scholarly journals Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 978
Author(s):  
Ioannis K. Argyros ◽  
Stepan Shakhno ◽  
Halyna Yarmola
Keyword(s):  
Large Scale ◽  
Inverse Operator ◽  
Vital Role ◽  
Systems Of Equations ◽  
Third Order ◽  
Large Scale Systems ◽  
Physical Systems ◽  
First Order ◽  
Finite Dimensional ◽  
Abstract Spaces

A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the solution. In this article, such methods of higher convergence order are studied. In particular, we develop a two-step iterative method to solve large scale systems that does not require finding an inverse operator. Instead of the operator’s inverting, it uses a two-step Schultz approximation. The convergence is investigated using Lipschitz condition on the first-order derivatives. The cubic order of convergence is established and the results of the numerical experiment are given to determine the real benefits of the proposed method.

COMPLEX BEHAVIOR IN COUPLED CHAOTIC CIRCUITS RELATED BY INTERMITTENCY AND ITS MODELING METHODS

2012 ◽  
Vol 22 (11) ◽  
pp. 1230037
Author(s):  
YOKO UWATE ◽  
YOSHIFUMI NISHIO
Keyword(s):  
Markov Chain ◽  
Large Scale ◽  
Complex Behavior ◽  
Chaotic Circuit ◽  
Large Scale Systems ◽  
First Order ◽  
Modeling Methods ◽  
Order Markov Chain ◽  
Modeling Accuracy ◽  
Chaotic Circuits

We discover a complex behavior when switching three different synchronization states in two coupled chaotic circuits related to intermittency. Two methods to model this interesting complex behavior are proposed. First, we model this complex behavior by a first-order Markov chain with four states. Second, the modeling method by using 1-D Poincaré map derived from one chaotic circuit is proposed. We confirm that both modeling methods agree very well with computer simulations. Finally, we discuss the characteristics of two modeling methods for calculation times, modeling accuracy and application of large-scale systems.

Introducing the Analysis of Bifurcation in Dynamical Systems by Symbolic Computation

2007 ◽  
Vol 44 (4) ◽  
pp. 289-306
Author(s):  
Ubirajara F. Moreno ◽  
Pedro L. D. Peres ◽  
Ivanil S. Bonatti
Keyword(s):  
Dynamical Systems ◽  
Symbolic Computation ◽  
Lorenz System ◽  
State Variables ◽  
Third Order ◽  
Physical Systems ◽  
First Order ◽  
Pitchfork Bifurcations ◽  
The Relationship ◽  
Order State

The aim of this paper is to introduce a few topics about nonlinear systems that are usual in electrical engineering but are frequently disregarded in undergraduate courses. More precisely, the main subject of this paper is to present the analysis of bifurcations in dynamical systems through the use of symbolic computation. The necessary conditions for the occurrence of Hopf, saddle-node, transcritical or pitchfork bifurcations in first order state space nonlinear equations depending upon a vector of parameters are expressed in terms of symbolic computation. With symbolic computation, the relationship between the state variables and the parameters that play a crucial role in the occurrence of such phenomena can be established. Firstly, the symbolic computation is applied to a third order dynamic Lorenz system in order to familiarise the students with the technique. Then, the symbolic routines are used in the analysis of the simplified model of a power system, bringing new insights and a deeper understanding about the occurrence of these phenomena in physical systems.

scholarly journals Evolution of clusters in large-scale dynamical networks

2018 ◽  
pp. 102-129 ◽  
Author(s):  
Anton V. Proskurnikov ◽  
Oleg N. Granichin
Keyword(s):  
Control Systems ◽  
Large Scale ◽  
Consensus Protocol ◽  
Large Scale Systems ◽  
Finite Dimensional ◽  
Tremendous Progress ◽  
Intellectual Control ◽  
Low Dimensionality ◽  
Multi Agent ◽  
Low Dimensional

Recent tremendous progress in electronics, complexity theory and network science provides new opportunities for intellectual control of complex large-scale systems operating in turbulent environment via networks of interconnected miniature devices, serving as actuators, sensors and data processors. Actual dynamics of the resulting control systems are too sophisticated to be examined controlled by traditional methods, which primarily deal with ordinary differential equations. However, their complexity can be dramatically reduced by fast processes, organizing the elementary units of the system (called agents) into relatively small number of clusters. The clusters emerge and deteriorate in response to changes in the environment, and the processes of their formation and destruction are very short in time. During the periods of the clusters’ existence, the system’s dynamics is essentially low-dimensional due to synchronization between the agents in each cluster. An enormously complicated system is thus reduced to a finite-dimensional model with time-varying structure of the state vector. The low-dimensionality of the reduced model allows to control it by using classical methods, e.g. model-predictive or adaptive control. This philosophy of complex systems control is illustrated on an experimental setup, called the “airplane with feathers”. The wings of this airplane are equipped with arrays of microsensors, microcomputers, and microactuators (“feathers”). The feathers self-organize into clusters by using a multi-agent consensus protocol; the aim of this coordination is to reduce the perturbing forces, affecting the airplane in a turbulent flow.

Sign in / Sign up

Export Citation Format

Share Document