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 A Partial Lagrangian Approach to Mathematical Models of Epidemiology

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
I. Naeem ◽  
F. M. Mahomed
Keyword(s):  
Mathematical Models ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
First Integrals ◽  
Order Equation ◽  
Second Order ◽  
Lagrangian Approach ◽  
Hiv Model ◽  
First Order ◽  
Partial Lagrangian

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

FEEDBACK LOOPS, STABILITY AND MULTISTATIONARITY IN DYNAMICAL SYSTEMS

1995 ◽  
Vol 03 (02) ◽  
pp. 409-413 ◽  
Author(s):  
ERIK PLAHTE ◽  
THOMAS MESTL ◽  
STIG W. OMHOLT
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Negative Feedback ◽  
Feedback Loop ◽  
Feedback Loops ◽  
Necessary Condition ◽  
Negative Feedback Loop ◽  
Positive Feedback Loop ◽  
First Order ◽  
First Order Differential Equations

By fairly simple considerations of stability and multistationarity in nonlinear systems of first order differential equations it is shown that under quite mild restrictions a negative feedback loop is a necessary condition for stability, and that a positive feedback loop is a necessary condition for multistationarity.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

2010 ◽  
Vol 20 (05) ◽  
pp. 1477-1483 ◽  
Author(s):  
KONSTANTIN E. STARKOV
Keyword(s):  
Hamiltonian System ◽  
Nonlinear Systems ◽  
Hamiltonian Systems ◽  
Positive Definite ◽  
First Integrals ◽  
Invariant Sets ◽  
Yang Mills ◽  
First Order ◽  
Special Case ◽  
Extremum Conditions

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

scholarly journals Exact Solutions for Certain Nonlinear Autonomous Ordinary Differential Equations of the Second Order and Families of Two-Dimensional Autonomous Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
M. P. Markakis
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Autonomous Systems ◽  
Closed Form Solution ◽  
First Integrals ◽  
Form Solution ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Abel Equations

Certain nonlinear autonomous ordinary differential equations of the second order are reduced to Abel equations of the first kind ((Ab-1) equations). Based on the results of a previous work, concerning a closed-form solution of a general (Ab-1) equation, and introducing an arbitrary function, exact one-parameter families of solutions are derived for the original autonomous equations, for the most of which only first integrals (in closed or parametric form) have been obtained so far. Two-dimensional autonomous systems of differential equations of the first order, equivalent to the considered herein autonomous forms, are constructed and solved by means of the developed analysis.

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.

Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

2014 ◽  
Vol 24 (07) ◽  
pp. 1430020 ◽  
Author(s):  
Paulo C. Rech
Keyword(s):  
Dynamical Systems ◽  
Continuous Time ◽  
Nonlinear Dynamical Systems ◽  
Angular Frequency ◽  
Two Dimensional ◽  
Periodic Forcing ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Nonlinear Dynamical ◽  
First Order

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

The Artificial Hamiltonian, First Integrals, and Closed-Form Solutions of Dynamical Systems for Epidemics

2018 ◽  
Vol 73 (4) ◽  
pp. 323-330 ◽  
Author(s):  
Rehana Naz ◽  
Imran Naeem
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Closed Form ◽  
Hamiltonian Systems ◽  
Physical Structure ◽  
First Integrals ◽  
Second Order ◽  
Hamiltonian Structures ◽  
Closed Form Solutions ◽  
First Order

AbstractThe non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

scholarly journals Parallel Numerical Creation of Phase-space Diagrams of Nonlinear Systems Using Maple

2018 ◽  
Vol 11 (3) ◽  
pp. 132-149
Author(s):  
F. Hajdu ◽  
Gy. Molnárka
Keyword(s):  
Numerical Calculation ◽  
Phase Space ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Parallel Algorithm ◽  
Phase Plane ◽  
Parallel Computers ◽  
Two Dimensional ◽  
Huge Amount ◽  
Large Systems

In this paper the numerical creation of phase-plane diagrams in parallel utilizing Maple is presented. One of the most effective method for studying nonlinear systems is the creation of detailed enough phase-plane diagrams. But in the case of large systems it requires huge amount of numerical calculation, which can be accelerated using parallel computers. Here we show some attempts for this using moderate size known problems. We demonstrate that detailed diagrams can be created fast and efficiently with a SIMD model based algorithm even using simple PC-s. We exhibit that the parallel algorithm taken for one- and two-dimensional problems can be expanded for 3D phase-space diagram creation without any loss of efficiency. In this paper a methodology is showed which can be followed in the study of large dynamical systems as well.

scholarly journals Formal First Integrals of General Dynamical Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Jia Jiao ◽  
Wenlei Li ◽  
Qingjian Zhou
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Invariant Manifolds ◽  
First Integrals ◽  
Higher Dimensional ◽  
Complete Study ◽  
Planar Systems

The goal of this paper is trying to make a complete study on the integrability for general analytic nonlinear systems by first integrals. We will firstly give an exhaustive discussion on analytic planar systems. Then a class of higher dimensional systems with invariant manifolds will be considered; we will develop several criteria for existence of formal integrals and give some applications to illustrate our results at last.

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.

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