Abstract
For sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form
where is a positive integer,is a function,is a function, andfor are continuous –periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.
Key words
periodic solution; Lienard differential equation; averaging theory; bifurcation theory
Resumo
Para suficientemente pequeno encontramos condições suficientes que garantem a existência de soluções periódicas para sistemas diferenciais de Lienard da forma
onde é um inteiro positivo,é uma função ,é uma função , epara são funções contínuas –periódicas. A principal ferramenta usada neste artigo é a teoria ’’averaging’’ de segunda ordem. Uma aplicação do resultado principal é feita.
Palavras-chave
solucão periódica; equação diferencial de Lienard; teoria da média; teoria de bifurcação
1 - Introduction and statement of the main results
In Ma and Wang (2013MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.) it was studied the existence of periodic solutions for the class of Lienard differential equations of the form
where are continuous and is –periodic. They also assumed that and are locally Lipschitz; and the function is a–isochronous potential, i.e. all nontrivial solutions of are –periodic, where is a positive integer. The authors provided sufficient bounded conditions related with the functions involved in equation (1) to ensure the existence of periodic solutions for this equation. We shall study a particular subclass of equations (1) in which such bounded conditions are not necessary.
In this paper we consider the subclass
of Lienard differential equations (1) where is a positive integer, is a small parameter,is a function in a neighborhood of ,is a function in a neighborhood of , and for are continuous –periodic functions. Note that here we are taking which is a –isochronous potential, already considered in Ma and Wang (2013MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.).
The objective of this paper is to give sufficient conditions on the functions, andto assure the existence of periodic solutions for the equation (2). Here the functions and do not need to satisfy the bounded conditions of Ma and Wang (2013MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.).
In general, to obtain analytically periodic solutions of a differential system, is a very difficult problem, many times impossible. Here using the averaging theory this difficult problem for the differential equations (2) is reduced to find the zeros of a nonlinear system of two functions with two unknowns. We must mention that the averaging theory for finding periodic solutions, in general, does not provide all the periodic solutions of the system. More precisely, the main tool used will be the averaging theory of second order for computing periodic orbits, see Llibre (2002LLIBRE J. 2002. Averaging theory and limit cycles for quadratic systems. Radovi Matematicki 11: 215-228.), BuicÄ and Llibre (2004BUICĂ A AND LLIBRE J. 2004. Averaging methods for finding periodic orbits via brouwer degree. Bull Sci Math 128: 7-22.), Giné et al. (2013GINÉ J, GRAU M AND LLIBRE J. 2013. Averaging theory at any order for computing periodic orbits. Phys D 250: 58-65.), Llibre et al. (2014LLIBRE J, NOVAES DD AND TEIXEIRA MA. 2014. Higher order averaging theory for finding periodic solutions via brouwer degree. Nonlinearity 27: 563-583.). This theory provides a quantitative relation between the solutions of some non–autonomous periodic differential system and the solutions of the averaged differential system, which is autonomous. In this way a finite dimensional function is computed, the simple zeros of this function correspond with the periodic orbits of the non–autonomous periodic differential system for values of a parameter sufficiently small. Here a simple zero of a function means that the Jacobian of at is not zero. For a general introduction to the averaging theory see, for instance, the book Sanders et al. (2007SANDERS JA, VERHULST F AND MURDOCK J. 2007. Averaging methods in nonlinear dynamical systems, volume 59 of Applied Mathematical Sciences Springer, New York, 2nd edition.).
In order to present our results we need some preliminary definitions and notations. We define the constants
and the two functions
and
where
Our main result is the following.
Theorem 1 . Assume that the functions and of the Lienard differential equation (2) satisfy that is a function in a neighborhood of , is a function in a neighborhood of , and. Suppose also that the constants . Then for sufficiently small and for every simple zero of the system
there exists a periodic solution of the differential equation (2) such that and .
Theorem 1 is proved in section 2.
In the next corollary we apply Theorem 1 to a given Lienard differential equation (2) and we show that such an equation has two periodic solutions.
Corollary 2 . Assume that
where
Then, for sufficiently small the Lienard differential equation (2) has two periodic solutions forsuch that
Corollary 2 is proved in section 2.
In Appendix we summarize the averaging theory of second order, for studying periodic solutions that we shall need for proving Theorem 1.
We remark that in Ma and Wang (2013MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.) the results about periodic solutions of system (1) are given, among other hypotheses, under the assumption that the function
is bounded on or that
for some . It is worth emphasizing that Theorem 1 assures the existence of periodic solutions of system (2) without making any bounding assumption. Consequently, the present result detects some periodic solutions which are not detected by the theorems given in Ma and Wang (2013MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.). Indeed, concerning the Corollary 2,. It is easy to see that the above conditions do not hold in this case.
2 - Proof of the results
In this section we shall prove Theorem 1 and Corollary 2.
Proof of Theorem 1. First we shall write the Lienard differential equations (2) in the normal form for applying the averaging theory, see Theorem 3 of the Appendix.
We change the variable by a new variable doing the rescaling. Then equation (3) becomes
Since is a function in a neighborhood of , is a function in a neighborhood of , and , we can write
Thus
We introduce a new variable . Then the differential equation (3) can be written as the differential system
Now we change the variables by the new variables defined through the equality
We do these changes so that the differential system in the new variablesstarts with terms of order and we can apply the averaging theory described in the Appendix. Thus the differential system (4) in the new variables becomes
where
In short, the differential system (6) is in the normal form for applying the averaging theory described in Theorem 3 of the Appendix. Using the notation of the Appendix we have .
Let for. Now we compute the function defined in the Appendix and we get, from the assumptions, that
Since the function we shall apply the averaging theory of second order. So we first compute
Now we are ready to compute the function
defined in the Appendix, and we get the functions and which appear in the statement of the theorem.
Now, from Theorem 3, we obtain that for every sufficiently small and for every simple zero of the system (7), i.e. satisfying that
there exists a periodic solution of the differential system (6) such that
Going back through the change of variables (5) the periodic solution of the differential system (6) becomes the periodic solution
of the differential system (4) such that
Finally, since the periodic solution of the differential system (4>) provides the periodic solution
of the Lienard differential equation (2) such that
Moreover, since the theorem follows.
∎
Proof of Corollary 2. We shall apply the results of Theorem 1 to the Lienard differential equation of the statement of Corollary 2. So we compute the functions and defined just before the statement of Theorem 1, and we obtain
respectively.
Doing the resultant of the functions and with respect to the variable we obtain a cubic polynomial in the variable which has the following three roots
In a similar way doing the resultant of the functions and with respect to the variable we obtain another cubic polynomial in the variable which has the following three roots
From the properties of the resultants it follows that all the solutionsof the system ,are of the form being and some of the above roots. Trying the nine possible solutions, we obtain only two solutions for the system ,, namely
Then applying Theorem 1 the corollary follows.
∎
For more information about the resultants see, for instance, Oliver (1999OLIVER P. 1999. Classical invariant theory, volume 44 of London Math. Soc. Student Texts Cambridge Univ. Press, New York, 2nd edition.).
Acknowledgements
The first author is partially supported by Ministerio de Economía y Competitividad (MiNECO) grant MTM2013-40998-P, by Agència de Gestió d’Ajuts Universitaris i de Recerca (AGAUR) grant number 2013SGR-568, and by Marie Curie International Research Staff Exchange Scheme (IRSES) grants numbers FP7-PEOPLE-2012-IRSES 318999 and 316338. The second author is partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grant number 2015/02517-6. The third author is partially supported by FAPESP grant number 2012/18780-0. The three authors are also supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) grant number 88881.030454/2013-01 from the program CSF-PVE. We thank to the referees for their helpful comments and suggestions.
REFERENCES
- BUICĂ A AND LLIBRE J. 2004. Averaging methods for finding periodic orbits via brouwer degree. Bull Sci Math 128: 7-22.
- GINÉ J, GRAU M AND LLIBRE J. 2013. Averaging theory at any order for computing periodic orbits. Phys D 250: 58-65.
- LLIBRE J. 2002. Averaging theory and limit cycles for quadratic systems. Radovi Matematicki 11: 215-228.
- LLIBRE J, NOVAES DD AND TEIXEIRA MA. 2014. Higher order averaging theory for finding periodic solutions via brouwer degree. Nonlinearity 27: 563-583.
- MA T AND WANG Z. 2013. Periodic solutions of lienard equations with resonant isochronous potentials. Disc Cont Dyn Sys 33: 1563-1581.
- OLIVER P. 1999. Classical invariant theory, volume 44 of London Math. Soc. Student Texts Cambridge Univ. Press, New York, 2nd edition.
- SANDERS JA, VERHULST F AND MURDOCK J. 2007. Averaging methods in nonlinear dynamical systems, volume 59 of Applied Mathematical Sciences Springer, New York, 2nd edition.
Appendix The averaging theory of second order
In this section we recall the averaging theory of second order to find periodic orbits.
Theorem 3 . Consider the differential system
where ,,are continuous functions, T-periodic in the first variable, and is an open subset of. Assume that the following hypothesis (i) and(ii) hold.
-
for all , , , and are locally Lipschitz with respect to . We define as
-
For an open and bounded set and for each , there exist such that and (see its definition later on).
Then for sufficiently small, there exists a periodic solution of the system such that when .
As usual we have denoted by , the Brouwer degree of the function at its fixed point . A sufficient condition for showing that the Brouwer degree of a function at its fixed point is non–zero, is that the Jacobian of the function at (when it is defined) is non–zero.
If the function is not identically zero, then the zeros of are mainly the zeros of for sufficiently small. In this case Theorem 3 provides the so-called averaging theory of first order.
If the function is identically zero and is not identically zero, then the zeros of are the zeros of . In this case Theorem 3 provides the so-called averaging theory of second order.
Publication Dates
-
Publication in this collection
04 Dec 2015 -
Date of issue
Oct-Dec 2015
History
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Received
21 Mar 2014 -
Accepted
20 Feb 2015