• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.441-444
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• 2013

It is shown that any two-dimensional globally hyperbolic spacetime with noncompact Cauchy surfaces has no null cut points.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1495-1499
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• 2013

In this short note we prove that if a symplectomorphism $f$ is $C^1$-stably shadowable, then $f$ is Anosov. The same result is obtained for volume-preserving diffeomorphisms.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.581-587
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• 2013

Let ${\Lambda}$ be a robustly transitive set of a diffeomorphism $f$ on a closed $C^{\infty}$ manifold. In this paper, we characterize hyperbolicity of ${\Lambda}$ in $C^1$-generic sense.

• Communications of the Korean Mathematical Society
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• v.34 no.3
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• pp.897-910
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• 2019

In this paper we introduce the notion of "robust special Anosov endomorphisms", and show that Anosov endomorphisms of tori which are not neither an Anosov diffeomorphism nor an expanding map, are not robust special.

• East Asian mathematical journal
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• v.37 no.3
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• pp.307-317
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• 2021

In this paper, we deal with a problem to determine the type of automorphisms of the unit disc in ℂ in terms of intrinsic geometry. We will characterize the hyperbolicity and parabolicity of automorphism by the distance function of the Poincaré metric.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.495-513
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• 2024

In the paper, we show that for a generic C¹ vector field X on a closed three dimensional manifold M, any isolated transitive set of X is singular hyperbolic. It is a partial answer of the conjecture in [13].

• Bulletin of the Korean Chemical Society
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• v.20 no.1
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• pp.35-41
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• 1999

Stability characteristics of hyperbolic reaction-diffusion equations with a reversible Brusselator model are investigated as an extension of the previous work. Intensive stability analysis is performed for three important parameters, Nrd, β and Dx, where Nrd is the reaction-diffusion number which is a measure of hyperbolicity, β is a measure of reversibility of autocatalytic reaction and Dx is a diffusion coefficient of intermediate X. Especially, the dependence on Nrd of stability exhibits some interesting features, such as hyperbolicity in the small Nrd region and parabolicity in the large Nrd region. The hyperbolic reaction-diffusion equations are solved numerically by a spectral method which is modified and adjusted to hyperbolic partial differential equations. The numerical method gives good accuracy and efficiency even in a stiff region in the case of small Nrd, and it can be extended to a two-dimensional system. Four types of solution, spatially homogeneous, spatially oscillatory, spatio-temporally oscillatory and chaotic can be obtained. Entropy productions for reaction are also calculated to get some crucial information related to the bifurcation of the system. At the bifurcation point, entropy production changes discontinuously and it shows that different structures of the system have different modes in the dissipative process required to maintain the structure of the system. But it appears that magnitude of entropy production in each structure give no important information related for states of system itself.

• Communications of the Korean Mathematical Society
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• v.21 no.3
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• pp.515-526
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• 2006

In this paper, we give a characterization of hyperbolic linear dynamical systems via the notions of various inverse shadowing. More precisely it is proved that for a linear dynamical system f(x)=Ax of ${\mathbb{C}^n}$, f has the ${\tau}_h$ inverse(${\tau}_h-orbital$ inverse or ${\tau}_h-weak$ inverse) shadowing property if and only if the matrix A is hyperbolic.

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.779-788
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• 1997

In this work, using the exact separatrix map which provides an efficient way to describe dynamics near the separatrix, we study the stochastic layer near the separatrix of a one-degree-of-freedom Hamilitonian system with time periodic perturbation. Applying the twist map theory to the exact separatrix map, T. Ahn, G. I. Kim and S. Kim proved the existence of the uniformly hyperbolic invariant set(UHIS) near separatrix. Using the theorems of Bowen and Franks, we prove this UHIS has measure zero.

• Journal of the Korean Mathematical Society
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• v.57 no.6
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• pp.1335-1345
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• 2020

We will show a rigidity of a Kähler potential of the Poincaré metric with a constant length differential.