Modeling of Neurodegenerative Diseases Using Discrete Chaotic Systems

Parkinson's and Huntington's diseases are two of the most common neurodegenerative disorders.Tremor, muscle stiffness, and slowness of movement are symptoms of Parkinson's disease. The symptoms of Huntington's disease are severe reduction in muscle control, emotional disturbance, and pathological disorders in brain cells.These diseases are caused by destruction of the cells that secrete a substance called dopamine. In this paper,a new discrete chaotic system is introduced,which can mimic the brain's behavior for neurodegenerative diseases such as Parkinson,Huntington, and Hypokinesia. This system is described based on the similarity between the brain's behavior in normal and abnormal conditions and the chaotic systems. Bifurcation analysis is carried out with respect to different parameters, providing full spectrum of the behavior for different parameter values.Our results can be used to mathematically study the mechanisms behind these diseases.     

Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion

This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L¹ to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result.     

Peristaltic motion of a magnetohydrodynamic generalized second‐order fluid in an asymmetric channel

We investigate the peristaltic motion of a magnetohydrodynamic (MHD) generalized second‐order fluid in an asymmetric channel. The governing equations are first modeled and then numerically solved under the long wavelength approximation. Attention has been focused to analyze the shear‐thinning and shear‐thickening effects of the investigated non‐Newtonian fluid, the influence of the magnetic force on the flow, especially the trapping, pumping characteristics caused by the peristalsis of the walls. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011