• Journal of computational fluids engineering
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• v.20 no.4
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• pp.58-62
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• 2015

Recent studies on wall-bounded shear flow have emphasized the significance of the stable manifold of simple nonlinear invariant solutions to the Navier-Stokes equation in the formation of the boundary between the laminar and turbulent regions in state space. In this paper we present newly discovered homoclinic orbits of the Kawahara and Kida(2001) periodic solution in plane Couette flow. We show that as the Reynolds number decreases a pair of homoclinic orbits move closer to each other until they disappear to exhibit homoclinic tangency.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.05a
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• pp.287-296
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• 2003

We investigate global bifurcation in the motion of an harmonically excited circular plate with one-to-one internal resonance. A perturbation method developed by Kovacic and Wiggins is used. Silnikov type homoclinic orbit has been pursued but it has turned out not to exist.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2003.11a
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• pp.504-514
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• 2003

We investigate global bifurcation in the subharmonic motion of a circular plate with one-to-one internal resonance. A system of autonomous equations are obtained from the partial differential equations governing the system by using Galerkin's procedure and the method of multiple scales. A perturbation method developed by Kovacic and Wiggins is used to find Silnikov type homoclinic orbits. The conditions under which the orbits occur are obtained.

• The Pure and Applied Mathematics
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• v.17 no.4
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• pp.389-396
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• 2010

We investigate the existence of homoclinic orbits of the following systems of $Li{\'{e}}nard$ type: $a(x)x^'=h(y)-F(x)$, $y^'$=-a(x)g(x), where $h(y)=m{\mid}y{\mid}^{p-2}y$ with m > 0 and p > 1 and a, F, 9 are continuous functions such that a(x) > 0 for all $x{\in}{\mathbb{R}}$ and F(0)=g(0)=0 and xg(x) > 0 for $x{\neq}0$. By a series of time and coordinates transformations of the above system, we obtain sufficient conditions for the positive orbits of the above system starting at the points on the curve h(y) = F(x) with x > 0 to approach the origin through only the first quadrant. The method of this paper is new and the results of this paper cover some early results on this topic.

• Journal of KSNVE
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• v.7 no.3
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• pp.489-498
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• 1997

In this paper, chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonliear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which has the external and parametric excitation with a same frequency. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Numerical simulations are performed to demonstrate theoretical results and show the strange attractor of the chaotic motion.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1996.10a
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• pp.288-294
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• 1996

In this paper, Chaotic motions of a curved pipe conveying oscillatory flow are theoretically investigated. The nonlinear partial differential equation of motion is derived by Newton's method. The transformed nonlinear ordinary differential equation is a type of Hill's equation, which have the parametric and external excitation. Bifurcation curves of chaotic motion of the piping systems are obtained by applying Melnikov's method. Poincare maps numerically demonstrate theoretical results and show transverse homoclinic orbit of the chaotic motion.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.17-53
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• 2014

Let ${\gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({\gamma})$ be the chain component of X which contains ${\gamma}$. We say that $C_X({\gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $\mathcal{U}$ of X such that for any $Y{\in}\mathcal{U}$, $C_Y({\gamma}_Y)$ is shadowable for $Y_t$, where ${\gamma}_Y$ denotes the continuation of ${\gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({\gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({\gamma})$ has no non-hyperbolic singularity.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.33 no.12
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• pp.1427-1432
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• 2009

The pitch motion of a generic gravity gradient satellite is investigated in terms of chaos. The Melnikov method is used for detecting the onset of chaotic behavior of the pitch motion of a gravity gradient satellite. The Melnikov method determines the distance between stable and unstable manifolds of a perturbed system. When stable and unstable manifolds transverse on the Poincare section, the resulting motion can be chaotic. The Melnikov analysis indicates that the pitch dynamics of a generic gravity gradient satellite can be chaotic when the orbit eccentricity is small.