• 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.

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.575-591
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• 2018

Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. Roughly, an Anosov family is a two-sided sequence of diffeomorphisms (or non-stationary dynamical system) with similar behavior to an Anosov diffeomorphisms. We show that the set consisting of Anosov families is an open subset of the set consisting of two-sided sequences of diffeomorphisms, which is equipped with the strong topology (or Whitney topology).

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.457-478
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• 2014

In this paper, a 3D autonomous system, which has only stable or non-hyperbolic equilibria but still generates chaos, is presented. This system is topologically non-equivalent to the original Lorenz system and all Lorenz-type systems. This motivates us to further study some of its dynamical behaviors, such as the local stability of equilibrium points, the Lyapunov exponent, the dissipativity, the chaotic waveform in time domain, the continuous frequency spectrum, the Poincar$\acute{e}$ map and the forming mechanism for compound structure of its special cases. Especially, with the help of the Project Method, its Hopf bifurcation of codimension one is in detailed formulated. Numerical simulation results not only examine the corresponding theoretical analytical results, but also show that this system possesses abundant and complex dynamical properties not solved theoretically, which need further attention.

• Structural Engineering and Mechanics
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• v.9 no.4
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• pp.397-416
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• 2000

The present study deals with the nonlinear dynamics and stability of autonomous dissipative either imperfect potential (limit point) systems or perfect (bifurcational) non-potential ones. Through a fully nonlinear dynamic analysis, performed on two simple 2-DOF models corresponding to the classes of systems mentioned above, and with the aid of basic definitions of the theory of nonlinear dynamical systems, new important phenomena are revealed. For the first class of systems a third possibility of postbuckling dynamic response is offered, associated with a point attractor on the prebuckling primary path, while for the second one the new findings are chaos-like (most likely chaotic) motions, consecutive regions of point and periodic attractors, series of global bifurcations and point attractor response of always existing complementary equilibrium configurations, regardless of the value of the nonconservativeness parameter.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.14 no.2
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• pp.91-97
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• 2014

Chaotic dynamics is an active research area in fields such as biology, physics, sociology, psychology, physiology, and engineering. Interest in chaos is also expanding to the social sciences, such as politics, economics, and societal events prediction. Most people pursue happiness, both spiritual and physical in many cases. However, happiness is not easy to define, because people differ in how they perceive it. Happiness can exist in mind and body. Therefore, we need to be happy in both simultaneously to achieve optimal happiness. To do this, we need to synchronize mind and body. In this paper, we propose a chaotic synchronization method in a mathematical model of happiness organized by a second-order ordinary differential equation with external force. This proposed mathematical happiness equation is similar to Duffing's equation, because it is derived from that equation. We introduce synchronization method from our mathematical happiness model by using the derived Duffing equation. To achieve chaotic synchronization between the human mind and body, we apply an idea of mind/body unity originating in Oriental philosophy. Of many chaotic synchronization methods, we use only coupled synchronization, because this method is closest to representing mind/body unity. Typically, coupled synchronization can be applied only to non-autonomous systems, such as a modified Duffing system. We represent the result of synchronization using a differential time series mind/body model.