Nonlinear Sciences

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular-automaton analogous of localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. They can be considered as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyse what types of interaction occur between gliders travelling on a cellular automaton `cyclotron' and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of cyclic tag systems.

A cellular non-linear network (CNN) is a uniform regular array of locally connected continuous-state machines, or nodes, which update their states simultaneously in discrete time. A microbial fuel cell (MFC) is an electro-chemical reactor using the metabolism of bacteria to drive an electrical current. In a CNN model of the MFC, each node takes a vector of states which represent geometrical characteristics of the cell, like the electrodes or impermeable borders, and quantify measurable properties like bacterial population, charges produced and hydrogen ions concentrations. The model allows the study of integral reaction of the MFC, including temporal outputs, to spatial disturbances of the bacterial population and supply of nutrients. The model can also be used to evaluate inhomogeneous configurations of bacterial populations attached on the electrode biofilms.

Chaos control in Random Boolean networks is implemented by freezing part of the network to drive it from chaotic to ordered phase. However, controlled nodes are only viewed as passive blocks to prevent perturbation spread. This paper proposes a new control method in which controlled nodes can exert an active impact on the network. Controlled nodes and frozen values are deliberately selected according to the information of connection and Boolean functions. Simulation results show that the number of nodes needed to achieve control is largely reduced compared to previous method. Theoretical analysis is also given to estimate the least fraction of nodes needed to achieve control.

A multispecies artificial ecosystem is formulated using cellular automata with species interactions and food chain hierarchy. The constructed finite state automaton can simulate the complexity and self-organized characteristics of the evolving multispecies living ecosystems. Numerical experiments show that a small perturbation or extinction event may affect many other species in the ecosystem in an avalanche manner. Both the avalanches and the extinction arising from these changes follow a power law, reflecting that the multispecies living ecosytems have the characteristics of self-organized criticality.

Consider the macroscale modelling of microscale spatiotemporal dynamics. Here we develop a new approach to ensure coarse scale discrete models preserve important self-adjoint properties of the fine scale dynamics. The first part explores the discretisation of microscale continuum dynamics. The second addresses how dynamics on a fine lattice are mapped to lattice a factor of two coarser (as in multigrids). Such mapping of discrete lattice dynamics may be iterated to empower us in future research to explore scale dependent emergent phenomena. The support of dynamical systems, centre manifold, theory ensures that the coarse scale modelling applies with a finite spectral gap, in a finite domain, and for all time. The accuracy of the models is limited by the asymptotic resolution of subgrid coarse scale processes, and is controlled by the level of truncation. As given examples demonstrate, the novel feature of the approach developed here is that it ensures the preservation of important conservation properties of the microscale dynamics.

A two-state, three-dimensional, deterministic, reversible cellular automaton is shown to be capable of approximately circular orbits, wavelike undulations, and particle-like configurations that decay in accordance with a half-life law.

There are many studies dealing with the protection or restoration of wetlands and the sustainable economic growth of cities as separate subjects. This study investigates the conflict between the two in an area where city growth is threatening a protected wetland area. We develop a stochastic cellular automaton model for urban growth and apply it to the Vecht area surrounding the city of Hilversum in the Netherlands, using topographic maps covering the past 150 years. We investigate the dependence of the urban growth pattern on the values associated with the protected wetland and other types of landscape surrounding the city. The conflict between city growth and wetland protection is projected to occur before 2035, assuming full protection of the wetland. Our results also show that a milder protection policy, allowing some of the wetland to be sacrificed, could be beneficial for maintaining other valuable landscapes. This insight would be difficult to achieve by other analytical means. We conclude that even slight changes in usage priorities of landscapes can significantly affect the landscape distribution in near future. Our results also point to the importance of a protection policy to take the value of surrounding landscapes and the dynamic nature of urban areas into account.

In order to develop systems capable of modeling artificial life, we need to identify, which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical systems. The method distinguishes between different asymptotic behaviors of a system's average computation time before entering a loop. When applied to elementary cellular automata, we obtain classification results, which correlate very well with Wolfram's manual classification. Further, we use it to classify 2D cellular automata to show that our technique can easily be applied to more complex models of computation. We believe this classification method can help to develop systems, in which complex structures emerge.

While the surjectivity of the global map in two-dimensional cellular automata (2D CA) is undecidable in general, in specific cases one can often decide if the rule is surjective or not. We attempt to classify as many 2D CA as possible by using a sequence of tests based on the balance theorem, injectivity of the restriction to finite configurations, as well as permutivity. We introduce the notion of slice permutivity which is shown to imply surjectivity in 2D CA. The tests are applied to 2D binary CA with neighbourhoods consisting of up to five sites, considering all possible contiguous shapes of the neighbourhood. We find that if the size of the neighbourhood is less than five, complete classification of all rules is possible. Among 5-site rules, those with von Neuman neighbourhoods as well as neighbourhoods corresponding to T, V, and Z pentominos can also be completely classified.

The present work aims to show one of the advantages of using the property of closure under coupling in the DEVS specification. The advantage concerned in this paper attempts to address the need for memory resources during the simulation of systems by cellular-DEVS. This improvement of performance is based on the usage of the property closure under coupling in the DEVS formalism. With this property and taking account of the iterative behavior of each cellular-DEVS atomic model, we provide simulation of many models simultaneously. The method starts with the specification of the cellular-DEVS coupled model which is then converted into its equivalent DEVS atomic model. Thus, the goal of this conversion is to transform large quantities of atomic models coupled together, which require huge computational resources, into one DEVS atomic model. A case study is presented at the end of the work on modeling and simulation of forest fire propagation using DEVS and cellular-DEVS. A specification by cellular-DEVS of the forest fire model and its non-modular equivalent DEVS atomic model are presented. Finally, a comparison between both methods is presented in term of consumption of resources.