Ivanov S. Analysis of local properties of dynamics of autonomous systems on a compact smooth manifold

The thesis is devoted to the research of actual problems in the field of analysis of autonomous systems such as the local structural stability (topologically orbitally equivalence), the local (in the neighborhood of the equilibrium point) diffeomorphity of dynamical systems (autonomous systems) on a compact smooth manifold, and the Kaplan-Yorke fractal dimension. A review of the literature on the study and detection of local properties of dynamics is given. The Grobman-Hartman theorem is discussed. Hyperbolic systems, conditions of its local diffeomorphity without zero and with zero among the eigenvalues of the Jacobian matrix calculated in the neighborhood of an equilibrium point are considered. Much attention has been paid to the Kaplan-Yorke dimension or the Lyapunov dimension for autonomous systems. There is equality of the Lyapunov dimension in topologically and topologically orbitally equivalent systems. The equality of the Lyapunov dimension and the information dimension is confirmed. A description of the Kaplan-Yorke dimension and its relationship with other dimensions are given. The entropy of the normalized vectors of norms of tangent vectors of autonomous systems is introduced. The extreme functional (Lagrangian) of the maximization of the sum of the introduced entropy, the function of exponential divergence (convergence), initial conditions, and normalization for each moment of time is considered. It is shown that the deduced functional corresponds to the principle of maximum (maximum uncertainty) which can also be considered as a propagation of the Bernoulli-Laplace principle of insufficient basis. The corresponding theorems are proved. The definition of introduced entropy and average entropy is given. Dynamic systems described by differential equations on a -measurable compact smooth manifold are not locally different from differential equations on . The Euclidean norm is used. The zero average entropy theorem is proved. The limit of the average entropy is considered. The formula for the relation between the average entropy and the Lyapunov exponents is derived. The maximum of the average entropy corresponds degenerate equilibrium points of the dynamic systems such as a scalar matrix or degenerate node (Jordan cage). When entropy is less than maximum such a degenerate equilibrium points are impossible. Thus, the possible change in the structure of the autonomous system is detected by the increase of the average entropy. That is the direction for a possible change in the type of phase portrait. When the phase space of larger dimension the number of types of such points is already greater, and they should combine the properties of the above types. It is noted that it is advisable to break into subspaces with stable points or unstable of a certain type. In diffeomorphic dynamical systems the Kaplan-Yorke dimension is the same. Takens's theorems on embedding for discrete and continuous time are considered. The delay time is estimated using an autocorrelation function for the time series of one variable of autonomous systems such as the Lorentz system, Ressler system, and Henon map. The Grasberger-Proccaccia dimension for estimating the size of embedding is described. The content of the Grassberger-Proccaccia dimension and the correlation integral is explained. The Lyapunov exponents decomposition shows the detection of the variability of the vector field of the autonomous dynamic system. The ordering of decomposition limits is considered. A numerical algorithm is provided to calculate these limits by a time series and average entropy. The method for estimating the local Jacobian matrix and calculating Lyapunov exponents is substantiated. The analysis and calculation of the Lyapunov exponents, the dimensions, and average entropy for geomagnetic indices Dst, Kp, and AE are given. It is noted that these geomagnetic indices have signs of hyperhaotic dynamics. Keywords: diffeomorphism, locally diffeomorphic systems, topological equivalence, compact smooth manifold, dynamic system, autonomous system, ordinary differential equations, Kaplan-Yorke dimension.