model were about 50-70% and 80-90% of the limit point, respectively.

• Coupled systems mechanics
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• v.2 no.2
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• pp.159-174
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• 2013

The present work aims at investigating the nonlinear dynamics, bifurcations, and stability of an axially accelerating beam with an intermediate spring-support. The problem of a parametrically excited system is addressed for the gyroscopic system. A geometric nonlinearity due to mid-plane stretching is considered and Hamilton's principle is employed to derive the nonlinear equation of motion. The equation is then reduced into a set of nonlinear ordinary differential equations with coupled terms via Galerkin's method. For the system in the sub-critical speed regime, the pseudo-arclength continuation technique is employed to plot the frequency-response curves. The results are presented for the system with and without a three-to-one internal resonance between the first two transverse modes. Also, the global dynamics of the system is investigated using direct time integration of the discretized equations. The mean axial speed and the amplitude of speed variations are varied as the bifurcation parameters and the bifurcation diagrams of Poincare maps are constructed.

• Journal of the Korean Mathematical Society
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• v.48 no.6
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• pp.1101-1123
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• 2011

Numerical schemes are routinely used to predict the behavior of continuous dynamical systems. All such schemes transform flows into maps, which can possess dynamical behavior deviating from their continuous counterparts. Here the common bifurcations of scalar dynamical systems are transformed under a class of algorithms known as linearized one-point collocation methods. Through the use of normal forms, we prove that each such bifurcation in an originating flow gives rise to an exactly corresponding one in its discretization. The conditions for spurious period doubling behavior under this class of algorithm are derived. We discuss the global behavioral consequences of a singular set induced by the discretizing methods, including loss of monotonicity of solutions, intermittency, and distortion of attractor basins.

• The Pure and Applied Mathematics
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• v.16 no.2
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• pp.243-254
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• 2009

The main purpose of this paper is to use the methods of Lattice Dynamical System to establish a global model, which extends the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets interacting each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and quadratic supplies with naive predictors, and investigate the spatially homogeneous global price dynamics and show that the dynamics is topologically conjugate to that of well-known logistic map and hence undergoes a period-doubling bifurcation route to chaos as a parameter varies through a critical value.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.689-700
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• 1998

We consider an infinite dimensional dynamical system what is called Lattice Dynamical System given by a discrete nonlinear diffusion equation. By assuming the nonlinearity to be a general nonlinear function with mild restrictions, we show that as the diffusion parameter changes the stationery state of the given system undergoes bifurcations from the zero state to a bounded invariant set or a 3- or 4-periodic state in the global phase space of the given system according to the values of the coefficients of the linear part of the given nonlinearity.

• Nonlinear Functional Analysis and Applications
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• v.26 no.3
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• pp.629-648
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• 2021

In this paper, we study a within-host mathematical model of HCV infection and carry out mathematical analysis of the global dynamics and bifurcations of the model in different parameter regimes. We explore the effect of reverse transcriptase inhibitors (RTI) on spontaneous HCV clearance. The model can produce all clinically observed patient profiles for realistic parameter values; it can also be used to estimate the efficacy and/or duration of treatment that will ensure permanent cure for a particular patient. From the results of the model, we infer possible measures that could be implemented in order to reduce the number of infected individuals.