• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.923-937
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• 2012

In this paper we study the dynamic bifurcation of the Swift-Hohenberg equation on a periodic cell ${\Omega}=[-L,L]$. It is shown that the equations bifurcates from the trivial solution to an attractor $\mathcal{A}_{\lambda}$ when th control parameter ${\lambda}$ crosses the critical value. In the odd periodic case $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$ and consists of eight singular points and thei connecting orbits. In the periodic case, $\mathcal{A}_{\lambda}$ is homeomorphic to $S^1$, an contains a torus and two circles which consist of singular points.

• Proceedings of the KIEE Conference
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• 1996.07b
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• pp.1083-1085
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• 1996

Controlling chaos is a new concept, which transform chaotic signal to fixed points, or low periodic orbits. In this paper we propose state feedback method in order to control chaotic signal in canonical Chua's circuit Canonical Chua's circuit is a simple electronic circuit consists of two linear resistors, a linear inductor, two linear capacitors, and only one nonlinear element so called Chua's diode. This nonlinear element supplies power to the circuit and drives the chaotic oscillations. Proposed control method is successful to control chaotic signal in canonical Chua's circuit Result shows that chaotic trajectory change rapidly its orbit to stable fixed points, 1 periodic orbit, or 2 periodic orbit when control signal applies.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.3 no.1
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• pp.1-6
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• 2003

The Delayed Feedback Control method (DFC) proposed by Pyragas applies an input based on the difference between the current state of the system, which is generating chaos orbits, and the $\tau$-time delayed state, and stabilizes the chaos orbit into a target. In DFC, the information about a position in the state space is unnecessary if the period of the unstable periodic orbit to stabilize is known. There exists the fault that DFC cannot stabilize the unstable periodic orbit when a linearlized system around the periodic point has an odd number property. There is the chaos control method using the prediction of the $\tau$-time future state (PDFC) proposed by Ushio et al. as the method to compensate this fault. Then, we propose a method such as improving the fault of the DFC. Namely, we combine DFC and PDFC with parameter W, which indicates the balance of both methods, not to lose each advantage. Therefore, we stabilize the state into the $\tau$ periodic orbit, and ask for the ranges of Wand gain K using Jury' method, and determine the quasi-optimum pair of (W, K) using a genetic algorithm. Finally, we apply the proposed method to a discrete-time chaotic system, and show the efficiency through some examples of numerical experiments.

• Communications of the Korean Mathematical Society
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• v.24 no.3
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• pp.451-458
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• 2009

The set of root classes plays a crucial role in the Nielsen root theory. Extending Brown et al.'s work on the set of root classes of iterates of maps, we rearrange it into the reduced orbit set and show that under suitable hypotheses, any reduced orbit has the full depth property as in the Nielsen type theory of periodic orbits.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.857-872
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• 1994

A PB-chain(Poincare-Birkhoff chain) is by definition a pair of elliptic and hyperbolic n-periodic orbits for a mapping and its existence has been well established numerically or analytically in many particular occasions such as in standard maps or twist maps [1, 8, 9] or Henon maps [1, 2, 12]. This paper gives focus on the investigaton of the appearance of such a PB-chain in a one-parameter family of general area-preserving maps and is in fact a generalization of the results given in [12] for a one-parameter family of specific area-preserving maps, so called Henon maps.

• 제어로봇시스템학회:학술대회논문집
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• 1993.10a
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• pp.1234-1239
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• 1993

We study how to design conventional feedback controllers to drive chaotic trajectories of the well-known systems to their equilibrium points or any of their inherent periodic orbits. The well-known chaotic systems are Heon map and Duffing's equation, which are used as illustrative examples. The proposed feedback controller forces the chaotic trajectory to the stable manifold as OGY method does. Simulation results are presented to show the effectiveness of the proposed design method.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.721-734
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• 2014

The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending our work on Reidemeister orbit sets, we obtain algebraic results such as addition formulae for irreducible Reidemeister orbit sets. Similar formulae for Nielsen type irreducible essential orbit numbers are also proved for fibre preserving maps.

• Communications of the Korean Mathematical Society
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• v.19 no.4
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• pp.745-757
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• 2004

The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Let f : G $\longrightarrow$ G be an endomorphism between the fundamental group of the mapping torus. Extending Jiang and Ferrario's works on Reidemeister sets, we obtain algebraic results such as addition formulae for Reidemeister orbit sets of f relative to Reidemeister sets on suspension groups. In particular, if f is an automorphism, an similar formula for Reidemeister orbit sets of f relative to Reidemeister sets on given groups is also proved.

• Journal for History of Mathematics
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• v.25 no.4
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• pp.53-67
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• 2012

Today, it is necessary to calculate orbits with high accuracy in space flight. The key words of Poincar$\acute{e}$ in celestial mechanics are periodic solutions, invariant integrals, asymptotic solutions, characteristic exponents and the non existence of new single-valued integrals. Poincar$\acute{e}$ define an invariant integral of the system as the form which maintains a constant value at all time $t$, where the integration is taken over the arc of a curve and $Y_i$ are some functions of $x$, and extend 2 dimension and 3 dimension. Eigenvalues are classified as the form of trajectories, as corresponding to nodes, foci, saddle points and center. In periodic solutions, the stability of periodic solutions is dependent on the properties of their characteristic exponents. Poincar$\acute{e}$ called bifurcation that is the possibility of existence of chaotic orbit in planetary motion. Existence of near exceptional trajectories as Hadamard's accounts, says that there are probabilistic orbits. In this context we study the eigenvalue problem in early 20th century in three body problem by analyzing the works of Darwin, Bruns, Gyld$\acute{e}$n, Sundman, Hill, Lyapunov, Birkhoff, Painlev$\acute{e}$ and Hadamard.

• The Bulletin of The Korean Astronomical Society
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• v.39 no.1
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• pp.67.1-67.1
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• 2014

Comets are mysterious travelers from outer Solar System. It is considered that comets loose their subsurface ice once they were injected into a snow-line of the solar system, at the same time, develop adiathermic dust layers on the surface in a time scale of ~10,000 years. They eventually become inactive (see also the presentation by Yoonyoung Kim et al.). Optical similarity between comets and asteroids in comet-like orbits suggests the existence of such dormant or inactive comets supporting the evolutionary scenario. However, unforeseen accidents cast a misgiving to modify the stereotype. A periodic comet, 17P/Holmes, is known as comet with very low activity before 2007. However, the comet suddenly exhibited an outburst in 2007 October, which is known as the most energetic cometary outburst since the beginning of modern astronomy. On the other hand, another periodic comet, P/2010 V1, was not known before 2010 November probably because of low activity and discovered while it experienced outburst. We investigated the time-evolution of the magnitudes and the morphological developments based on the dynamical theory of dust grains, and derived the energy per unit mass of ~10,000 J/kg. From these observational evidences, we suggest that crystallization of buried amorphous ice (even in low-activity comets) can be responsible for the dramatic cometary outbursts.