• Korean Journal of Mathematics
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• 제24권3호
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• pp.375-395
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• 2016

In this paper, we study the long-time dynamical behavior for the extensible suspension bridge equations with past history. We prove the existence of the global attractors by using the contraction function method. Furthermore, the regularity of global attractor is achieved.

• Journal of applied mathematics & informatics
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• 제26권1_2호
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• pp.275-282
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• 2008

Our aim in this paper is to investigate the global attractivity of the recursive sequence $x_{n+1}$ = $\frac{\alpha-{\beta}x_{n-1}}{\gamma+g(x_n)}$ under specified conditions. We show that the positive (or zero for $\alpha$ = 0) equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients and the function g(x).

• 대한수학회논문집
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• 제33권3호
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• pp.1039-1048
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• 2018

In this paper we study the dynamics of a general ${\omega}-periodic$ model. Necessary and sufficient conditions for the global stability of zero steady state of the model are given. The conditions under which there exists a unique periodic solutions to the model are determined. We also show that the unique periodic solution is the global attractor of all other positive solutions. Some applications to mathematical models for cancer and tumor growth are given.

• Structural Engineering and Mechanics
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• 제9권4호
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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.

• Journal of applied mathematics & informatics
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• 제13권1_2호
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• pp.287-300
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• 2003

In this paper we shall consider the nonlinear delay differential equation (equation omitted) where m is a positive integer, ${\beta}$(t) and $\delta$(t) are positive periodic functions of period $\omega$. In the nondelay case we shall show that (*) has a unique positive periodic solution (equation omitted), and show that (equation omitted) is a global attractor all other positive solutions. In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about (equation omitted), and establish sufficient conditions for the global attractivity of (equation omitted). Our results extend and improve the well known results in the autonomous case.

• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.243-249
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• 2004

Our aim in this paper is to investigate the global attractivity of the recursive sequence $x_{n+1}\;=\;\frac{{\alpha}\;-\;{\beta}x_{n-\kappa}}{{\gamma}+x_n}$, where ${\alpha},\;{\beta},\;{\gamma}\;>\;0\;and\;{kappa}\;=\;1,\;2,\;{\ldots}$ We show that the positive equilibrium point of the equation is a global attractor with a basin that depends on certain conditions posed on the coefficients.

• 대한수학회보
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• 제52권5호
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• pp.1445-1465
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• 2015

In this paper, the long time behavior of the convective Cahn-Hilliard equation in two dimensions is considered, semidiscrete and completely discrete spectral approximations are constructed, error estimates of optimal order that hold uniformly on the unbounded time interval $0{\leq}t<{\infty}$ are obtained.

• 대한수학회논문집
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• 제28권4호
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• pp.751-766
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• 2013

We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

• 대한수학회논문집
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• 제34권4호
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• pp.1365-1388
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• 2019

In this paper we consider a class of nonlinear nonlocal parabolic equations involving p-Laplacian operator where the nonlocal quantity is present in the diffusion coefficient which depends on $L^p$-norm of the gradient and the nonlinear term is of polynomial type. We first prove the existence and uniqueness of weak solutions by combining the compactness method and the monotonicity method. Then we study the existence of global attractors in various spaces for the continuous semigroup generated by the problem. Finally, we investigate the existence and exponential stability of weak stationary solutions to the problem.

• 대한수학회지
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• 제48권6호
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• pp.1101-1123
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• 2011

Numerical schemes are routinely used to predict the behavior of continuous dynamical systems. All such schemes transform flows into maps, which can possess dynamical behavior deviating from their continuous counterparts. Here the common bifurcations of scalar dynamical systems are transformed under a class of algorithms known as linearized one-point collocation methods. Through the use of normal forms, we prove that each such bifurcation in an originating flow gives rise to an exactly corresponding one in its discretization. The conditions for spurious period doubling behavior under this class of algorithm are derived. We discuss the global behavioral consequences of a singular set induced by the discretizing methods, including loss of monotonicity of solutions, intermittency, and distortion of attractor basins.