Topics in dynamical systems
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The thesis consists of two parts. the first one is dealing with isosspectral transformations and the second one with the phenomenon of local immunodeficiency. Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors. Chapter 1 analyzes what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks. In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristics (attributes) of the network's nodes (edges). Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. Chapter 2 shows that any orbit of this dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. This attractor is a smaller network where the chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral compressions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that a collection of networks may be spectrally equivalent with respect to some network characteristic but nonequivalent with respect to another. These results suggest a new constructive approach which allows to analyze and compare the topologies of different networks. Some basic aspects of the recently discovered phenomenon of local immunodeficiency generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated in chapter 3. We prove that stable with respect to perturbations local immunodeficiency (LI) already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large, but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topologies. In particular, the scale-free property of a CR network, assumed previously, is not required.