# <p>Network and stability

This is a model of joint dynamics of a finite number of elementaty dynamic systems interconnected by communication channels. This model seems to be a discrete analogue of a number of wellknown models of distributive reactions and actually coincides with them when the number of the systems is large and the connection structures are of the " lattice " type. (This connection structure is a model of diffusion). The essential feature of the suggested model is that the connection structure can be an arbitrary graph (or a graph from some set). This permits to investigate the role of the connection structure in determining peculiarities of the given dynamics or, on the contrary, the peculiarities of reactions of different types of the elementary systems to a certain connection structure. In this model loss (or decrease) of stability was investigated in a finite set of two dimensional elementary systems under the condition that the exchange between them is small. A criterion for stability decrease and a complete classification of elementary systems was obtained, depending on whether they increase or decrease stability when connections arise between them. The results were, in particular, used to analyze the Wolkenstein- -Kuznetsov model of the multiple dormant tumour. Several particular cases of large dimensionality were studied, for instance, the Belousov - Zhabotinski model.

Alecseeva E.I., Kirzhner V.(1987). **Stability in finite structures
with small exchange**, Preprint of Comput.Centre of Acad. Sci.USSR, 1-16. (in Russian)

Alekseeva E.I., Kirzhner V. (1988). **Dynamics of a system with a
chain of internal transformations.Immobile point stability and
effect of models interrelations**. Preprint of Comput.Centre of
Acad.Sci. USSR, 1-26. (in Russian)

*The authors consider coupled systems of autonomous ordinary differential equations. The coupling is given by a graph and formally leads to another ODE. The basic problem is to investigate the effect of coupling for the stability of a stationary point, especially the case when coupling improves stability properties.*

Alekseeva E.I., Kirzhner V. (1989). **On stability of dynamic systems on graphs**. In: *Operation Research*, Comp. Centre of Acad. Sci. USSR, p. 30-44. (in Russian)

Alekseeva E.I., Kirzhner V. (1990). **Dependence of the stability of a dynamic system set on the interrelation structure**. *Soviet Doklady*, **313**, 3, 521-524. (English translation: Soviet Math. Dokl. 42 (1991), 1, 10-13).

Alekseeva E.I., Kuznetsov V, Kirzhner V.(1990) *Collective stability*.
- Moskow, Znanie, 46p. (in Russian)

*In this paper G-systems are investigated. These are dynamic models which describe a network of a finite number of pointwise systems connected with each other by a certain interaction structure. These systems are close to a description of continuous spatially distributed phenomena by equations of the "diffusion-kinetics" type, but at variance of diffusion models, interconnection here is not obligatory local and therefore, the connection structure is an important model parameter. The results achieved in literature for certain connection structures and for some types of point systems show the essential influence of the structure upon collective behaviour.The stability of G-systems with arbitrary structure and low interconnection intensities has been studied. An analytical criterion for the stabilizing influence of a connecting structure among an arbitrary number of two-dimensional systems has been obtained. On this basis, a classification of possible types of connecting structures and of point systems differently reacting to the interaction, is given.*

Alekseeva E.I., Kirzhner V. (1991). **The influence of a connecting structure on stability of a finite set of dynamic systems**. CND, IASI, Ed. Sci. KLIM-ROMA, 1-24.

Alekseeva E. & Kirzhner V. (1994). **Migration on networks and its stability
consequences**. *System Dynamics Rev*., **10**, 63-85

*The behavior of coupled dynamic systems is investigated. The model of continuous/discrete medium which contains a network of a finite number of localized (pointwise) systems (elements)connected by means of certain contacts between elements is considered. These elements are distributed quite dense along 1-(or 2-) dimensional continuum or a certain domain in it. Each pointwise system has its own state and is considered as an active element, which influences the surrounding medium, and ice versa. The nearness of these elements has to be enough providing hat local changes of a media caused by activity of one element could influence neighboring elements. Elements could interact y means of "impulses" transmission at short distance or at long-range distance (using "signal channels").*

Alekseeva E., Kirzhner V. **The influence of diffusion on stability in continuous/discrete media** In: Nonlinear Analysis, Volume 30, Issue 8, December 1997, Pages 4799-4804.(aricle.nonl)

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