• Title, Summary, Keyword: bifurcation

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AN ESCAPE CRITERION FOR THE COMPLEX POLYNOMIAL, WITH APPLICATIONS TO THE DEGREE-n BIFURCATION SET

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.1
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    • pp.7-14
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    • 2003
  • Let $P_c(z)=z^n+c$ be a complex polynomial with an integer $n{\geq}2$. We derive a criterion that the critical orbit of $P_c$ escapes to infinity and investigate its applications to the degree-n bifurcation set. The intersection of the degree-n bifurcation set with the real line as well as with a typical symmetric axis is explicitly written as a function of n. A well-defined escape-time algorithm is also included for the improved construction of the degree-n bifurcation set.

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Bifurcation Behaviours of Composite Tubes With Two Different Materials Subjected To Uniform Radial Shrinkage At The External Surface (외주에 균일한 압축을 받는 두꺼운 복합원관의 분지거동)

  • ;;Tomita,Y.
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.14 no.2
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    • pp.269-275
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    • 1990
  • Nonaxisymmetric bifurcation behaviours of composite tubes two different materials subjected to uniform radial shrinkage at the external surface have been investigated and compared with those of single tube. The effect of material parameters normalized with respect to those of outer tube upon the bifurcation point and corresponding mode has been clarified. The parameters substantially affect the bifurcation mode with long-wavelength so that the composite tube with low hardening exponent or with high yield stress of inner tube destabilizes the overall deformation of the tube. However surface type bifurcation, short-wavelength mode, shown on the traction-free inner surface is hardly affected by the material parameters. The surface type bifurcation completely depends on the material characteristics of inner tube and the bifurcation point of composite tube almost coincides with the of single tube.

BIFURCATIONS OF A PREDATOR-PREY SYSTEM WITH WEAK ALLEE EFFECTS

  • Lin, Rongzhen;Liu, Shengqiang;Lai, Xiaohong
    • Journal of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.695-713
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    • 2013
  • We formulate and study a predator-prey model with non-monotonic functional response type and weak Allee effects on the prey, which extends the system studied by Ruan and Xiao in [Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), no. 4, 1445-1472] but containing an extra term describing weak Allee effects on the prey. We obtain the global dynamics of the model by combining the global qualitative and bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the supercritical and the subcritical Hopf bifurcations, and the homoclinic bifurcation, as the values of parameters vary. In the generic case, the model has the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation).

A Study on the Concept and Spatial Organization of Bifurcation (분기(Bifurcation)의 개념과 공간조직에 관한 연구)

  • Kim Jong-Jin
    • Korean Institute of Interior Design Journal
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    • v.14 no.1
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    • pp.20-27
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    • 2005
  • This study is focused on the concept and spatial organization of bifurcation. After discussing the concept of bifurcation used in Borges' literature and Deleuze's Fold philosophy, case examples in contemporary architecture are analysed to comparatively investigate the relationship between the concept and space. In Deleuze's philosophy, bifurcation as well as pleats, inflection are used to form the world of fold that goes to infinity while, in Borges' literature, the structure of bifurcation is the key method to create the labyrinth of time. There are various projects in contemporary architecture based on the Deleuzian concept of bifurcation. Rem Koolhaas's Two Libraries for Jussieu University and UN Studio's Arnhem Central are selected and researched for further comparison study. In Jussieu project, the bifurcating spatial organization is 'intentionally' used to construct the indeterminant space whereas in Arnhem Central, bifurcation can be found in both the ever-bifurcating design process as well as the final spatial organization'unintentionally'generated from the process. This study is concluded with the comparative analysis between the representation and actualization of a concept that are crucially different.

INTERSECTION OF THE DEGREE-n BIFURCATION SET WITH THE REAL LINE

  • Geum, Young-Hee;Kim, Young-Ik
    • The Pure and Applied Mathematics
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    • v.9 no.2
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    • pp.113-118
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    • 2002
  • Definition and some properties of the degree-n bifurcation set are introduced. It is proved that the interval formed by the intersection of the degree-n bifurcation set with the real line is explicitly written as a function of n. The functionality of the interval is computationally and geometrically confirmed through numerical examples. Our study extends the result of Carleson & Gamelin [2].

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BIFURCATION THEORY FOR A CIRCULAR ARCH SUBJECT TO NORMAL PRESSURE

  • Bang, Keumseong;Go, JaeGwi
    • Korean Journal of Mathematics
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    • v.14 no.1
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    • pp.113-123
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    • 2006
  • The arches may buckle in a symmetrical snap-through mode or in an asymmetry bifurcation mode if the load reaches a certain value. Each bifurcation curve develops as pressure increases. The governing equation is derived according to the bending theory. The balance of forces provides a nonlinear equilibrium equation. Bifurcation theory near trivial solution of the equation is developed, and the buckling pressures are investigated for various spring constants and opening angles.

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General Asymptotic Formulation for the Bifurcation Problem of Thin Walled Structures in Contact with Rigid Surfaces

  • Kwon, Young-Joo;Triantafyllidis, N.
    • Journal of Mechanical Science and Technology
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    • v.14 no.1
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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.

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Dangerous Border-collision Bifurcation for a Piecewise Smooth Nonlinear System

  • Kang, Hunseok
    • Kyungpook Mathematical Journal
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    • v.52 no.4
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    • pp.459-472
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    • 2012
  • A piecewise smooth system is characterized by non-differentiability on a curve in the phase space. In this paper, we discuss particular bifurcation phenomena in the dynamics of a piecewise smooth system. We consider a two-dimensional piecewise smooth system which is composed of a linear map and a nonlinear map, and analyze the stability of the system to determine the existence of dangerous border-collision bifurcation. We finally present some numerical examples of the bifurcation phenomena in the system.

Bifurcation analysis of over-consolidated clays in different stress paths and drainage conditions

  • Sun, De'an;Chen, Liwen;Zhang, Junran;Zhou, Annan
    • Geomechanics and Engineering
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    • v.9 no.5
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    • pp.669-685
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    • 2015
  • A three-dimensional elastoplastic constitutive model, also known as a UH model (Yao et al. 2009), was developed to describe the stress-strain relationship for normally consolidated and over-consolidated soils. In this paper, an acoustic tensor and discriminator of bifurcation for the UH model are derived for the strain localization of saturated clays under undrained and fully and partially drained conditions. Analytical analysis is performed to illustrate the points of bifurcation for the UH model with different three-dimensional stress paths. Numerical analyses of cubic specimens for the bifurcation of saturated clays under undrained and fully and partially drained conditions are conducted using ABAQUS with the UH model. Analytical and numerical analyses show the similar bifurcation behaviour of overconsolidated clays in three-dimensional stress states and various drainage conditions. The results of analytical and numerical analyses show that (1) the occurrence of bifurcation is dependent on the stress path and drainage condition; and (2) bifurcation can appear in either a strain-hardening or strain-softening regime.