• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.155-163
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• 2002

The purpose of this paper is to establish discrete versions of the well-known Lyapunov's stability theorem and LaSalle's invariance theorem for a non-autonomous discrete dynamical system. Our proofs for these theorems are constructive in the sense that they are made by explicitly building a Lyapunov function for the system. A comparison between non-autonomous discrete dynamical systems and continuous dynamical systems is conducted.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.289-295
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• 1994

Let (A,G,.alpha.) be a $C^{*}$-dynamical system. In [3] the classical notions of ergodic properties of topological dynamical systems such as topological transitivity, minimality, and uniquely ergodicity are extended and analyzed in the context of non-abelian $C^{*}$-dynamical systems. We showed in [2] that if G is a compact group, then minimality, topological transitivity, uniquely ergodicity, and weakly ergodicity of the $C^{*}$-dynamical system (A,G,.alpha.) are equivalent.alent.

• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.1-11
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• 1990

The purpose of this paper is to study the chain recurrent sets under persistent dynamical systems, and give a necessary condition for a persistent dynamical system to be topologically stable. Moreover, we show that the various recurrent sets depend continuously on persistent dynamical system.

• Interaction and multiscale mechanics
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• v.3 no.1
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• pp.1-22
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• 2010

In this paper one introduces a method of multiscale modelling called collection of dynamical systems with dimensional reduction. The method is suggested to be an appropriate approach to theoretical modelling of phenomena in mechanics of materials having in mind especially dynamics of processes. Within this method one formalizes scale of averaging of processes during modelling. To this end a collection of dynamical systems is distinguished within an elementary dynamical system. One introduces a dimensional reduction procedure which is designed to be a method of transition between various scales. In order to consider continuum models as obtained by means of the dimensional reduction one introduces continuum with finite-dimensional fields. Owing to geometrical elements associated with the elementary dynamical system we can formalize scale of averaging within continuum mechanics approach. In general presented here approach is viewed as a continuation of the rational mechanics.

• Journal of the Korean Data and Information Science Society
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• v.23 no.3
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• pp.595-604
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• 2012

In this article, we consider chaotic behavior happened in nonsmooth dynamical systems. To quantify such a behavior, a computation of Lyapunov exponents for chaotic orbits of a given nonsmooth dynamical system is focused. The Lyapunov exponent is a very important concept in chaotic theory, because this quantity measures the sensitive dependence on initial conditions in dynamical systems. Therefore, Lyapunov exponents can decide whether an orbit is chaos or not. To measure the sensitive dependence on initial conditions for nonsmooth dynamical systems, the calculation of Lyapunov exponent plays a key role, but in a theoretical point of view or based on the definition of Lyapunov exponents, Lyapunov exponents of nonsmooth orbit could not be calculated easily, because the Jacobian derivative at some point in the orbit may not exists. We use an algorithmic calculation method for computing Lyapunov exponents using time series for a two dimensional piecewise smooth dynamic system.

• Bulletin of the Korean Mathematical Society
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• v.41 no.4
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• pp.665-676
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• 2004

In this paper we characterize asymptotic stability via Lyapunov function in general dynamical systems on c-first countable space. We give a family of examples which have first countable but not c-first countable, also c-first countable and locally compact space but not metric space. We obtain several necessary and sufficient conditions for a compact subset M of the phase space X to be asymptotic stability.

• Journal of the Chungcheong Mathematical Society
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• v.35 no.2
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• pp.171-176
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• 2022

In this article, we investigate the transitivity and chain transitivity on multi-valued dynamical systems. For compact-valued continuous dynamics, we prove that the notion of transitivity is expressed by the notions of the shadowing property and chain transitivity under locally maximal condition.

• 제어로봇시스템학회:학술대회논문집
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• 2002.10a
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• pp.94.5-94
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• 2002

The current study of complex dynamical networks is pervading all kinds of sciences today, ranging from physical to biological, even to social sciences. its impact on modern engineering and technology is prominent and will be far-reaching. Typical complex dynamical networks include the World Wide Web, the Internet, various wireless communication networks, meta-bolic networks, biological neural networks, social connection networks, scientific cooperation and citation networks, and so on. Research on fundamental properties and dynamical features of such complex networks have become overwhelm ing. This talk will provide a brief overview of some basic concepts about com plex dynamical netwo...

• 제어로봇시스템학회:학술대회논문집
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• 1998.10a
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• pp.379-379
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• 1998

The report presents an approach to constructing or control algorithms for finite dimensional dynamical systems under the deficit of information about dynamical disturbances. The approach is based on the constructions of the extremal shift strategy of the differential game theory.

• Journal of the Korean Mathematical Society
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• v.49 no.1
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• pp.165-185
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• 2012

In this paper, the topological entropy and measure-theoretic entropy for nonautonomous dynamical systems are studied. Some properties of these entropies are given and the relation between them is discussed. Moreover, the bounds of them for several particular nonautonomous systems, such as affine transformations on metrizable groups (especially on the torus) and smooth maps on Riemannian manifolds, are obtained.