• Journal of the Chungcheong Mathematical Society
• /
• v.20 no.4
• /
• pp.577-585
• /
• 2007

We introduce the inverse shadowing property of geometric Lorenz flows and prove that the geometric Lorenz flows do not have the inverse shadowing property.

• Journal of the Korean Mathematical Society
• /
• v.61 no.2
• /
• pp.395-408
• /
• 2024

There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a C² strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.

• Journal of the Chungcheong Mathematical Society
• /
• v.5 no.1
• /
• pp.35-47
• /
• 1992

In this paper we give a necessary condition for a homeomorphism restricted to its nonwandering set to have the shadowing property. Also, we consider a homeomorphism which cannot have the shadowing property.

• Journal for History of Mathematics
• /
• v.18 no.1
• /
• pp.93-102
• /
• 2005

In this paper, we introduce the history of dynamical systems and the notion of the H-shadowing property. And we study some relationships between the H-shadowing property and other dynamical properties such as expansivity and topological stability.

• Bulletin of the Korean Mathematical Society
• /
• v.45 no.4
• /
• pp.645-650
• /
• 2008

Let M be a generalized homogeneous compact space, and let Z(M) denotes the space of homeomorphisms of M with the $C^0$ topology. In this paper, we show that if the interior of the set of weak stable homeomorphisms on M is not empty then for any open subset W of Z(M) containing only weak stable homeomorphisms the orbital shadowing property is generic in W.

• Journal of the Chungcheong Mathematical Society
• /
• v.31 no.1
• /
• pp.167-175
• /
• 2018

In this paper we show that two notions of the average shadowing property and the hyperbolicity are equivalent in linear dynamical systems on the vector space ${\mathbb{C}}^n$.

• Journal of the Korean Mathematical Society
• /
• v.58 no.2
• /
• pp.439-449
• /
• 2021

In this paper, we introduce the notions of expansiveness, shadowing property and topological stability for group actions on metric spaces and give a version of Walters's stability theorem for group actions on locally compact metric spaces. Moreover, we show that if G is a finitely generated virtually nilpotent group and there exists g ∈ G such that if Tg is expansive and has the shadowing property, then T is topologically stable.

• Journal of the Chungcheong Mathematical Society
• /
• v.35 no.4
• /
• pp.329-333
• /
• 2022

In this paper we investigate some properties concerning the set of shadowable points for homeomorphisms. Then we show that the almost shadowing property is preserved by a topological conjugacy between homeomorphisms. Also, we give an example to illustrate our results.

• Bulletin of the Korean Mathematical Society
• /
• v.60 no.4
• /
• pp.1131-1139
• /
• 2023

In this paper, we introduce the concept of ω-expansive of random map on compact metric spaces 𝓟. Also we introduce the definitions of positively, negatively shadowing property and shadowing property for two-sided RDS. Then we show that if 𝜑 is ω-expansive and has the shadowing property for ω, then 𝜑 is topologically stable for ω.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.1
• /
• pp.27-33
• /
• 2012

Let $f$ be a difeomorphism of a closed $n$ -dimensional smooth manifold M, and $p$ be a hyperbolic periodic point of $f$. Let ${\Lambda}(p)$ be a closed set which containing $p$. In this paper, we show that (i) if $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$, then ${\Lambda}(p)$ is the chain component which contains $p$. (ii) suppose $f$ has the asymptotic average shadowing property on ${\Lambda}(p)$. Then if $f|_{\Lambda(p)}$ has the $C^{1}$-stably shadowing property then it is hyperbolic.