• Journal of the Korean Physical Society
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• v.73 no.9
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• pp.1385-1392
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• 2018

A network with coupled biological neurons provides various forms of collective synchronous dynamics. Such phase-locking dynamics states resemble eigenvectors in a linear coupling system in that the forms are determined by the symmetry of the coupling strengths. However, the states behave as attractors in a nonlinear dynamics system. We here study the collective synchronous dynamics in a neural system by using a novel theory. We exhibit how the period and the stability of individual phase-locking dynamics states are determined by the characteristics of synaptic couplings. We find that, contrary to common sense, the firing rate of a synchronized state decreases with increasing synaptic coupling strength.

• Communications of the Korean Mathematical Society
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• v.34 no.4
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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.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.9
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• pp.1711-1721
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• 1992

A nonlinear dissipative dynamical system can often have multiple attractors. In this case, it is important to study the global behavior of the system by determining the global domain of attraction of each attractor. In this paper we study the global behavior of a forced beam with two mode interaction. The governing equation of motion is reduced to two second-order nonlinear nonautonomous ordinary differential equations. When .omega. /=3.omega.$_{1}$ and .ohm.=.omega $_{1}$, the system can have two asymptotically stable steady-state periodic solutions, where .omega./ sub 1/, .omega.$_{2}$ and .ohm. denote natural frequencies of the first and second modes and the excitation frequency, respectively. Both solutions have the same period as the excitation period. Therefore each of them shows up as a period-1 solution in Poincare map. We show how interpolated mapping method can be used to determine the two four-dimensional domains of attraction of the two solutions in a very effective way. The results are compared with the ones obtained by direct numerical integration.

• Journal of the Korean Mathematical Society
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• v.61 no.4
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• pp.797-836
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• 2024

We consider the following strongly damped wave equation on ℝ³ with memory u_tt - αΔu_t - βΔu + λu - ∫₀^∞ κ'(s)∆u(t - s)ds + f(x, u) + g(x, u_t) = h, where a quite general memory kernel and the nonlinearity f exhibit a critical growth. Existence, uniqueness and continuous dependence results are provided as well as the existence of regular global and exponential attractors of finite fractal dimension.