• 충청수학회지
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• 제16권1호
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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.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.339-350
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• 2006

The bifurcation point where a satellite component buds from another component is characterized by the existence of the common tangent line between the two osculating components appearing in the degree-n bifurcation set. We investigate the existence, location and number of bifurcation points for satellite components budding from the main component in the degree-n bifurcation set as well as a parametric boundary equation of the main component of the degree-n bifurcation set. Cusp points are also located on the boundary of the main component. Typical degree-n bifurcation sets and their components are illustrated with some computational results.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제9권2호
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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］.

• 한국산학기술학회:학술대회논문집
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• 한국산학기술학회 2002년도 춘계학술발표논문집
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• pp.115-117
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• 2002

In this paper, the degree-n bifurcation set as well as the Julia sets is defined by extending the concept of the Mandelbrot set to the complex polynomial $z^{n}{\;}+{\;}c(c{\;}\in{\;}C,{\;}n{\;}\geq{\;}2)$. Some properties of the degree-n bifurcation set and the Julia sets have been theoretically investigated including the symmetry, periodicity, boundedness, connectedness and the bifurcation points as well as the governing equation for the component centers. An efficient algorithm constructing both the degree-n bifurcation set and the Julia sets is proposed using theoretical results. The mouse-operated software calico "MANJUL" has been developed for the effective construction of the degree-n bifurcation set and the Julia sets in graphic environments with C++ programming language under the windows operating system. Simple mouse operations can construct and magnify the degree-n bifurcation set as well as the Julia sets. They not only compute the component period, bifurcation points and component centers but also save the images of the degree-n bifurcation set and the Julia sets to visually confirm various properties and the geometrical structure of the sets. A demonstration has verified the useful versatility of MANJUL.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.221-229
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• 2004

It is known that the parametric boundary equation for the main component in the Mandelbrot set represents a cardioid. We derive an epicy-cloidal boundary equation of the main component in the degree-n bifurcation set by extending the parameter which describes the cardioid in the Mandelbrot set. Computational results as well as some useful properties are presented together with the programming source codes written in Mathematica. Various boundaries are displayed for $2\leqn\leq7$7 and show a good agreement with the theory presented here. The known boundary equation enables us to significantly reduce the construction time for the degree-n bifurcation set.

• 한국산학기술학회논문지
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• 제3권3호
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• pp.177-182
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• 2002

이 논문에서는 맨델브로트집합의 개념을 n차 복소 다항식 Zⁿ+c(c∈C, n≥2)에 확장하여 n차 분기집합 및 줄리아 집합을 정의하고, 이 집합의 대칭성, 유계성 및 연결성 등에 관하여 이론적으로 연구하였다. 그 연구결과를 이용하여 n차 분기집합 및 줄리아 집합을 효율적으로 작성하는 알고리즘을 고안하고, C++컴퓨터 언어를 사용하여 마이크로소프트사의 윈도우 운영체제하에서 사용자가 마우스를 조작하여 n차 분기집합 및 줄리아 집합을 구성할 수 있도록 소프트웨어 MANJUL을 개발하는 것이 본 논문의 목적이다. MANJUL 소프트웨어의 중요한 특징으로서 CUI(graphical user interfaces) 환경에서 단순한 마우스 조작을 통하여 n차 분기집합 및 줄리아 집합을 작성하고 그 일부분을 확대함은 물론, n차 분기집합 성분의 주기등을 계산 및 저장함으로써, 이 집합들의 다양한 이론적 성질과 기하학적 구조를 시각적으로 확인할 수 있도록 하였다.

• 한국산학기술학회논문지
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• 제7권6호
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• pp.1421-1424
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• 2006

본 논문은 맨델브로트 집합을 9차 복소 다항식에 확장시켜 새로운 프랙탈 도형을 나타내는 9차 분기집합을 정의하고, 2주기 성분의 경계방정식을 매개함수로 표현한다. 또한, 2주기 성분을 작도하는 알고리즘을 고안하고, 매스매티카를 활용하여 2주기 성분의 기하학적 구조에 관한 결과를 제시하고자 한다.

• Journal of applied mathematics & informatics
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• 제10권1_2호
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• pp.63-73
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• 2002

The governing equation locating component centers in the degree-n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor(n-1). Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2 $\leq$ n $\leq$ 25 and show a remarkably improved computation.

• 충청수학회지
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• 제16권2호
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• pp.43-57
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• 2003

The boundary of a typical period-2 component in the degree-3 bifurcation set is formulated by a parametrization of its image which is the unit circle under the multiplier map. Some properties on the geometry of the boundary are investigated including the root point, the cusp and the length as well as the area bounded by the boundary curve. The centroid of the area for the period-2 component was numerically found with high accuracy and compared with its center. An algorithm drawing the boundary curve with Mathematica codes is proposed and its implementation exhibits a good agreement with the analysis presented here.

• Journal of applied mathematics & informatics
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• 제25권1_2호
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• pp.109-118
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• 2007

A parametric boundary equation is established for the principal period-2 bulb in the cubic Mandelbrot set. Using its geometry, an efficient escape-time algorithm which reduces the construction time for the period-2 bulbs in the cubic Mandelbrot set is introduced and the implementation graphic results display the fascinating fractal beauty.