On a basis of local-topological approach we propose modification of a correlation integral approach that solves a problem of minimizing computer resources for fractal analysis implementation and makes the employed algorithm insensitive to enlarging phase space dimension. The effective approach for estimating characteristic equation components for extended Jacobian-matrix is developed, that allows determination of characteristic exponents with respect to nonlinear feedback modeling. Application of developed methods to neural systems is explored.
We propose to use characteristic exponents of averaged instability for diagnostics of complex systems. It is shown that these exponents can be calculated from eigenvalues of the product matrix obtained from Jacobian-matrixes of the multi-step transformation. The effective scheme for calculating the product matrix in a finite time interval is developed, it allows to reduce calculation complexity due to exact factorization of the Jacobian-matrix in restricted time intervals. The detailed analytical analysis of the reconstruction process is implemented with respect to delay nonlinear equations for both Runge-Kutta and Euler approaches.
The method of multifractal analysis for multidimensional attractor is proposed, the adaptive segmentation of phase trajectories is used analogously generalized correlation integral approach. A rarefied sequence of points of phase trajectory just forms the centers of segmentation cells, the upper boundary of such rarefying is estimated from statistical analysis of quasi-periodicity along phase trajectories of multidimensional attractor. This method allows to reduce a required computer time in comparison with traditional correlation integral approach and provides a good convergence. So far as the attractor reconstruction was implemented in Takens phase space, analytical approaches for product scheme of functional matrixes derived from delay differential iterations is developed for such phase space. Numerical results for eigenvalues evolution of obtained functional matrixes and for multifractal investigation are represented.
For integrated instability investigation, a sequence of functional matrixes multiplication is transformed by separation of elements describing linear and nonlinear transformations in delay difference map. A quantity of multiplications of diagonal matrixes is minimized by means of dividing initial functional matrix into diagonal, triangle and the matrix of unit cyclic shift. The analytical expressions for product matrix have been obtained and different approaches of reducing the product matrix to rarefied form have been investigated. The analytical scheme for calculating rarefied product matrix has been constructed, its operations are invariant to changing a common length of investigated time series and transformed segments. The developed model can be used for characteristic polynomial determination and for increasing efficiency of computational algorithm of Lyapunov exponents calculation.
The development of locally topological analysis resulting in reduction of required computer resources and experimental data has been proposed. Calculating local nonuniformity for obtained topological sequence provides a high reliability and accuracy of statistical parameters definition for the attractor under investigation and for complex system diagnostics. Segment variance exploration by partition into both fixed length regions and variable left boundary segments with adaptive transformations allows to increase accuracy of obtained results. The generalized model of adaptive transformations formalism has been represented. Numerical experiments with chaotic time series obtained from delay differential equation confirm the validity of developed methods.
The methods of delay nonlinear system exploration for minelike objects detection have been developed by means of analytical and scheme construction of delay discrete transformation. Conditions of stability for proposed transformation have been defined from eignevalues analysis for corresponding Jacobian-matrix of delay functional operator. The evolution of the object under investigation is considered as dissipativity described by the vector field divergency. The corresponding numerical simulations confirmed the reliability of proposed methods.
The analytical approach based on modeling Takens phase space temporal transformation for nonlinear delay discrete iterations is proposed. This model allows to obtain expressions for functional matrix of the vector map reconstructed phase trajectories from delay discrete iterations in multidimensional phase space. On a base of the suggested mode, analytical expressions for embedding parameters of investigated attractor have been calculated. Some computer experiments have been implemented modeling of the vector feedback in nonlinear processes under investigation.
The conditions of topological stabilization for Takens attractor are investigated when enlarging phase space dimension by means of topological dependences analysis and asymptotical estimations calculation. It has been shown that ideal topological stabilization is equivalent to presence of linearized segmentation properties in time series under investigation, these properties being invariant concerning scale of partition. The exact ideal topological stabilization appears to be impossible for nonlinear time series, so we have to consider asymptotical estimations of convergence to exact stabilization. Some statistical characteristics of investigated attractor have been defined by relatively- difference investigation of obtained topological curves.
Multidimensional phase space of a system attractor is transformed into the space constructed by distance topological sequences created from measured digitized signal. This transformation allows to reduce calculation complexity of attractor characteristics determination and experimental data quantity. It has been proved that representation of the attractor by means of distance topological sequences is sufficient for determination of all parameters of phase trajectories. Numerical simulations showed high convergence and reliability of proposed method.
The method of minimal embedding dimension definition of the chaotic attractor on the basis of topological structure dynamics analysis of phase trajectories is proposed. It is shown that the suggested method yields reduction of experimental data quantity and computer resources about an order in comparison with traditional methods. This reduction is achieved due to localizing topological analysis of the phase trajectories. The proposed method can be used of investigation of nonlinear dynamical systems with chaotic behavior in technique, biology, medicine.