• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1503-1510
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• 2014

The aim of this work is to construct a general family of two dimensional differential systems which admits homoclinic solutions near a non-hyperbolic fixed point, such that a Jacobian matrix at this point is zero. We then discretize it by using Euler's method and look after the persistence of the homoclinic solutions in the obtained discrete system.

• 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.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.3
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• pp.171-180
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• 2023

Investigating local dynamics requires precise control to effectively manage the subtle differences that distinguish it from global dynamics. This paper aims to study the localized perspective of the recently proposed continuum-wise expansive measures [13]. Let f : M → M be a diffeomorphism on a closed smooth manifold M and let p be a hyperbolic periodic point of f. We prove that if the homoclinic class H_f (p) of f associated to p is C¹-robustly measure continuum-wise expansive then it is hyperbolic.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.137-161
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• 2022

In this article, we study the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation (1) u⁽⁴⁾(x) + ωu''(x) + a(x)u(x) = f(x, u(x)), ∀x ∈ ℝ where a(x) is not required to be either positive or coercive, and F(x, u) = ∫^u₀ f(x, v)dv is of subquadratic or superquadratic growth as |u| → ∞, or satisfies only local conditions near the origin (i.e., it can be subquadratic, superquadratic or asymptotically quadratic as |u| → ∞). To the best of our knowledge, there is no result published concerning the existence and multiplicity of homoclinic solutions for (1) with our conditions. The proof is based on variational methods and critical point theory.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.693-703
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• 2022

In this paper, we are concerned with the existence of infinitely many fast homoclinic solutions for the following damped vibration system $$(1){\hspace{32}}{\ddot{u}}(t)+q(t){\dot{u}}(t)-L(t)u(t)+{\nabla}W(t,u(t))=0,\;{\forall}t{\in}{\mathbb{R}},$$ where q ∈ C(ℝ, ℝ), L ∈ C(ℝ, ${\mathbb{R}}^{N^2}$) is a symmetric and positive definite matix-valued function and W ∈ C¹(ℝ×ℝ^N, ℝ). The novelty of this paper is that, assuming that L is bounded from below unnecessarily coercive at infinity, and W is only locally defined near the origin with respect to the second variable, we show that (1) possesses infinitely many homoclinic solutions via a variant symmetric mountain pass theorem.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.723-738
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• 2015

In this paper, the prescribed mean curvature Rayleigh p-Laplacian equation with a deviating argument

• Journal of the Chungcheong Mathematical Society
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• v.36 no.2
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• pp.93-97
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• 2023

Let f : M → M be a diffeomorphism of two dimensional manifold M. If f has a homoclinic tangency associated to a hyperbolic periodic point p then there is a diffeomorphism g : M → M with C¹ close to f such that g is not continuum-wise expansive.

• 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.

• 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.

• 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.