• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.703-713
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• 2012

In this paper we study ${\omega}$-limit sets and attraction of non-autonomous discrete dynamical systems. We introduce some basic concepts such as ${\omega}$-limit set and attraction for non-autonomous discrete system. We study fundamental properties of ${\omega}$-limit sets and discuss the relationship between ${\omega}$-limit sets and attraction for non-autonomous discrete dynamical systems.

• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.13-16
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• 1998

For continuous maps f of the circle to itself, we show that (1) every $\omega$-limit point is recurrent (or almost periodic) if and only if every $\omega$-limit set is minimal, (2) every $\omega$-limit set is almost periodic, then every $\omega$-limit set contains only one minimal set.

• Communications of the Korean Mathematical Society
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• v.15 no.3
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• pp.549-553
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• 2000

For a continuous map of the circle to itself, we give necessary and sufficient conditions for the $\omega$-limit set of each nonwandering point to be minimal.

• Kyungpook Mathematical Journal
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• v.54 no.2
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• pp.333-339
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• 2014

In this paper, we describe the characterizations of omega limit sets (= ${\omega}$-limit set) on $\mathbb{R}^2$ in detail. For a local real analytic flow ${\Phi}$ by z' = f(z) on $\mathbb{R}^2$, we prove the ${\omega}$-limit set from the basin of a given attractor is in the boundary of the attractor. Using the result of Jim$\acute{e}$nez-L$\acute{o}$pez and Llibre [9], we can completely understand how both the attractors and the ${\omega}$-limit sets from the basin.

• Interaction and multiscale mechanics
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• v.4 no.4
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• pp.291-311
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• 2011

To simulate the self excited torsional vibrations of rotating drill strings (DSs) in vertical bore-holes, the nonlinear wave models of homogeneous and sectional torsional pendulums are formulated. The stated problem is shown to be of singularly perturbed type because the coefficient appearing before the second derivative of the constitutive nonlinear differential equation is small. The diapasons ${\omega}_b\leq{\omega}\leq{\omega}_l$ of angular velocity ${\omega}$ of the DS rotation are found, where the torsional auto-oscillations (of limit cycles) of the DS bit are generated. The variation of the limit cycle states, i.e. birth (${\omega}={\omega}_b$), evolution (${\omega}_b<{\omega}<{\omega}_l$) and loss (${\omega}={\omega}_l$), with the increase in angular velocity ${\omega}$ is analyzed. It is observed that firstly, at birth state of bifurcation of the limit cycle, the auto-oscillation generated proceeds in the regime of fast and slow motions (multiscale motion) with very small amplitude and it has a relaxation mode with nearly discontinuous angular velocities of elastic twisting. The vibration amplitude increases as ${\omega}$ increases, and then it decreases as ${\omega}$ approaches ${\omega}_l$. Sectional drill strings are also considered, and the conditions of the solution at the point of the upper and lower section joints are deduced. Besides, the peculiarities of the auto-oscillations of the sectional DSs are discussed.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.837-851
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• 2014

We consider ${\omega}$-chaos as defined by S. H. Li in 1993. We show that c-dense ${\omega}$-scrambled sets are present in every transitive system with two-sided limit shadowing property (TSLmSP) and that every transitive map on topological graph has a dense Mycielski ${\omega}$-scrambled set. As a preliminary step, we provide a characterization of dynamical properties of maps with TSLmSP.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.3
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• pp.523-530
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• 2014

In this article, we deal with the notion of ­${\Omega}$-limit sets in dynamical systems. We show that the ${\Omega}$­-limit set of a compact subset of a phase space is quasi-attracting.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.101-107
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• 2000

For a continuous map f of the circle to itself, we show that if P(f) is closed, then ${\Gamma}(f)$ is closed, and ${\Omega}(f)={\Omega}(f^n)$ for all n > 0.

• Journal of the Chungcheong Mathematical Society
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• v.5 no.1
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• pp.65-74
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• 1992

In this note, we show that the ${\omega}$-limit mapping is continuous and the Lipschitz constants vary continuously if the flow (x, ${\pi}$) is Lipschitz stable. Moreover we analyse the ${\omega}$-limit sets under the generalized locally Lipschitz stable flows.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.1
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• pp.73-80
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• 2009

This paper aims to study local and global structure of holomorphic flows on $\mathbb{C}^1$. At a singular point of a holomorphic flow, we locally sector the flow into parabolic or elliptic types. By the local structure of holomorphic flows, we classify all the possible types of topologies of $\omega$-limit sets.