Geometrical Framework Application Directions in Identification Systems. Review

The approaches review of the framework application in identification problems is fulfilled. It is showed that this concept can have different interpretations of identification problems. In particular, the framework is understood as a frame, structure, system, platform, concept, and basis. Two directions of this concept application are allocated: 1) the framework integrating the number of methods, approaches or procedures; b) the mapping describing in the generalized view processes and properties in a system. We give the review of approaches that are the basis of the second direction. They are based on the analysis of virtual geometric structures. These mappings (frameworks) differ in the theory of chaos, accidents, and the qualitative theory of dynamic systems. Introduced mappings (frameworks) are not set a priori, and they are determined based of the experimental data processing. The main directions analysis of geometrical frameworks application is fulfilled in structural identification problems of systems. The review includes following directions: i) structural identification of nonlinear systems; ii) an estimation of Lyapunov exponents; iii) structural identifiability of nonlinear systems; iv) the system structure choice with lag variables; v) system attractor reconstruction.

Time-Reversible Chaotic System with Conditional Symmetry

When the polarity reversal induced by offset boosting is considered, a new regime of a time-reversible chaotic system with conditional symmetry is found, and some new time-reversible systems are revealed based on multiple dimensional offset boosting. Numerical analysis shows that the system attractor and repellor have their own dynamics in respective time domains which constitutes the fundamental property in a time-reversible system. More remarkably, when the conditional symmetry is destroyed by a slightly mismatched offset controller, the system undergoes different bifurcations to chaos, and the corresponding coexisting attractors and repellors shape their own phase trajectories.

Maximizing Sensitivity Vector Fields: A Parametric Study

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

Some Algebraic Properties of Continuous Time Dynamical Systems

Number of zero Lyapunov exponents of a system is directly related to the dimension of the manifold of the system attractor. Moreover, this attractor dimension is governed by the algebraic structure of the manifold it lives on. In this work we try to establish a basis for the description of this manifold, which we aim to use in determining zero Lyapunov exponents of a continuous time dynamical system.

Homoclinics in the Reconstruction of Dynamic Systems From Experimental Data

The role of homoclinic effects in solution of a reconstruction problem of system attractors and model equations from experimental observable in the presence of external noise is investigated numerically. It is shown that the possibility of reconstruction essentially depends on character of origin system homoclinic trajectories and noise intensity. If the homoclinic structure belongs to the attractor, then the reconstruction results in restoration origin system attractors. A small noise influence causes in this case a small perturbation of attractors probability measure and practically disappears due to filtering properties of the reconstruction algorithm. The homoclinic structure does not belong to the attractor, then in the absence of noise the probability measure concentrates at the attractor, the structure of which is not defined by the homoclinics. The noise perturbation induces new regimes. Then the attractor structure essentially depends on the homoclinics structure and noise level. In this case the model system attractor of which reproduces “invisible” homoclinic structure, is obtained as a result of reconstruction.

LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA

We review the idea of Lyapunov exponents for chaotic systems and discuss their evaluation from observed data alone. These exponents govern the growth or decrease of small perturbations to orbits of a dynamical system. They are critical to the predictability of models made from observations as well as known analytic models. The Lyapunov exponents are invariants of the dynamical system and are connected with the dimension of the system attractor and to the idea of information generation by the system dynamics. Lyapunov exponents are among the many ways we can classify observed nonlinear systems, and their appeal to physicists remains their clear interpretation in terms of system stability and predictability. We discuss the familiar global Lyapunov exponents which govern the evolution of perturbations for long times and local Lyapunov exponents which determine the predictability over a finite number of time steps.