First Integrals and Integrating Factors of Second-Order Autonomous Systems

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Tamás Kalmár-Nagy ◽  
Balázs Sándor
Keyword(s):  
Harmonic Oscillator ◽  
Autonomous Systems ◽  
Dynamical Symmetry ◽  
Nonlinear Oscillators ◽  
First Integrals ◽  
Second Order ◽  
Point Symmetry ◽  
New Approach ◽  
First Order ◽  
Integrating Factors

We present a new approach to the construction of first integrals for second-order autonomous systems without invoking a Lagrangian or Hamiltonian reformulation. We show and exploit the analogy between integrating factors of first-order equations and their Lie point symmetry and integrating factors of second-order autonomous systems and their dynamical symmetry. We connect intuitive and dynamical symmetry approaches through one-to-one correspondence in the framework proposed for first-order systems. Conditional equations for first integrals are written out, as well as equations determining symmetries. The equations are applied on the simple harmonic oscillator and a class of nonlinear oscillators to yield integrating factors and first integrals.

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.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

Reduction approach to second order perturbed state-dependent sweeping process

2019 ◽  
Vol 28 (2) ◽  
Author(s):  
MUSTAPHA FATEH YAROU
Keyword(s):  
Fixed Point Theory ◽  
Point Theory ◽  
Second Order ◽  
Sweeping Process ◽  
New Approach ◽  
Standard Methods ◽  
First Order ◽  
Finite Dimensional ◽  
Perturbed State ◽  
State Dependent

In this paper, we present a new approach to solving second order nonconvex perturbed sweeping process in finite dimensional setting. It consists in a reduction of the problem to a first order one without use of the standard methods of fixed point theory. The perturbation, that is the external force applied on the system is not necessary with bounded values.

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.

scholarly journals On the complete integrability and linearization of certain second-order nonlinear ordinary differential equations

2005 ◽  
Vol 461 (2060) ◽  
pp. 2451-2477 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Free Particle ◽  
First Integrals ◽  
Second Order ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Integrals Of Motion ◽  
Integrating Factors

A method for finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle–Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. We also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second-order nonlinear ordinary differential equations to free particle equations. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

scholarly journals Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Emrullah Yaşar
Keyword(s):  
First Integral ◽  
First Integrals ◽  
The Other ◽  
First Order ◽  
Integrating Factor ◽  
Damped Oscillators ◽  
Damped Oscillator ◽  
Integrating Factors ◽  
Oscillator Equations

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute -symmetries and the associated reduction algorithm. The knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

On the Inverse Problem of Competitive Modes and the Search for Chaotic Dynamics

2017 ◽  
Vol 27 (10) ◽  
pp. 1730032 ◽  
Author(s):  
Lewis Ruks ◽  
Robert A. Van Gorder
Keyword(s):  
Dynamical System ◽  
Inverse Problem ◽  
Autonomous Systems ◽  
Nonlinear Dynamical System ◽  
Second Order ◽  
Order System ◽  
Nonlinear Dynamical ◽  
First Order ◽  
Competitive Modes ◽  
Hyperchaotic Systems

Generalized competitive modes (GCM) have been used as a diagnostic tool in order to analytically identify parameter regimes which may lead to chaotic trajectories in a given first order nonlinear dynamical system. The approach involves recasting the first order system as a second order nonlinear oscillator system, and then checking to see if certain conditions on the modes of these oscillators are satisfied. In the present paper, we will consider the inverse problem of GCM: If a system of second order oscillator equations satisfy the GCM conditions, can we then construct a first order dynamical system from it which admits chaotic trajectories? Solving the direct inverse problem is equivalent to finding solutions to an inhomogeneous form of the Euler equations. As there are no general solutions to this PDE system, we instead consider the problem for restricted classes of functions for autonomous systems which, upon obtaining the nonlinear oscillatory representation, we are able to extract at least two of the modes explicitly. We find that these methods often make finding chaotic regimes a much simpler task; many classes of parameter-function regimes that lead to nonchaos are excluded by the competitive mode conditions, and classical knowledge of dynamical systems then allows us to tune the free parameters or functions appropriately in order to obtain chaos. To find new hyperchaotic systems, a similar approach is used, but more effort and additional considerations are needed. These results demonstrate one method for constructing new chaotic or hyperchaotic systems.

A Driving-Gear Dynamic Analysis

2018 ◽  
Vol 931 ◽  
pp. 422-427
Author(s):  
Yevgeniy M. Kudryavtsev
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Dynamic Analysis ◽  
Second Order ◽  
New Approach ◽  
First Order ◽  
Gear Design ◽  
Third Stage ◽  
Mathcad Software ◽  
High Level

A new approach of mechanical driving-gear dynamic analysis, which includes several modelling stages is observed in the article. On the first stage driving-gear is represented in the form of consistently connected rotation bodies. The driving-gear is represented in a graphic kind by means of the marked graph. On the second stage mathematical model of driving-gear performance with using of mnemonic rule is created. Mathematical model of mechanical driving-gear is a system of second-order regular differential equations (RDEs). The system of second-order regular differential equations is transformed into a system of first-order regular differential equations. There is a standard method for writing a higher-order RDE as a system of the first-order RDEs. On the third stage computer model of driving-gear performance using system Mathcad is created and initial data is defined. On the fourth stage the mechanical driving-gear modelling is performed and calculation data in numerical and graphical forms is obtained. This approach provides high level of the driving-gear dynamic analysis, including the received results presentation, which is especially important on the earliest stages of mechanical driving-gear design. The proposed procedure of mechanical driving-gear dynamic analysis using Mathcad software significantly decreases time and working costs on execution of such computations and helps to execute investigations related with changing of driving-gear elements parameters efficiently.

First integrals and exact solutions of some compartmental disease models

2019 ◽  
Vol 74 (4) ◽  
pp. 293-304
Author(s):  
Burhan Ul Haq ◽  
Imran Naeem
Keyword(s):  
Hamiltonian System ◽  
Exact Solutions ◽  
First Integrals ◽  
Second Order ◽  
Disease Models ◽  
Disease Dynamics ◽  
Hamiltonian Approach ◽  
First Order ◽  
Hamiltonian Operators ◽  
Second Order Equations

AbstractThe notions of artificial Hamiltonian (partial Hamiltonian) and partial Hamiltonian operators are used to derive the first integrals for the first order systems of ordinary differential equations (ODEs) in epidemiology, which need not be derived from standard Hamiltonian approaches. We show that every system of first order ODEs can be cast into artificial Hamiltonian system $\dot{q}=\frac{{\partial H}}{{\partial p}}$, $\dot{p}=-\frac{{\partial H}}{{\partial q}}+\Gamma(t,\;q,\;p)$ (see [1]). Moreover, the second order equations and the system of second order ODEs can be written in the form of artificial Hamiltonian system. Then, the partial Hamiltonian approach is employed to derive the first integrals for systems under consideration. These first integrals are then utilized to find the exact solutions of models from the epidemiology for a distinct class of population. For physical insights, the solution curves of the closed-form expressions obtained are interpreted in order for readers understand the disease dynamics in a much deeper way. The effects of various pertinent parameters on the prognosis of the disease are observed and discussed briefly.

scholarly journals A Variational Approach to an Inhomogeneous Second-Order Ordinary Differential System

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
B. Muatjetjeja ◽  
C. M. Khalique
Keyword(s):  
Positive Solutions ◽  
Differential System ◽  
Variational Approach ◽  
First Integrals ◽  
Second Order ◽  
Point Of View ◽  
Lagrangian Formulation ◽  
Multiple Positive Solutions ◽  
First Order ◽  
Ordinary Differential System

This paper studies the coupled inhomogeneous Lane-Emden system from the Lagrangian formulation point of view. The existence of multiple positive solutions has been discussed in the literature. Here we aim to classify the system with respect to a first-order Lagrangian according to the Noether point symmetries it admits. We then obtain first integrals of the various cases which admit Noether point symmetries.

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