Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited

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 Lienard-type equations and in Bazykin’s predator-prey system.


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 2-dimensional systems without the use of normal forms while in [7], the authors presentdimensional 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 2 Abstract and Applied Analysis made in Section 5. Concluding comments about the results obtained here are in Section 6.
The coefficients − , ( ) for = 2, 3, . . . and = 0, . . . , 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 (x, y, ) → (x, y, ) and (10), it follows that 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 = 0 . This linear differential equation has the solution where 0 ∈ C. For = 0 , it follows that 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 2 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 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 2 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  ) .
A property of the terms of the sequence { ( , ])} ∈N , widely used in this theory of approximation of periodic orbits of vector fields in R 2 , is obtained in Proposition 1.  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, Thus, which is a periodic function of period 2 in the variable . In fact, the terms of the sequence { ( , ])} ∈N are solutions of certain boundary value problems which appear when (29) is substituted into the differential equation (28). For each = 1, 2, . . ., the boundary value problem is of the following form: where +1 ( , (]), (])) = +1 ( + 2 , (]), (])).
The following theorem guarantees the existence of the solutions of the boundary value problem (38).
Continuing the process and using the result (37), the coefficient of the term in 2 provides the boundary value problem By applying Theorem 2 to the function ( , 1 (]), 1 (])) → 2 ( , 1 (]), 1 (])), it follows that and by separating the real and imaginary parts of (45), we have 1 (]) = 0 and 1 (]) = 0. 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 3 , we have the following boundary value problem: and the coefficient 2,1 ( 0 ) is defined as Expression (50) is identical to the one given in [1].

Remark 4. A Hopf point of codimension one for
for 0 ∈ . In a neighborhood of a transversal Hopf point of codimension one (0, 0 ) ∈ × , with 1 ( 0 ) ̸ = 0, the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form: where = sign( 1 ( 0 )). The sign of the first Lyapunov coefficient determines the stability of the family of periodic orbits that appears (or disappears) from (0, 0 ) ∈ × as will be seen later.
Applying Theorem 2 to the boundary value problem for = 3, it follows that 3 (]) = 0, 3 (]) = 0 and Abstract and Applied Analysis From the boundary value problem for = 5, it follows that 8 Abstract and Applied Analysis Rewriting the coefficient 3,2 ( 0 ) in a convenient way, expression (64) is exactly the one that appears in [1].

Remark 6. A Hopf point of codimension two for
In a neighborhood of a transversal Hopf point of codimension two (0, 1 ) ∈ × , with 2 ( 1 ) ̸ = 0, the dynamic behavior of differential equation (1) is orbitally topologically equivalent to the following complex normal form: where = sign( 2 ( 1 )). 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 ≥ 0.
The function ( , ]) → 5 ( , ]) 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  (a) The vector ( 0 ) ∈ C 2 is the solution of the following nonsingular 3-dimensional system: with the condition ⟨ ( 0 ), (b) The vector ( 0 ) ∈ C 2 is the solution of the following nonsingular 3-dimensional system: with the condition ⟨ ( 0 ), (c) The partial derivative with respect to of the real part of the eigenvalue ( ), evaluated at = 0 , is given by (d) The partial derivative with respect to of the imaginary part of the eigenvalue ( ), evaluated at = 0 , is given by (e) The second-order partial derivative with respect to of the real part of the eigenvalue ( ), evaluated at = 0 , is given by (f) The second-order partial derivative with respect to of the imaginary part of the eigenvalue ( ), evaluated at = 0 , is given by Proof. Differentiating (6) with respect to the parameter and evaluating at = 0 , we have Using the hypotheses, the previous equation is rewritten as Taking the inner product of ( 0 ) ∈ C 2 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 = 0 . The proofs of items (e) and (f) consist of calculating the second-order partial derivative of (6) with respect to the parameter , evaluated at = 0 , and to use the Fredholm alternative.
The following statements hold.
(a) The partial derivative with respect to of the coefficient (b) The partial derivative with respect to of the coefficient 1,1 ( ), evaluated at = 0 , is given by (c) The partial derivative with respect to of the coefficient 0,2 ( ), evaluated at = 0 , is obtained as (d) The partial derivative with respect to of the coefficient 2,1 ( ), evaluated at = 0 , is calculated as The family of periodic orbits ( , , ]) → ( , , ]) has formal Taylor series around = 0 of the following form: and the theory developed previously and the Taylor expansion of (87), around = 0, show that 4 ( , ]) = ( 0 ) 4 ( , ]) + ( 0 ) 4 ( , ]) The stability of the approximate family of periodic orbits is studied in the next section by means of the Floquet exponent.

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 R 2 , is a function ( , ]) → ( , ]) such that  (96) Adding equations (95) and (96) and taking into account that = 1 + 2 , it follows that 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 { (])} ∈N . For = 1, . . . , 4, the next theorem provides these terms.

Theorem 10. Let
with 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. (a) For a fixed ( , ]) ∈ , ∈ R sufficiently small, and Re( 2,1 ( 0 )) ̸ = 0, the stability of the periodic orbit of the differential equation (1) is given by the sign of Re( 2,1 ( 0 )). When Re( 2,1 ( 0 )) < 0 for 0 ∈ , the periodic orbit in the phase portrait of differential equation (1) is stable. As for ∈ R, sufficiently small, if ( 0 ) > 0, the periodic orbit in the phase portrait of (1) exists for > 0, and if ( 0 ) < 0, the periodic orbit in the phase portrait exists for < 0. If Re( 2,1 ( 0 )) > 0, the periodic orbit in the phase portrait of the differential equation (1) is unstable.
(b) Suppose that for 1 = (0, 0), Re( 2,1 ( 1 )) = 0 and Re( 3,2 ( 1 )) ̸ = 0. Then, for ∈ R sufficiently small, the stability is given by the sign of Re( 3,2 ( 1 )). When Re( 3,2 ( 1 )) < 0, the periodic orbit in the phase portrait of the differential equation (1) is stable. As, in this case, ( 1 ) > 0, the periodic orbit in the phase portrait of the differential equation (1) exists for > 0, and if ( 1 ) < 0, the periodic orbit in the phase portrait of the differential equation (1) exists for < 0. If Re( 3,2 ( 1 )) > 0, the periodic orbit in the phase portrait of the differential equation (1) From the set −1 (0) and the Implicit Function Theorem, the parameter ] can be obtained as a function of the parameter . Therefore, the curve NH follows from functions → ] = ( ) and ( , ]) → = ( , ]); that is, the curve NH 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 = 0 of the exponent characteristic is such as in (99), and, therefore, where the third-order Taylor It is easy to see that Ψ −1 (0) = {( , ]) ∈ : Ψ( , ]) = 0} ⊂ −1 (0). Thus, the study of the curve of nonhyperbolic periodic orbits in the parameter plane ( , ]) ∈ R 2 , 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 Ψ −1 (0).
The next lemma, whose proof is given in [3], guarantees the existence of the function → ] = ( ).
The local representations (122) and (123) for ] ≤ ] 1 , 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 predatorprey system and show how local representations of the the curve NH are obtained.

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 V , and V , 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 → V = ( ), where V 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 X( ) = (( / √ ) ). Suppose that In the literature, the differential equation (139) satisfying (140) is known as the Liénard-type equation.
The comparison between the curve of nonhyperbolic periodic orbits NH of (155) obtained numerically with the software MATCONT and the approximation (165) is shown in Figure 5.

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 R 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 R restricted to the center manifold.