• Bulletin of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.689-700
• /
• 1998

We consider an infinite dimensional dynamical system what is called Lattice Dynamical System given by a discrete nonlinear diffusion equation. By assuming the nonlinearity to be a general nonlinear function with mild restrictions, we show that as the diffusion parameter changes the stationery state of the given system undergoes bifurcations from the zero state to a bounded invariant set or a 3- or 4-periodic state in the global phase space of the given system according to the values of the coefficients of the linear part of the given nonlinearity.

• The Pure and Applied Mathematics
• /
• v.16 no.1
• /
• pp.133-145
• /
• 2009

The purpose of this paper is to extend and globalize the Walrasian evolutionary cobweb model in an independent single local market of Brock and Hommes ([3]), to the case of the global market evolution over an infinite chain of many local markets interacting each other through a diffusion of prices between them. In the case of decreasing demands and increasing supplies with a weighted average of rational and naive predictors, we investigate, via the methods of Lattice Dynamical System, the spatial-temporal behaviors of global market dynamics and show that some kind of bounded dynamics of global market do exist and can be controlled by using the parameters in the model.

• The Pure and Applied Mathematics
• /
• v.16 no.4
• /
• pp.427-441
• /
• 2009

The method of Lattice Dynamical System is used to establish a global model on an infinite chain of many local markets interacting each other through a diffusion of prices between them. This global model extends the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution. We assume that each local market has linear decreasing demands and quadratic supplies with naive predictors, and investigate the stationary behaviors of global price dynamics and show that their dynamics are conjugate to those of $H{\acute{e}}non$ maps and hence can exhibit complicated behaviors such as period-doubling bifurcations, chaos, and homoclic orbits etc.

• The Pure and Applied Mathematics
• /
• v.17 no.1
• /
• pp.93-106
• /
• 2010

We employ the methods of Lattice Dynamical System to establish a global model extending the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets with interaction of each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and logistic supplies with naive predictors, and investigate the traveling wave behaviors of global price dynamics and show that their dynamics are conjugate to those of H$\acute{e}$non maps and hence can exhibit complicated behaviors such as period-doubling bifurcations, chaos, and homoclic orbits etc.

• The Pure and Applied Mathematics
• /
• v.16 no.2
• /
• pp.243-254
• /
• 2009

The main purpose of this paper is to use the methods of Lattice Dynamical System to establish a global model, which extends the Walrasian evolutionary cobweb model in an independent single local market to the global market evolution over an infinite chain of many local markets interacting each other through a diffusion of prices between them. For brevity of the model, we assume linear decreasing demands and quadratic supplies with naive predictors, and investigate the spatially homogeneous global price dynamics and show that the dynamics is topologically conjugate to that of well-known logistic map and hence undergoes a period-doubling bifurcation route to chaos as a parameter varies through a critical value.

• Coupled systems mechanics
• /
• v.7 no.2
• /
• pp.141-162
• /
• 2018

This work presents a novel model for analysis of the loading rate influence onto structure response. The model is based on the principles of nonlinear system dynamics, i.e., consists of a system of nonlinear differential equations. In contrast to classical linearized models, this one comprises mass and loading as integral parts of the model. Application of the Kelvin and the Maxwell material models relates the novel formulation to the existing material formulations. All the analysis is performed on a proprietary computer program based on Wolfram Mathematica. This work can be considered as an extended proof of concept for the application of the nonlinear solid model in material response to dynamic loading.

• Applied Microscopy
• /
• v.39 no.4
• /
• pp.341-348
• /
• 2009

The crystal structure of nano-crystalline, $BaTiO_3$, with the average particle size of 100 nm was investigated using electron diffraction techniques. We characterized the precession electron diffraction system and then carried out the structure determination using precession electron diffraction and conventional selected area electron diffraction. As a result, it was revealed that $BaTiO_3$ nano-crystalline exist as a mixture of tetragonal structure and cubic structure by precession electron diffraction technique. In addition, it could be turned out that $BaTiO_3$ nano-crystalline is a core-shell structure consisted of a tetragonal phased core and a cubic phased surface layer by theoretical calculation. The thickness of the cubic surface layer was approximately 8.5 nm and the lattice parameters of cubic and tetragonal phases were a=3.999${\AA}$ and a=3.999${\AA}$, c=4.022${\AA}$, respectively. Finally, it is expected that precession electron diffraction is more useful technique for structure determination of complicated nano-crystalline materials because of its higher spatial resolution and minimization of dynamical scattering effect.