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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 212340, 19 pages
http://dx.doi.org/10.1155/2013/212340
Research Article

Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited

1Campus Avançado de Itabira, Universidade Federal de Itajubá, Rua São Paulo 377, Bairro Amazonas, 35.900–373 Itabira, MG, Brazil
2Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, 37.500–903 Itajubá, MG, Brazil
3Instituto de Sistemas Elétricos e Energia, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, 37.500–903 Itajubá, MG, Brazil

Received 23 June 2013; Accepted 8 July 2013

Academic Editor: Sakthivel Rathinasamy

Copyright © 2013 Denis de Carvalho Braga et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main goal of this paper is to present a theory of approximation of periodic orbits of vector fields in the plane. From the theory developed here, it is possible to obtain an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two. Applications of the developed theory are made in Liénard-type equations and in Bazykin’s predator-prey system.

1. Introduction

The existence of a curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two can be demonstrated with the theories presented in [1, 2]. However, these theories do not allow us to find or even approximate the curve of nonhyperbolic periodic orbits, except in very special cases as in [3]. On the other hand, good approximations to this curve are essential not only to mathematicians, but primarily for engineers, physicists, and other users of mathematics.

In general, the curve of nonhyperbolic periodic orbits is obtained by numerical methods as in [4] or through specific softwares such as [5], for instance. An analytical alternative proposed in this paper is to generalize the theory of approximation of periodic orbits of [6], using some results and notations of [1, 2], in order to obtain an approximation to the curve of nonhyperbolic periodic orbits of a family of differential equations that has transversal Hopf bifurcations of codimension two. Furthermore, the theory developed here does not need normal forms of the vector field in the neighborhood of the Hopf points.

Article [7], among other cases, treats also the generalized Hopf bifurcation in general as -dimensional systems. In particular, it provides quadratic asymptotics for the bifurcation parameter values corresponding to the nonhyperbolic limit cycle, and for this cycle itself. Moreover, these asymptotics are implemented into the standard software MATCONT [5], allowing to automatically initialize the continuation of the cycle-saddle-node curve from the generalized Hopf point. However, the authors believe that the constructions presented here are independent and self-contained. More precisely, both articles give an approximation to the curve of nonhyperbolic periodic orbits of a family of differential equations that has transversal Hopf bifurcations of codimension two. Here we present this theory for -dimensional systems without the use of normal forms while in [7], the authors present -dimensional systems using normal forms.

This paper is organized as follows. In Section 2, the theory of approximation of periodic orbits for vector fields in the plane is developed. The stability of the approximate periodic orbits is discussed in Section 3. In Section 4, applications of the theory in Liénard-type differential equations are made, while applications to the Bazykin's predator-prey system are made in Section 5. Concluding comments about the results obtained here are in Section 6.

2. Approximation of Periodic Orbits

Consider a family of the differential equations where , is an open set in , , and is the parameter vector. Let be an equilibrium point of (1); that is, for . Suppose the following assumption: (H1)the linear part of the vector field , evaluated at and denoted by , has eigenvalues and , with . For , , , and , where

There is no loss of generality in considering that for all , and . Just make a translation of the equilibrium point and of the critical parameter to their origins and adjust in a convenient way the sets and . By doing this, (1) can be rewritten as where is a smooth vector field with Taylor expansion around , starting with second-order terms at least, as follows: where are the components of symmetric multilinear functions , , , and .

Let be an eigenvector corresponding to the eigenvalue , and let be an adjoint eigenvector corresponding to the eigenvalue satisfying and the normalization where is the standard inner product in and is the transpose of the matrix . The set is a basis of and the subspace of defined by is isomorphic to the vector space . Taking into account the isomorphism between and , if , then the notation used is . Thus, every vector can be uniquely represented as a linear combination of elements of ; that is, there is such that

It is easy to show that and . So (1) can be written as a complex family of differential equations as follows: for sufficiently small, where and The function has formal Taylor series where for and .

The coefficients for and play an important role in the method of approximation of a family of periodic orbits of (1). A simple way to calculate these coefficients, alternative to (14), is through the symmetric multilinear functions. From the symmetric bilinear function and (10), it follows that and, therefore, Similarly, for the symmetric trilinear function , and so on for other symmetric multilinear functions.

The aim of the theory of approximation of periodic orbits in [6] is to build an approximation for a periodic orbit of the complex differential equation (11), from the solution of the linear differential equation for . This linear differential equation has the solution where . For , it follows that and making the change in time , this solution is periodic of period in the variable . To formalize the method, consider the functions , and the change of coordinates and time where

Note that the parameter , as defined in (22), is a complex number or, more precisely, a complex function whose independent variable is . However, it is possible, through a change of variables, to consider the parameter as a real number. In fact, as it follows that Thus, making the change of variable in (24) and setting , since the function is periodic of period in the variable . Therefore, by (25), the parameter as defined in (22) will be considered a real parameter.

The generalization of the theory of approximation of periodic orbits introduced in [6] consists in achieving an approximation to the two-parameter family of periodic orbits where .

The change in time is essential, since the period of the family of periodic orbits (26) is unknown and, therefore, the change in time is used only to provide an approximation of the known period for the family of periodic orbits (26). If denotes the period of the family of periodic orbits, then In other words, the knowledge of the function completely determines the period of the family of periodic orbits of (26).

By changing the coordinates and time (21) and applying the chain rule, the complex differential equation (11) is rewritten as

Approximations to the functions , and are obtained through (28) and the formal power series

A property of the terms of the sequence , widely used in this theory of approximation of periodic orbits of vector fields in , is obtained in Proposition 1.

Proposition 1. Each term of the sequence satisfies

Proof. Setting , the proof is an immediate consequence of the definition of linear map and the formal power series in the variable of the function , because

The terms of the sequences , and are determined through a process that involves analysis of the powers in , obtained by replacing (29) into the differential equation (28). Note that, for and , the coefficients of powers in are determined by expanding the composition in the Taylor series around . Such an expansion, up to the fifth-order terms, is of the following form: with the same being valid for the composition .

The coefficient of the term in leads to the following boundary value problem: The solution of the differential equation in (34) is and as by Proposition 1, , it follows that Thus, which is a periodic function of period in the variable . In fact, the terms of the sequence are solutions of certain boundary value problems which appear when (29) is substituted into the differential equation (28). For each , the boundary value problem is of the following form: where .

The following theorem guarantees the existence of the solutions of the boundary value problem (38).

Theorem 2. For each , the boundary value problem (38) admits solution if and only if

Proof. For fixed , suppose that is the solution of (38). Thus, and by integrating by parts the left member of (40), it follows that . Now suppose that for a fixed . The general solution of the differential equation in (38) is of the following form: where . This solution will be periodic of period if ; that is, if Thus, using the hypothesis , it follows that , and, therefore, for each fixed , the function is the solution of the boundary value problem (38).

The previous theorem shows that, for , the solution of (38) is obtained by solving the differential equation in (38) with conditions and .

Continuing the process and using the result (37), the coefficient of the term in provides the boundary value problem where By applying Theorem 2 to the function , it follows that and by separating the real and imaginary parts of (45), we have and . Under these conditions, Theorem 2 guarantees the existence of the solution of the boundary value problem (43), which is given by

For the coefficient of the term in , we have the following boundary value problem: with where and the coefficient is defined as Expression (50) is identical to the one given in [1].

Continuing the process and calculating , it follows that And by separating the real and imaginary parts,

Once the coefficients and are determined, the solution of the boundary value problem (47) has the following form: where

Definition 3. The real number
is called the first Lyapunov coefficient.

Remark 4. A Hopf point of codimension one for (1) is an equilibrium point , with , such that has eigenvalues and , with , , , and the first Lyapunov coefficient, , is different from zero. A transversal Hopf point of codimension one is a Hopf point of codimension one such that for . In a neighborhood of a transversal Hopf point of codimension one , with , the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form: where . The sign of the first Lyapunov coefficient determines the stability of the family of periodic orbits that appears (or disappears) from as will be seen later.

When , for , there is the possibility of Hopf bifurcations of codimension two. In this case, it is necessary to obtain an expression for .

Applying Theorem 2 to the boundary value problem for , it follows that , and where

From the boundary value problem for , it follows that where

Rewriting the coefficient in a convenient way, expression (64) is exactly the one that appears in [1].

Definition 5. The real number where is given in (64), is called the second Lyapunov coefficient.

Remark 6. A Hopf point of codimension two for (1) is an equilibrium point , where , that satisfies the definition of a point Hopf of codimension one, except that . Moreover, it satisfies an additional condition; the second Lyapunov coefficient is nonzero. A Hopf point of codimension two is transversal if In a neighborhood of a transversal Hopf point of codimension two , with , the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form: where . In the bifurcation diagram of (67), there exists a curve of nonhyperbolic periodic orbits that has the exact representations as a curve parameterized by or as a graph of the function for .

The function will not be shown here because it is a long expression and it is not necessary in this work. In many results in this section and, particularly in (63), the following expressions , , , , , , , and appear. These expressions are calculated according to Propositions 7 and 8.

Proposition 7. Consider the differential equation (1) with an equilibrium point , such that the linear part of the map , evaluated at , , has eigenvalues and , where , and . Let also be an eigenvector corresponding to the eigenvalue , and let be an adjoint eigenvector corresponding to the eigenvalue , satisfying (6), (7), and (8). The following statements hold. (a)The vector is the solution of the following nonsingular 3-dimensional system:with the condition , where(b)The vector is the solution of the following nonsingular 3-dimensional system:with the condition , where(c)The partial derivative with respect to of the real part of the eigenvalue , evaluated at , is given by(d)The partial derivative with respect to of the imaginary part of the eigenvalue , evaluated at , is given by(e)The second-order partial derivative with respect to of the real part of the eigenvalue , evaluated at , is given by(f)