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 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.

scholarly journals Consensus Conditions for High-Order Multiagent Systems with Nonuniform Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Mengji Shi ◽  
Kaiyu Qin ◽  
Ping Li ◽  
Jun Liu
Keyword(s):  
Multiagent Systems ◽  
Time Delays ◽  
Second Order ◽  
High Order ◽  
Sixth Order ◽  
Multiple Time ◽  
State Variables ◽  
Third Order ◽  
First Order ◽  
The Third

Consensus of first-order and second-order multiagent systems has been wildly studied. However, the convergence of high-order (especially the third-order to the sixth-order) state variables is also ubiquitous in various fields. The paper handles consensus problems of high-order multiagent systems in the presence of multiple time delays. Obtained by a novel frequency domain approach which properly resolves the challenges associated with nonuniform time delays, the consensus conditions for the first-order and second-order systems are proven to be nonconservative, and those for the third-order to the sixth-order systems are provided in the form of simple inequalities. The method revealed in this article is applicable to arbitrary-order systems, and the results are less conservative than those based on Lyapunov approaches, because it roots in sufficient and necessary criteria of stabilities. Simulations are carried out to validate the theoretical results.

scholarly journals Consensus Analysis for Third-Order Multiagent Systems in Directed Networks

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yanfen Cao ◽  
Yuangong Sun
Keyword(s):  
Dynamical Systems ◽  
Multiagent Systems ◽  
Sufficient Conditions ◽  
Laplacian Matrix ◽  
Consensus Problem ◽  
Third Order ◽  
Consensus Analysis ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Theoretical Results

We investigate consensus problem for third-order multiagent dynamical systems in directed graph. Necessary and sufficient conditions to consensus of third-order multiagent systems have been established under three different protocols. Compared with existing results, we focus on the relationship between the scaling strengths and the eigenvalues of the involved Laplacian matrix, which guarantees consensus of third-order multiagent systems. Finally, some simulation examples are given to illustrate the theoretical results.

scholarly journals First Integrals of Two-Dimensional Dynamical Systems via Complex Lagrangian Approach

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1244
Author(s):  
Muhammad Umar Farooq ◽  
Chaudry Masood Khalique ◽  
Fazal M. Mahomed
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
First Integrals ◽  
Lagrangian Approach ◽  
Noether Symmetry ◽  
Symmetry Condition ◽  
Two Dimensional ◽  
Physical Systems ◽  
First Order

The aim of the present work is to classify the Noether-like operators of two-dimensional physical systems whose dynamics is governed by a pair of Lane-Emden equations. Considering first-order Lagrangians for these systems, we construct corresponding first integrals. It is seen that for a number of forms of arbitrary functions appearing in the set of equations, the Noether-like operators also fulfill the classical Noether symmetry condition for the pairs of real Lagrangians and the generated first integrals are reminiscent of those we obtain from the complex Lagrangian approach. We also investigate the cases in which the underlying systems are reducible via quadrature. We derive some interesting results about the nonlinear systems under consideration and also find that the algebra of Noether-like operators is Abelian in a few cases.

scholarly journals CARACTERIZAÇÃO MORFOMÉTRICA DA MICROBACIA HIDROGRÁFICA DO CÓRREGO DA FAZENDA GLÓRIA, MUNICÍPIO DE TAQUARITINGA, SP.

Irriga ◽  
2008 ◽  
Vol 13 (3) ◽  
pp. 310-322 ◽  
Author(s):  
Flávia Mazzer Rodrigues ◽  
Teresa Cristina Tarlé Pissara ◽  
Sérgio Campos
Keyword(s):  
Remote Sensing ◽  
Fourth Order ◽  
Land Development ◽  
Morphometric Characteristics ◽  
Third Order ◽  
First Order ◽  
The Relationship ◽  
Number Of Segments ◽  
Morphometric Characterization

Caracterização morfométrica da microbacia hidrográfica do córrego da Fazenda Glória, Município de Taquaritinga, SP.  Flavia Mazzer Rodrigues1; Teresa Cristina Tarlé Pissarra1; Sérgio Campos2 1Departamento de Engenharia Rural, Faculdade de Ciências Agronômicas e Veterinária, Universidade Estadual Paulista, Jaboticabal, SP, [email protected] de Engenharia Rural, Faculdade de Ciências Agronômicas, Universidade Estadual Paulista, Botucatu, SP   1 RESUMO  Com a análise das características morfométricas, procura-se entender a relação solo-superfície, em decorrência dos processos erosivos sobre estruturas e litologias variadas. Neste trabalho, objetivou-se caracterizar, no período de 1983 e 2000, as características morfométricas na Microbacia Hidrográfica do Córrego da Fazenda Glória, de 4a ordem de magnitude, Município de Taquaritinga - SP. Esta microbacia hidrográfica foi dividida em 7 microbacias hidrográficas de 2a ordem e 2 microbacias hidrográficas de 3a ordem. As características morfométricas demonstraram que ocorreu uma redução do número de segmentos de rios de 1.a ordem e comprimento da rede de drenagem ao longo do período analisado, estando relacionadas às diversas influências que a evolução do modelado sofreu, tendo em vista o uso e ocupação do solo, indicando comportamento hidrológico desigual. Os resultados permitiram inferir que o comprimento do segmento de rio de 4a ordem se manteve constante ao longo do período analisado. UNITERMOS: sensoriamento remoto, análise morfométrica, microbacias hidrográficas.  RODRIGUES, F. M.; PISSARRA, T. C. T.; CAMPOS, S. Morphometric characterization of Glória FARM Watershed, Taquaritinga, state of São Paulo, brazil.  2 ABSTRACT The analysis of morphometric characteristics is used to understand the relationship between soil and surface as a result of erosive processes on different structures and lithologies. The objective of this study was to study the morphometric characteristics of Fazenda Gloria watershed from 1983 to2000, afourth-order watershed in  TaquaritingaMunicipality,São PauloState. The study was based on photointerpretation techniques. Drainage net and the respective watersheds were selected and the morphometric variables were determined. The watersheds consisted of 7 second-order watersheds and 2 third-order watersheds. The morphometric characteristics showed a reduction in the number of segments of first-order rivers and in the length of the drainage net during the study period. These findings could be related to several influences on land development considering the occupation and use of land. A different hydrological behavior could also be observed. The analysis of Fazenda Glória Watershed showed that the length of the segment of fourth order river remained constant during the study period. KEY WORDS: remote sensing, morphometric analysis, watershed.

scholarly journals An Overview of the Hamilton–Jacobi Theory: the Classical and Geometrical Approaches and Some Extensions and Applications

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 85
Author(s):  
Narciso Román-Roy
Keyword(s):  
Dynamical Systems ◽  
Classical Field ◽  
Field Theories ◽  
Geometric Description ◽  
Hamiltonian Approach ◽  
Physical Systems ◽  
First Order ◽  
Generic Framework ◽  
Lagrangian And Hamiltonian Formalisms ◽  
Classical Field Theories

This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton–Jacobi theory. The relation with the “classical” Hamiltonian approach using canonical transformations is also analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation.

Higher order rogue waves for the(3 + 1)-dimensional Jimbo–Miwa equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammed K. Elboree
Keyword(s):  
Symbolic Computation ◽  
Fourth Order ◽  
Bilinear Form ◽  
Rogue Waves ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Hirota Bilinear Form ◽  
Hirota Bilinear

Abstract Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Ehsan Mirzakhalili ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical Systems ◽  
Probability Density Function ◽  
Probability Density ◽  
Lorenz System ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
Pitchfork Bifurcations ◽  
The Lorenz System

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

Back-Stepping Control of Quadrotor

2010 ◽  
pp. 484-513 ◽  
Author(s):  
Abhijit Das ◽  
Frank L. Lewis ◽  
Kamesh Subbarao
Keyword(s):  
Neural Network ◽  
Higher Order ◽  
Underactuated System ◽  
Control Law ◽  
State Variables ◽  
Unknown Parameters ◽  
Control Variables ◽  
First Order ◽  
Virtual Control ◽  
Order State

The dynamics of a quadrotor is a simplified form of helicopter dynamics that exhibit the same basic problems of strong coupling, multi-input/multi-output design, and unknown nonlinearities. The Lagrangian model of a typical quadrotor that involves four inputs and six outputs results in an underactuated system. There are several design techniques are available for nonlinear control of mechanical underactuated system. One of the most popular among them is backstepping. Backstepping is a well known recursive procedure where underactuation characteristic of the system is resolved by defining ‘desired’ virtual control and virtual state variables. Virtual control variables is determined in each recursive step assuming the corresponding subsystem is Lyapunov stable and virtual states are typically the errors of actual and desired virtual control variables. The application of the backstepping even more interesting when a virtual control law is applied to a Lagrangian subsystem. The necessary information to select virtual control and state variables for these systems can be obtained through model identification methods. One of these methods includes Neural Network approximation to identify the unknown parameters of the system. The unknown parameters may include uncertain aerodynamic force and moment coefficients or unmodeled dynamics. These aerodynamic coefficients generally are the functions of higher order state polynomials. In this chapter we will discuss how we can implement linear in parameter first order neural network approximation methods to identify these unknown higher order state polynomials in every recursive step of the backstepping. Thus the first order neural network eventually estimates the higher order state polynomials which is in fact a higher order like neural net (HOLNN). Moreover, when these NN placed into a control loop, they become dynamic NN whose weights are tuned only. Due to the inherent characteristics of the quadrotor, the Lagrangian form for the position dynamics is bilinear in the controls, which is confronted using a bilinear inverse kinematics solution. The result is a controller of intuitively appealing structure having an outer kinematics loop for position control and an inner dynamics loop for attitude control. The stability of the control law is guaranteed by a Lyapunov proof. The control approach described in this chapter is robust since it explicitly deals with unmodeled state dependent disturbances without needing any prior knowledge of the same. A simulation study validates the results such as decoupling, tracking etc obtained in the paper.

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