Analysis of Lienard II-type oscillator equation by symmetry-transformation methods
- Özlem Orhan^{1} and
- Teoman Özer^{2}Email author
https://doi.org/10.1186/s13662-016-0966-4
© Orhan and Özer 2016
Received: 21 October 2015
Accepted: 6 September 2016
Published: 12 October 2016
Abstract
In this study, we consider a Lienard II-type harmonic nonlinear oscillator equation as a nonlinear dynamical system. Firstly, we examine the first integrals in the form \(A(t,x)\dot{x}+B(t,x)\), the corresponding exact solutions and the integrating factors. In addition, we analyze other types of the first integrals via the λ-symmetry approach. We show that the equation can be linearized by means of a nonlocal transformation, the so-called Sundman transformation. Furthermore, using the modified Prelle-Singer approach, we point out that explicit time-independent first integrals can be identified for the Lienard II-type harmonic nonlinear oscillator equation.
Keywords
1 Introduction
In addition, the modified Prelle-Singer procedure [17, 18] is used to apply it to a class of second-order nonlinear ordinary differential equations, to solve several physically interesting nonlinear systems, and to identify a number of important linearization procedures. Prelle and Singer have proposed an algorithmic procedure to find the integrating factor for the system of first-order ordinary differential equations. Once the integrating factor for the equation is determined, it leads to a time-independent integral of motion for the first-order ordinary differential equation. The Prelle-Singer method guarantees that if a first-order ordinary differential equation has a first integral in terms of elementary functions, then this first integral can be found. This method has been generalized to incorporate the integrals with nonelementary functions. Recently, this theory is generalized to obtain general solutions for second- and higher-order ordinary differential equations without any integration [18].
This study is organized as follows. In Section 2, we present some fundamental definitions and theorems. In Section 3, we discuss the nonlinear Lienard II-type harmonic nonlinear oscillator equation and the corresponding linearization methods. Furthermore, the first integral, the λ-symmetry, the integrating factor, and the transformation pair are presented. In Section 4, we apply the modified Prelle-Singer method to the Lienard II-type harmonic nonlinear oscillator equation to obtain Lie symmetries, the first integrals, λ-symmetries, the integrating factors and the Lagrangian-Hamiltonian functions. The last section summarizes some important results and discussions in the study.
2 Preliminaries
2.1 The first integrals of the form \(A(t,x)\dot{x}+B(t,x)\)
We can say that if \(S_{1}=0\), then \(S_{2}=0\). Equation (1.1) is S-linearizable if and only if \(S_{2}=0\). By these definitions we have the following theorem to determine \(A(t,x)\) and \(B(t,x)\).
Theorem 1
[11]
2.2 The λ-symmetries and the integrating factors
Theorem 2
[11]
2.3 The nonlocal transformations
Theorem 4
[12]
2.4 Lagrangian and Hamiltonian description
3 The first integral, λ-symmetry, and the integrating factor of Lienard II-type harmonic nonlinear oscillator equation
Definition 1
Proposition 1
A dynamical system is called isochronous if it features in its phase space an open, fully-dimensional region where all its solutions are periodic in all its degrees of freedom with the same, fixed, period. In order the Lienard II-type harmonic nonlinear oscillator equation to belong to this class, it must satisfy the isochronous condition to be linearized. Thus, if we apply (3.3) to equation (3.1), then we see that Lienard II-type harmonic nonlinear oscillator equation satisfies the isochronous condition.
Proof
3.1 The first integral of the form \(A(t,x)\dot{x}+B(t,x)\) and the invariant solution
It can be shown that the Lienard II-type harmonic nonlinear oscillator equation (3.1) has the first integral of the form \(A(t,x)\dot {x}+B(t,x)\) by determining the functions A and B using a procedure given above. Then, the equation can be integrated by using this first integral, and the exact solution of the equation can be obtained.
Proposition 2
Proof
In the literature, the phase plane method refers to graphically determining the existence of limit cycles in the solutions of the oscillator equations. The solutions to a nonlinear differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field. The vectors representing the derivatives of the points with respect to a parameter time t at representative points are drawn. With enough of these arrows in place, the system behavior over the regions of plane in analysis can be visualized, and the limit cycles can be identified. Then a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve or point.
The phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the state space. This reveals information such as whether an attractor, a repeller, or a limit cycle is presented for the chosen parameter value. The concept of topological equivalence is important in classifying the behavior of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. A phase portrait graph of a dynamical system depicts the system trajectories.
Remark 1
Remark 2
Remark 3
A contour plot is a graphical technique for representing a three-dimensional surface by plotting constant z slices, called contours, on a two-dimensional format. That is, given a value for z, lines are drawn for connecting the \((x,y)\) coordinates where that z value occurs. The contour plot is an alternative to a 3-D surface plot. The independent variables are usually restricted to a regular grid. An additional variable may be required to specify the z values for drawing the isolines. If the function does not form a regular grid, you typically need to perform a 2-D interpolation to form a regular grid. The contour plot is used to answer the question ‘How does z change as a function of x and y?’
3.2 The λ-symmetry and the nonlocal transformation pair
Proposition 3
We consider equation (3.1) and the functions \(S_{1}\), \(S_{2}\) defined by (2.1), (2.2). The condition \(S_{1}=S_{2}=0\) is satisfied if and only if \(\partial _{x}\) is a λ-symmetry of (3.1) for \(\lambda=\frac {1}{3}\omega\tan(\frac{1}{3}(-\omega t+9 \omega c_{1}))-\frac{2\dot {x}}{3x}\).
Proof
Proposition 4
Equation (3.1) has a transformation pair F and G, and the equation can be linearized using this pair. Then the first integral is obtained from this transformation pair.
Proof
4 The extended Prelle-Singer method and λ-symmetry relation
In this section, we consider other types of the first integrals and the exact solutions by using the Prelle-Singer method and its relation to λ-symmetry. This method provides not only the first integrals but also integrating factors. Moreover, we can define the Hamiltonian and Lagrangian forms of the differential equations by using the extended Prelle-Singer method. In this section, we consider the first integrals and exact solutions of the Lienard II-type harmonic nonlinear oscillator equation by the approach related to the Prelle-Singer symmetry, λ-symmetry, and Lie point symmetry as different concepts from the mathematical point of view.
4.1 The time-independent first integrals
4.2 The exact solution of the equation using the λ-symmetries based on a linearization method
- 1.Find a first integral \(w(t,x,\dot{x})\) of \(\upsilon^{[\lambda,(1)]}\), that is, a particular solution of the equationwhere subscripts denote partial derivatives with respect to that variable, and \(\upsilon^{[\lambda,(1)]}\) is the first-order λ-prolongation of the vector field υ.$$ w_{x}+\lambda w_{\dot{x}}=0, $$(4.23)
- 2.Evaluate \(A(w)\) and express \(A(w)\) in terms of \((t,w)\) as \(A(w)=F(t,w)\), and the operator A is defined in the form$$ A=\partial_{t}+\dot{x}\partial_{x}+ \phi(t,x, \dot{x})\partial _{\dot{x}}. $$(4.24)
- 3.
Find a first integral G of \(\partial_{t}+F(t,w)\partial_{w}\).
- 4.
Evaluate \(I(t,x,\dot{x})=G(t,w(t,x,\dot{x}))\). Then \(I(t,x,\dot{x})\) is a first integral, and \(\mu(t,x,\dot {x})=I_{\dot{x}}\) is an integrating factor of the given second-order equation.
Furthermore, it is possible to show that we can find other forms of the first integrals and the integrating factors rather then the forms given by (4.31) and (4.32) for the same null function S. With this aim, we consider again (4.23) and substitute this form of S into equation (4.7):
5 Concluding remarks
The Lienard II-type nonlinear harmonic oscillator equation has a natural generalization in three dimensions and can be interpreted as an oscillator constrained to move on a three-sphere. Such a problem is highly nonlinear. In this study, we analyze the first integral of the form \(A(t,x)\dot {x}+B(t,x)\), the λ-symmetries, and the integrating factors of the Lienard II-type nonlinear harmonic oscillator equation, which is a second-order nonlinear ordinary differential equation.
Firstly, we have characterized the second-order nonlinear ordinary differential equations, and this characterization is given by the coefficients of the equation and also determines the first integral, the λ-symmetry, and the integrating factor. Thus, the Lienard II-type nonlinear harmonic oscillator equation is classified by using functions \(S_{1}\) and \(S_{2}\), and the first integral of the form \(A(t,x)\dot{x}+B(t,x)\) is obtained by an algorithm. Moreover, we presented some properties and characterization of the equation that admits a vector field as λ-symmetry. Linearization, the symmetries, and the transformation of equations play a crucial role. Furthermore, the nonlinear second-order ordinary differential equations can be linearized by a Sundman transformation. Finally, we apply a Sundman transformation to the Lienard II-type nonlinear harmonic oscillator equation.
We have identified the time-independent first integrals for the Lienard II-type nonlinear harmonic oscillator equation using the modified Prelle-Singer approach. Moreover, we have constructed appropriate Lagrangian and Hamiltonian functions from the time-independent first integrals and transformed the corresponding Hamiltonian forms to standard Hamiltonian forms. The important point of the Prelle-Singer procedure lies in finding explicit solutions satisfying all three determining equations (4.6)-(4.8). In our study, we have taken specific ansatz forms to determine the null forms S and the integrating factor R. Finally, from our detailed analysis we have shown these results with the phase portraits depending on the choice of parameters, and using these phase portraits, we interpreted geometric meanings of the solutions. Using the Hamiltonian and the conjugate momentum functions, we demonstrated relations among the solutions, Hamiltonians, and conjugate momentum functions by contour plot portraits.
Declarations
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Authors’ Affiliations
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