# General Description

Proposed theme: One of the research areas of applied nonlinear dynamics mathematics is the analysis of certain process with origins in problems from chaos theory and which can be figural by means of nonhomogeneous, degenerated and singular partial equations. Such models can be represented by particular classes of partial equations which can be graphically modulated by fractals. Systems of contraction mappings had been considered previously, by a numbered author for various purposes. Fractal Image Coding has numerous open problems as block-based fractal image coding and fractal image quantum compression, but in practical implementation we can observe that the necessary computation time is discouraging. It is need of a new mathematical model part of the way to treat it. For example, in the classically Mandelbrot set (sets with non-integral Hausdorff dimension), in a less amount of parameters change the graphically representation change almost all. For the applications means big entropy which can lead in a good data compression representation but for a partial equation solution means instability at first sight.

Research Subjects The term of nonlinear describes a wide class of systems. We intend to study those new one which had none linearly output proportional to the strength of an applied stimulus. It can be applied in medicine, information theory (cryptography and coding theory) and (but not restrictive) to condensed matter physics. In this context, one of the project's purpose is to study some new properties concerning the spaces mentioned before and to introduce new spaces of this kind, adequate to the study of some particular problems. We note that our attention has already been captured by the study of some related problems mentioned above. The differential equation study in the area of fractal, named “fractal analysis”, deals with some analytic questions in which the underlying space is fractal. One of the main examples in this domain is to develop an improved theory of function spaces on fractal subset of Rn to the set, from the initial idea of the Whitney extension theorem (the existence of orthonormal bases of complex exponents with frequencies in a spectrum). Another approach based on calculus, is due to Jun Kigami with detailed and formal presentation of operation on functions are defined as limits of discrete analogs. One the actual research in the domain due to set a small eigenvalues bounded by an open neighborhood of 0, for a specific function F where hard to compute delta functions (to develop fractal specifications) has to be defined. The classical case of Dirichlet spectrum can be used in the next parameter adjustments: the constant function is made using Neumann eigenfunction with all lm = 0 and l = 0, there is no 2-series, the series has multiplicity increased by 3, namely the eigenfunctions associated to the boundary points, the 5-series has multiplicity reduced by 2, since we retain all eigenfunctions associated with loops, but discard the two extra ones that chain from one boundary point to another. The 5-series begin with m0 = 0. It is well know that to localize a specific number of eigenfunctions used to improve the inter fractal parameters is an open problem for certain particular case. It is necessary to define dn eigenvalues in the bottom part of the Dirichlet spectrum consisting of the first (3^(m+1)-3)/2 eigenvalues that are identical to the proportion of d_n eigenfunctions in the Dirichlet spectrum of d_m. If we denote the total number of dn eigenfunctions in the Dirichlet spectrum of d_m. For the initial eigenvalue l_m=6, all (3^m-3)/2 eigenfunctions are d_n. To develop a solution in this area it is necessary to be able to compute inner products of eigenfunctions. In addition, it is necessary to have Dirichlet boundary conditions that change the parameter m into an adequate value. We intend to study the cases in which exists positive solutions, unique normalized solution and if the unique normalized solution attracting exists. This can be interpreted like the solution in the limit by starting anywhere and iterating the renormalization map, followed by normalizing.

Main research points consists in the construction of multidimensional nonlinear evolutive integrable equations and to study the geometrical properties of nonlinear dynamical systems and to found a system of generators for the multiple solutions with applications (for example in the key generation which is necessary for the cryptographic process). These models are based on topological properties of the solution space for the parameters generated by these types of equations.

# Young Researchers

By his structure, the project focus on development of research capacities of young people and their training in areas of high importance of theoretical and applied mathematics. Fulfilling this goal is apparent in the articles described in section results, which were accepted in publications of international circulation.

All the project's objectives, according with the schedule, were fullfield, more than that, were overcounted. Beside these, were created new research lines. All of these can be shown from Rezults section