1.  Symmetry and Symmetry-Breaking of the Emergent Dynamics of the Discrete Stochastic Majority-Voter Model Full text (PDF)

K. G. Spiliotis, L. Russo and C. I. Siettos

Pages: 1-20

Abstract:  We analyse the emergent dynamics of the so called majority voter model evolving on complex networks. In particular we study the influence of three characteristic types of networks, namely Random Regular, Erdôs-Rényi (ER), Watts and Strogatz (WS, small-world) and Barabasi (scale- free) on the bifurcating stationary coarsegrained solutions. We first prove analytically some simple properties about the symmetry and symmetry breaking of the macroscopic dynamics with respect to the network topology. We also show how one can exploit the Equation-free framework to bridge in a computational strict manner the micro to macro scales of the dynamics of stochastic individualistic models on complex random graphs. In particular, we show how systems-level tasks such as bifurcation analysis of the coarse-grained dynamics can be performed bypassing the need to extract macroscopic models in a closed form. A comparison with the mean-field approximations is also given illustrating the merits of the Equation-Free approach, especially in the case of scale-free networks exhibiting a heavy-tailed connectivity distribution.

2. Convergent Numerical Solutions of Unsteady Problems Full text (PDF)

Lun-Shin Yao

This paper contain some discussions and open problems

Pages: 21-31

Abstract:  Von Neumann established that discretized algebraic equations must be consistent with the differential equations, and must be stable in order to obtain convergent numerical solutions for the given differential equations. The ”stability” is required to satisfactorily approximate a differential derivative by its discretized form, such as a finite-difference scheme, in order to compute in computers. His criterion is the necessary and sufficient condition only for steady or equilibrium problems. It is also a necessary condition, but not a sufficient condition for unsteady transient problems; additional care is required to ensure the accuracy of unsteady solutions.

3. Nonlinear Control and Chaotic Vibrations of Perturbed Trajectories of Manipulators  Full text (PDF)

P. Szumiński T. Kapitaniak

Pages: 32-54

Abstract:  We study different types of manipulators’ attractors and propose a motion control method. In our analysis the manipulator’s motion is perturbed and its stability investigated using the nonlinear equations of perturbations and linearized equations for practical control. In order to realize a practical control the common areas of stability for nonlinear and linear models are identified. The maps of stability calculated as functions of model parameters are proposed as a tool for motion control. The spectrum of Lyapunov exponents is introduced as a practical measure of motion quality. The procedure allows choosing a way of reaching system stability in order to avoid undesired attractors. Additionally, the possibility of the occurrence of strange chaotic attractors in manipulators, ways they appear, and codimension 2 bifurcations have been analyzed.