Advances in Complex Systems

We give a new introduction to hyperstructures and higher order structures in general. A specific implementation of a hyperstructure we call Abstract Matter, and we argue that this is a suitable model for studying and simulating various kinds of both organic and inorganic matter — even cognitive matter.

Our earlier language model is modified to allow for the survival of a minority language without higher status, just because of the pride of its speakers in their linguistic identity. An appendix studies the roughness of the interface for linguistic regions when one language conquers the whole territory.

The design and mathematical modeling of the programing electric field in a neural switch is carried out. The specified function for the switch is to operate as a synaptic processor behaving in an adaptive manner and suitable to be used as a compact programable device with other artificial neural network hardware. Modeling of the switch is carried out by means of complex mathematical analysis employing the Schwarz–Christoffel transform. The effect of inter-electrode separation on the field strength is analyzed in two dimensions. The realized power law function of the programing field is discussed and explained.

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

We investigate a three-species food chain system with density-dependent birth rate and impulsive effect concerning biological and chemical control strategy — periodic releasing of natural enemies or spraying pesticide at different fixed times. Conditions for the extinction of the prey and top predator are given. By using the Floquet theory of impulsive differential equations and small amplitude perturbation skills, we consider the local stability of the prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system.

In this paper, we develop a mathematical model for malaria with impulsive effect and infection-age, and consider the effect of blood transfusion and infected vector transmission. We derive the condition under which malaria goes to extinction.

Emergent behavior in interconnected systems (complex systems) is of fundamental significance in natural and engineering sciences. A commonly investigated problem is how complicated dynamics take place in dynamical systems consisting of (often simple) subsystems. It is shown though numerical experiments that emergent order such as periodic behavior can likely take place in coupled chaotic dynamical systems. This is demonstrated for the particular case of coupled chaotic continuous time Hopfield neural networks. In particular, it is shown that when two chaotic Hopfield neural networks are coupled by simple sigmoid signals, periodic behavior can emerge as a consequence of this coupling.

We introduce and study a chemostat model with plasmid-bearing, plasmid-free competition and impulsive effect. According to the stability analysis of the boundary periodic solution, we obtain the invasion threshold of the plasmid-free organism and plasmid-bearing organism. Furthermore, by using standard techniques of bifurcation theory, we prove the system has a positive τ-periodic solution, which shows that the impulsive effect destroys the equilibria of the unforced continuous system and initiates the periodic solution. Our results can be applied to control bioreactors aimed at producing commercial products through genetically altered organisms.

In this paper, some decentralized control problems composed of two subsystems are addressed from a special perspective. First, it is pointed out that some subsystems must be unstable to stabilize the overall interconnected system in some special cases. Then, a special kind of decentralized control problem is studied. This kind of problem can be viewed as harmonic control among independent subsystems. Research results show that two unstable systems can generate a stable system through some effective cooperations. And a linear matrix inequality method is provided for decentralized controller design by using the parameter-dependent Lyapunov function method. Two examples are given to illustrate the results.

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R² or R³ (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R³ to Rⁿ (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model.

This model uses a "mapping partial derivative" and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in Rⁿ. Therefore, the result is very general. Moreover, the multi-dimensional piece-wise self-affine fractal interpolation model in tensor form is more terse than in the usual matrix form.