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)The second-order partial derivative with respect to of the imaginary part of the eigenvalue , evaluated at , is given by
Proof. Differentiating (6) with respect to the parameter and evaluating at , we have Using the hypotheses, the previous equation is rewritten as Taking the inner product of on both sides of the above equation and using (8), it follows that Items (a), (c), and (d) follow from the above equation, the Fredholm alternative (see [1]), and the results of [8]. The proof of part (b) is equal to the previous proof; that is, it is sufficient to differentiate (7) with respect to the parameter and to evaluate at . The proofs of items (e) and (f) consist of calculating the second-order partial derivative of (6) with respect to the parameter , evaluated at , and to use the Fredholm alternative.
Proposition 8. Consider the coefficients of the formal Taylor series of the map , The following statements hold. (a)The partial derivative with respect to of the coefficient , evaluated at , is(b)The partial derivative with respect to of the coefficient , evaluated at , is given by(c)The partial derivative with respect to of the coefficient , evaluated at , is obtained as(d)The partial derivative with respect to of the coefficient , evaluated at , is calculated as
Proof. Observing how the symmetric multilinear functions are defined, the proofs of items (a) to (d) consist in differentiating each expression in (81) with respect to the parameter and evaluating at .
The theory built up to this point approximates a family of periodic orbits of the complex differential equation (11). In the hypotheses of the Hopf bifurcation, if is a family of periodic orbits of (11), then is a family of periodic orbits associated with the differential equation (1), where or, in a more simple way,
The family of periodic orbits has formal Taylor series around of the following form: and the theory developed previously and the Taylor expansion of (87), around , show that
The stability of the approximate family of periodic orbits is studied in the next section by means of the Floquet exponent.
3. Stability of the Family of Periodic Orbits
According to the Floquet theory (see [9]), the stability of a periodic orbit can be determined through the characteristic exponent that, in this context and for differential equations in , is a function such that where . The next proposition provides a simple way to compute (90) in terms of the map .
Proposition 9. Through a change in time , the characteristic exponent associated with the differential equation is of the following form: where
Proof. The differential equation (1) can be written as (11), where , and . Thus, through the changes and , the characteristic exponent (90) can be rewritten as where Adding equations (95) and (96) and taking into account that , it follows that Therefore, , with
By the formal Taylor series in the variable of the function , the theory of approximation of a family of periodic orbits developed in the previous section and Proposition 9 allow us to obtain the terms of the sequence . For , the next theorem provides these terms.
Theorem 10. Let be the formal Taylor series of the characteristic exponent associated with the differential equation . Then, where , , and are given by (50), (63), and (64), respectively.
Proof. From (13) and (14), we have Thus, formally, the map has the Taylor series Doing the fourth-order Taylor expansion of the map around , and taking into account that , it results that with where for , the functions are such as in (37), (46), (54), and (59) and the expressions and are given by (52), and (61), respectively. Thus, from (99) and (105) and by Proposition 9, Therefore, which proves the theorem.
It follows from Theorem 10 a corollary that deals with the stability of a family of periodic orbits of the differential equation (1) which exists due to a Hopf bifurcation.
Corollary 11. Let be the fourth-order Taylor expansion around of the characteristic exponent associated with the differential equation , and let be the fourth-order Taylor expansion, around , of the function . The following statements hold. (a)For a fixed , sufficiently small, and , the stability of the periodic orbit of the differential equation (1) is given by the sign of . When for , the periodic orbit in the phase portrait of differential equation (1) is stable. As for , sufficiently small, if , the periodic orbit in the phase portrait of (1) exists for , and if , the periodic orbit in the phase portrait exists for . If , the periodic orbit in the phase portrait of the differential equation (1) is unstable. (b)Suppose that for , and . Then, for sufficiently small, the stability is given by the sign of . When , the periodic orbit in the phase portrait of the differential equation (1) is stable. As, in this case, if , the periodic orbit in the phase portrait of the differential equation (1) exists for , and if , the periodic orbit in the phase portrait of the differential equation (1) exists for . If , the periodic orbit in the phase portrait of the differential equation (1) is unstable.
Proof. As the sign of the Floquet exponent provides the stability of a periodic orbit, by (109), and (110) the proof is immediate.
Corollary 11 does not deal with the case where for a set of points . The theory developed up to this point enables us to study the curve of nonhyperbolic periodic orbits in the parameter plane , associated with a transversal Hopf point of codimension two. This curve is the set
From the set and the Implicit Function Theorem, the parameter can be obtained as a function of the parameter . Therefore, the curve follows from functions and ; that is, the curve can be locally represented as a curve parameterized by or can be locally represented as the graph of a function
In fact, the Taylor expansion around of the exponent characteristic is such as in (99), and, therefore, where the third-order Taylor expansion around of the function is of the following form: It is easy to see that . Thus, the study of the curve of nonhyperbolic periodic orbits in the parameter plane , associated with the differential equation (1), and in the hypotheses of a transversal Hopf bifurcation of codimension two is reduced to the study of the set .
The next lemma, whose proof is given in [3], guarantees the existence of the function .
Lemma 12. Let be a smooth function, where . Suppose that for , , the function in (118) satisfies the following assumptions: (A1); (A2); (A3); (A4). Then, there exists a unique smooth function such that and . Moreover, the function has the following representation: where
The following theorem can be stated now.
Theorem 13. Let be a transversal Hopf point of codimension two of (1). Then, the curve of nonhyperbolic periodic orbits , in the parameter plane , associated with the differential equation (1), has the following local representations: where
Proof. As Lemma 12 guarantees the existence of a smooth function such that , or even, . Moreover, the function has the second-order Taylor expansion around of the following form: where Thus, and substituting (128) into the function results in the following Taylor expansion: So, there is a curve in the parameter plane, , that can be parameterized by and represented as in (122). Another representation for this curve is obtained when the Implicit Function Theorem is applied to the following function: By substituting (130) into (110), the curve can also be represented locally as Therefore, there exists a curve in the parameter plane that locally has the representation (122) or (123). By the hypotheses of the transversal Hopf bifurcation of codimension two, and equation are locally topologically equivalent, around , to the complex differential equation (67). Therefore, the curve of nonhyperbolic periodic orbits has the representation (122) or (123).
Example 14. For the complex differential equation (67), we have So, by Theorem 13, the curve of nonhyperbolic periodic orbits has the following representations: which agree with (68) and (69), respectively.
The local representations (122) and (123) in Theorem 13 are valid when the Hopf curve is the set . If the Hopf curve is the set and the transversal Hopf bifurcation of codimension two occurs for , it is easy to show that the local representations are given by for , where
The next two sections present applications of the theory developed here in an extension of the van der Pol equation known as the Liénard equation and in Bazykin's predator-prey system and show how local representations of the the curve are obtained.
4. Liénard Equation
One of the pioneers in nonlinear electrical circuits was, undoubtedly, Balthasar van der Pol, through studies with triodes (vacuum tubes). Balthasar van der Pol showed that in circuits with triodes, the electrical quantities can exhibit nonlinear oscillations under certain conditions. Nowadays, it is known that the model of this circuit with triode presents a Hopf bifurcation. In a simple and theoretical way, the electric circuit of van der Pol consists of a triode, a capacitor of capacitance , and an inductor of inductance , according to the diagram of Figure 1.
Let , and , be the models of voltage and current in the capacitor and inductor, respectively. The triode of van der Pol, by the hypothesis, satisfies the generalized Ohm's law , where and are the models of voltage and current of the triode of van der Pol, respectively. Applying Kirchhoff's laws to the van der Pol electrical circuit model and using the capacitor and inductor equations, it follows that Therefore, the van der Pol circuit model is of the following form:
The study of differential equation (137) is simplified by the change of coordinates and time which leads to the differential equation where . Suppose that In the literature, the differential equation (139) satisfying (140) is known as the Liénard-type equation.
The Liénard equation has a unique equilibrium point , with , and the linear part of the vector field, evaluated at , has eigenvalues and , with for . When , , and , which indicates the occurrence of Hopf bifurcations. The eigenvectors and , where is normalized with respect to according to (8), are chosen as
In the case of the Liénard equation, the symmetric multilinear functions are given by Thus, for and , the only nonzero coefficients are
The eigenvectors and and the coefficients , , , and , computed by Propositions 7 and 8, are such that
Thus, from the previous results and by (50), (63), and (64), it follows that
Therefore, the first Lyapunov coefficient is given by and since the Liénard equation presents a transversal Hopf bifurcation of codimension one for and . From Corollary 11, and if , then there exists a unique unstable periodic orbit in the phase portrait of the Liénard equation when , and if , the periodic orbit is stable and there exists for .
For , the Liénard equation has a transversal Hopf point of codimension two, and the second Lyapunov coefficient is given by where . Since by Corollary 11 and Theorem 13, the Liénard equation has a bifurcation diagram as shown in Figure 2.
The curve of nonhyperbolic periodic orbits has the following local representations: as a curve parameterized by or as the graph of the function for .
Figure 3 emphasizes the comparison between the curve of nonhyperbolic periodic orbits of (139) obtained numerically with the software MATCONT (see [5]) and the quadratic approximation (154).
5. Bazykin's Predator-Prey System
Consider the dynamics of a predator-prey ecosystem, whose model is where (fixed), , and are parameters. Model (155) is known in the literature as Bazykin's predator-prey system. See [1] or [10].
Taking , the equilibrium point of interest is For the linear part of the vector field, evaluated at , has eigenvalues and , where , and .
The eigenvectors and are chosen as and by Proposition 7,
The symmetric multilinear functions are given by
Therefore, from the previous results,
When , . Thus, for , Bazykin's system (155) has a transversal Hopf point of codimension two, since , , and
Using (134), the curve of nonhyperbolic periodic orbits has the following local representations: as a curve parameterized by or as a graph of the function for .
Figure 4 emphasizes the comparison between the curve of nonhyperbolic periodic orbits of (155) obtained numerically with the software MATCONT and the quadratic approximation (164).
The comparison between the curve of nonhyperbolic periodic orbits of (155) obtained numerically with the software MATCONT and the approximation (165) is shown in Figure 5.
6. Concluding Comments
This paper shows how to obtain approximations of periodic orbits of a family of differential equations in the plane that has a transversal Hopf point. Moreover, if the family of differential equations has a transversal Hopf point of codimension two, then it is also possible to build an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram. These results are summarized in Corollary 11 and Theorem 13. Example 14, the study of the Liénard equation (139) in Section 4, and Bazykin's predator-prey system in Section 5 demonstrate the applicability of the theory. See also Figures 3, 4, and 5.
Although the theory is formulated for a family of differential equations in the plane, it can be applied to any family of differential equations in that presents a transversal Hopf bifurcation of codimension two. For this, it is necessary to use the Center Manifold Theorem, or more precisely, to apply the proposed theory to the family of differential equations in restricted to the center manifold.
Acknowledgments
The second author is partially supported by CNPq Grant 301758/2012-3 and by FAPEMIG Grant PPM-0092-13. The third author thanks CNPq and INERGE for partially supporting this paper. This work was initiated when the second author visited the Laboratoire de Mathématiques, Informatique et Applications of Université de Haute Alsace (2011).