**
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.

**
**