Vol. 4, No. 2 (2015): Regular Issue

We show that the dependence of the mimimal Poincaré return time on the vicinity size is universal for the golden and silver ratios in the circle map and can be referred to as the “Fibonacci stairs”. The rigorous result for the Afraimovich-Pesin dimension equality αc = 1 is confirmed for irrational rotation numbers with the measure of irrationality μ = 2. It is shown that some transcendental number are Diophantine and have the measure μ = 2. It is also confirmed that the gauge function 1/t cannot be applied for Liouvillian numbers. All the obtained features hold for both the linear and the nonlinear circle map.

In this paper, periodic motions in the Duffing oscillator are investigated through the mapping structures of discrete implicit maps. The discrete implicit maps are obtained from differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically through nonlinear algebraic equations of implicit maps, and the corresponding stability and bifurcation analysis of periodic motion in the bifurcation trees are carried out. The bifurcation trees of periodic motions are also presented through the harmonic amplitudes of the discrete Fourier series. Finally, from the analytical prediction, numerical simulation results of periodic motions are performed to verify the analytical prediction. The harmonic amplitude spectra are also presented, and the corresponding analytical expression of periodic motions can be obtained approximately. The method presented in this paper can be applied to other nonlinear dynamical systems for bifurcation trees of periodic motions to chaos.

This study is concernedwith destruction of Anderson localization by a nonlinearity of the power-law type. We suggest using a nonlinear Schr¨odinger model with random potential on a lattice that quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. For super-quadratic nonlinearity the borderline spreading corresponds to diffusion processes on finite clusters. We have proposed an analytical method to predict and explain such transport processes. Our method uses a topological approximation of the nonlinearAnderson model and, if the exponent of the power nonlinearity is either integer or half-integer, will yield the wanted value of the transport exponent via a triangulation procedure in an Euclidean mapping space. A kinetic picture of the transport arising from these investigations uses a fractional extension of the diffusion equation to fractional derivatives over the time, signifying non-Markovian dynamics with algebraically decaying time correlations.

Recently, it has been great interest in the development of methods for solving nonlinear differential equations directly. Here, it is shown an algorithm based on Pad′e approximants for solving nonlinear partial differential equations without requiring a one-dimensional reduction. This method is applied to the λφ4 model in 4 dimensions and new solutions are obtained.

Finite time synchronization of chaotic Micro-Electro-Mechanical Sys-tems (MEMS) is considered. In particular, a Lyapunov-based adaptive controller is developed such that convergence of synchronization error is guaranteed globally in the presence unknown perturbations. The system under consideration suffers from bounded parametric uncertainties, additive external disturbances as well as dead zone input nonlinearities. We establish the controller on being resistance against hard nonlinearities by a novel scheme which can be developed to general chaotic systems even. We provide rigorous stability analysis to come up with sufficient conditions that guarantee finite time error convergence of perturbed system. Several simulation scenarios are carried out to verify the effectiveness of obtained theoretical results.

A simple two-dimensional mathematical approach for source localization of contaminants in the vicinity of individual simple two-dimensional obstacles is proposed. The approach consists of scaling analysis of advectiondiffusion potential flows that can be used in the vicinity of two-dimensional cylindrical obstacles. Three different modeling scenarios are developed in order to simulate the effects of wind. Particularly, the model incorporates the cases of anisotropic diffusion and spatially and temporary inhomogeneous airflow speeds.

A coarse graining process is applied to a Ising like model with a conservative and a reversible dynamics. It is shown that, under some assumptions, this coarse graining leads to a tractable probability transfer matrix of finite size which provides a master equation for a coarse graining probability distribution. Some examples are discussed.

The selective decay phenomena has been observed by physicists for many dynamic flows such as Navier-Stoke flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. Rigorous mathematical works have been carried out for both Navier-Stoke and barotropic geophysical flows. In our previous work, we have rigorously showed the existence of selective states for 2D MHD flows. In this paper, we present a partial result on instability of the selective states.