#### Abstract

We study the local codimension one, two, and three bifurcations which occur in a recently proposed Van der Pol-Duffing circuit (ADVP) with parallel resistor, which is an extension of the classical Chua's circuit with cubic nonlinearity. The ADVP system presents a very rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations. Aiming to contribute to the understand of the complex dynamics of this new system we present an analytical study of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. Then, by studying the continuation of such periodic orbits, we numerically find a sequence of period doubling and symmetric homoclinic bifurcations which leads to the creation of strange attractors, for a given set of the parameter values.

#### 1. Introduction and Statement of the Main Results

In this paper we study the local codimension one, two, and three bifurcations and the respective qualitative changes in the dynamics of the following system of nonlinear equations:

where are the state variables and , , , and are real parameters.

As far as we know, system (1.1) was proposed and firstly studied in [1], and it can be obtained from the system

by the following changes in variables, parameters, and rescaling in time:

In (1.2) and are the voltages across the capacitors and , respectively, is the current through the nonlinear diode , is the current through the inductor, and are the capacitances, is the inductance, is the linear resistor, is the parallel resistor, , and the prime denotes derivatives with respect to the time . It is assumed that the current through the nonlinear diode is given by , with and . See Figure 1 and [1] for more details.

Despite the simplicity of the electronic circuit shown in Figure 1, the related system (1.1) has a rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations, depending on the parameter values. The study of simple nonlinear electronic circuits presenting chaotic behavior was initiated by Chua in the mid-1980s (see [2]) and the interest on the subject has grown in the last decades. In fact, it has useful applications like chaos synchronization which is applied in industry and secure communications, beyond other areas of sciences (see [1] and references therein and also [3, 4]). System (1.1) with reduces to a system equivalent to the classical Chua's differential equations with cubic nonlinearity. The authors have recently developed a complete study of degenerate Hopf bifurcations for this case in [5], answering a challenge proposed by Moiola and Chua in [6].

System (1.1) has the equilibrium point , which exists for any parameter values. For it also has the symmetric equilibria . Codimension one Hopf bifurcation which occurs at the equilibrium point was studied in [1], by using a result of HsÃ¼ and Kazarinoff in [7].

In this paper by using the classical projection method which allows one the calculation of the Lyapunov coefficients associated to the Hopf bifurcations we study all possible bifurcations (generic and degenerate ones) which occur at the equilibria and of system (1.1). In this way the analyses presented in [1] are extended and completed. More precisely, we prove the following statements.

(a)For the equilibrium the Hopf hypersurface is obtained in the space of parameters and the first Lyapunov coefficient is calculated. It is shown that this coefficient vanishes on a -dimensional surface contained in , giving rise to codimension two Hopf bifurcations (see Figure 2(a)). Then the second Lyapunov coefficient is calculated and it is established that this coefficient is always negative on the surface .(b)For the symmetric equilibria the Hopf hypersurface is obtained in the space of parameters , the first Lyapunov coefficient is calculated, and it is shown that this coefficient vanishes on a 2-dimensional surface contained in (see Figure 2(b)), giving rise to codimension two bifurcations. The second Lyapunov coefficient is calculated and it is established that this coefficient also vanishes on a 2-dimensional surface contained in (see Figure 2(b)), giving rise to codimension three bifurcations. The third Lyapunov coefficients along the curve given by the intersection of the surfaces and are obtained and found to be positive.**(a)**

**(b)**

The corresponding bifurcation diagrams are traced for the above bifurcations (see Figures 3, 4, 7, and 8 of Section 4). From statement (a) it follows that the maximum number of small periodic orbits bifurcating from the origin is 2. Furthermore, from statement (b) one can deduce that there is a region in the parameter space for which an attracting periodic orbit and two other unstable periodic orbits coexist with the attracting equilibrium points . See Figures 7 and 8 in Section 4.

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**(b)**

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We go further and perform a numerical study of the continuation of periodic orbits which arise from the Hopf bifurcations at the points . Through this analysis we have found period doubling and homoclinic bifurcations which seems to lead to the creation of strange attractors of system (1.1), for some parameter values (see Section 5).

The rest of this paper is organized as follows. In Section 2 through a linear analysis of system (1.1) we obtain the Hopf hypersurfaces for and . In Section 3 following [8â€“10] we present a brief review of the methods used to study codimension one, two, and three Hopf bifurcations, describing in particular how to calculate the Lyapunov coefficients related to the stability of the equilibrium point as well as of the periodic orbits which appear in these bifurcations. In general the Lyapunov coefficients are very difficult to be obtained. These methods are used in Section 4 to prove the main results of this paper, described in statements (a) and (b). In Section 5 we present numerical simulations for particular values of the parameters which illustrate and corroborate the analytical results obtained. Finally, in Section 6 we make some concluding remarks.

#### 2. Linear Analysis of System (1.1)

In a vectorial notation which will be useful in the next calculations system (1.1) can be written as

where

The origin is an equilibrium point of system (1.1) for all values of the parameters. The characteristic polynomial of the Jacobian matrix of the function given in (2.1) at is given by

If then the term in the above polynomial is negative and this implies that is an unstable equilibrium point.

For the sake of completeness we state the following lemma (Routh-Hurwitz stability criterion) whose proof can be found in [11, page 58].

Lemma 2.1. *The polynomial , , with real coefficients, has all roots with negative real parts if and only if the numbers , , are positive and the inequality is satisfied.*

The following proposition is a direct consequence of Lemma 2.1.

Proposition 2.2. *Consider in system (1.1). If then the equilibrium point is asymptotically stable for all and . If and
**
then system (1.1) has an asymptotically stable equilibrium point at . If and then is unstable.*

For the origin becomes a nonhyperbolic equilibrium point and the situation is highly degenerate. This case, which will not be considered in this work, can be studied by using the methods presented for instance in [12].

For two new equilibria appear. System (1.1) is invariant under the change of coordinates . So the stability of the equilibrium can be obtained from the stability of .

The characteristic polynomial of the Jacobian matrix of the function given in (2.1) at is given by

The following proposition is also a direct consequence of Lemma 2.1.

Proposition 2.3. *Consider in system (1.1). If then the equilibrium point is asymptotically stable for all and . If and
**
then system (1.1) has an asymptotically stable equilibrium point at . If and then is unstable.*

The equations and in (2.4) and (2.6) give the equations of the Hopf hypersurfaces and in the parameter space . These equations will be used in Section 4 in the study of degenerate Hopf bifurcations which occur at the equilibria and of system (1.1).

#### 3. Outline of the Hopf Bifurcation Methods

This section is a review of the projection method described in [8] for the calculation of the first and second Lyapunov coefficients associated to Hopf bifurcations, denoted by and , respectively. This method was extended to the calculation of the third and fourth Lyapunov coefficients in [9, 10]. Other equivalent definitions and algorithmic procedures to write the expressions of the Lyapunov coefficients for two-dimensional systems can be found in Andronov et al. [13] and Gasull and Torregrosa [14], among others, and can be adapted to the three-dimensional system of this paper if it is restricted to the center manifold. See also [15].

Consider the differential equation

where and are, respectively, vectors representing phase variables and control parameters. Assume that is of class in . Suppose that (3.1) has an equilibrium point at and, denoting the variable also by , write

as

where and, for ,

and so on for , , , and .

Suppose that is an equilibrium point of (3.1) where the Jacobian matrix has a pair of purely imaginary eigenvalues , , and admits no other eigenvalue with zero real part. Let be the generalized eigenspace of corresponding to . By this it is meant the largest subspace invariant by on which the eigenvalues are .

Let be vectors such that

where is the transpose of the matrix . Any vector can be represented as , where . The two-dimensional center manifold associated to the eigenvalues can be parameterized by the variables and by means of an immersion of the form , where has a Taylor expansion of the form

with and . Substituting this expression into (3.1) we obtain the following differential equation:

where is given by (3.2). The complex vectors are obtained solving the system of linear equations defined by the coefficients of (3.7), taking into account the coefficients of (see [9, Remark â€‰3.1, page 27]), so that system (3.7), on the chart for a central manifold, writes as follows:

with .

The *first Lyapunov coefficient * is defined by

where , and .

The *second Lyapunov coefficient* is defined by

where . The expression for can be found in [9, equation (36), page 28].

The *third Lyapunov coefficient* is defined by

where . The expression for can be found in [9, equation (44), page 30].

A *Hopf point * of system (3.1) is an equilibrium point where the Jacobian matrix has a pair of purely imaginary eigenvalues , , and the other eigenvalue . From the Center Manifold Theorem, at a Hopf point a two-dimensional center manifold is well defined, it is invariant under the flow generated by (3.1) and can be continued with arbitrary high class of differentiability to nearby parameter values (see [8]).

A Hopf point is called *transversal* if the parameter dependent complex eigenvalues cross the imaginary axis with nonzero derivative. In a neighborhood of a transversal Hopf point with the dynamic behavior of the system (3.1), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form , where , , , and are real functions having derivatives of arbitrary higher order, which are continuations of , , and the first Lyapunov coefficient at the Hopf point. See [8, 9] for details. When one family of stable (unstable) periodic orbits can be found on this family of manifolds, shrinking to an equilibrium point at the Hopf point.

A *Hopf point of codimension 2* is a Hopf point where vanishes. It is called *transversal* if and have transversal intersections, where is the real part of the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2 with the dynamic behavior of the system (3.1) reduced to the family of parameter-dependent continuations of the center manifold is orbitally topologically equivalent to , where and are unfolding parameters. See [8, 9]. The bifurcation diagrams for can be found in [8, page 313] and in [16].

A *Hopf point of codimension 3* is a Hopf point of codimension 2 where vanishes. It is called *transversal* if , , and have transversal intersections. In a neighborhood of a transversal Hopf point of codimension 3 with the dynamic behavior of the system (3.1), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to , where , , and are unfolding parameters. The bifurcation diagram for can be found in Takens [16].

#### 4. Hopf Bifurcations in System (1.1)

##### 4.1. Hopf Bifurcations at

In this subsection we study the stability of under the conditions and , that is, on the Hopf hypersurface complementary to the range of validity of Proposition 2.2.

The Jacobian matrix of the function given in (2.1) at is given by

Then, using the notation of the previous section (see expression (3.3)) the multilinear symmetric functions corresponding to can be written as

The eigenvalues of are

and from (3.5) one has where

Using these calculations we prove the next two theorems, from which statement (a) of the introduction follows.

Theorem 4.1. *Consider the four-parameter family of differential equations (1.1). The first Lyapunov coefficient associated to the equilibrium is given by
**
If
**
is different from zero then system (1.1) has a transversal Hopf point at for and .**More precisely, if (see Figure 2(a)), then system (1.1) has a Hopf point of codimension 1 at . If then is asymptotically stable and for each , but close to , there exists a stable periodic orbit near the unstable equilibrium point . If then is unstable and for each , but close to , there exists an unstable periodic orbit near the asymptotically stable equilibrium point .*

*Proof. *As the function then the expression in (3.9) reduces to . Performing the calculations we get
Substituting the expressions of and into the above expression of and taking the real part we have, from (3.9), the expression of the first Lyapunov coefficient given in (4.6). Note that the sign of is given by the sign of the function defined in (4.7).

It remains only to verify the transversality condition of the Hopf bifurcation. In order to do so, consider the family of differential equations (1.1) regarded as dependent on the parameter . The real part, , of the pair of complex eigenvalues at the critical parameter verifies
Therefore, the transversality condition at the Hopf point holds.

The sets and illustrated in Figure 2(a) are defined implicitly as and , respectively. The theorem is proved.

Theorem 4.2. *Consider the four-parameter family of differential equations (1.1) restricted to and . Then the second Lyapunov coefficient associated to the equilibrium is negative on the Hopf hypersurface where the first Lyapunov coefficient vanishes.*

*Proof. *Following the notation introduced in Section 3, we have Then
The expressions of the complex vectors and are too long to be put in print. The interested author can encounter such expressions and all the details of the calculations presented in this paper in the website [17].

By direct calculations the second Lyapunov coefficient associated to can be written as

where has the form

Along the surface we have, from (4.7) of Theorem 4.1, , or equivalently, . Substituting this expression into (4.11) one has

Here the transversality condition is equivalent to the transversality of the hypersurfaces and . The bifurcation diagram for a typical point on the surface is depicted in Figure 3.

##### 4.2. Hopf Bifurcations at

In this subsection we study the stability of under the conditions and , that is, on the Hopf hypersurface complementary to the range of validity of Proposition 2.3.

The Jacobian matrix of the function given in (2.1) calculated at is given by

Then, using the notation of Section 3 (see expression (3.3)) the multilinear symmetric functions corresponding to can be written as

The eigenvalues of are

and from (3.5) one has

where

Using these calculations we prove the next three theorems, from which statement (b) of introduction follows.

Theorem 4.3. *Consider the four-parameter family of differential equations (1.1). The first Lyapunov coefficient associated to the equilibrium is given by
**
where
**
If is different from zero then system (1.1) has a transversal Hopf point at for and .**More precisely, if (see Figure 2(b)) then system (1.1) has a Hopf point of codimension 1 at . That is, if then is asymptotically stable and for each , but close to , there exists a stable periodic orbit near the unstable equilibrium point ; if then is unstable and for each , but close to , there exists an unstable periodic orbit near the asymptotically stable equilibrium point .*

*Proof. *The theorem follows by expanding the expressions in definition of the first Lyapunov coefficient (3.9). It relies on extensive calculations involving the vector in (4.17), the vector in (4.18) and the multilinear functions and . The inclusion here of the enormous expressions obtained from the mentioned calculations would not bring any contribution in clarifying the reading of the text. Then the detailed calculations related to this proof, corroborated by Computer Algebra, have been posted in [17], including its implementation using the software MATHEMATICA 5 [18].

Notice that the sign of the first Lyapunov coefficient is given by the sign of the function defined in (4.21) since the denominator of and the expression are negative. In Figure 2(b) the surface and the regions and are illustrated. These regions are defined implicitly as and , respectively.

It remains only to verify the transversality condition of the Hopf bifurcation. In order to do so, consider the family of differential equations (1.1) regarded as dependent on the parameter . The real part, , of the pair of complex eigenvalues at the critical parameter verifies
Therefore, the transversality condition holds at the Hopf point. The theorem is proved.

Theorem 4.4. *Consider the four-parameter family of differential equations (1.1) restricted to and . Then the second Lyapunov coefficient associated to the equilibrium is given by
**
where the function is given by
**
and the function in the denominator is given by
**
If (see Figure 2(b)), then system (1.1) has a transversal Hopf point of codimension 2 at . More specifically, if then the Hopf point at is asymptotically stable and the bifurcation diagram is illustrated in Figure 3 for a typical point . If then the Hopf point at is unstable and the bifurcation diagram is drawn in Figure 4 for a typical point . *

*Proof. * The theorem follows by expanding the expressions in definition (3.10) of the second Lyapunov coefficient. It relies on extensive calculation involving the vector in (4.17), the vector in (4.18) and the multilinear functions , , , and . The calculations in this proof, corroborated by Computer Algebra, have been posted in [17].

In Figure 2 we present a geometric synthesis interpreting the long calculations involved in this proof. The sign of gives the sign of the second Lyapunov coefficient since the function in the denominator of is positive. The graph of on the surface , which determines the curve , and the signs of the first and second Lyapunov coefficients are also illustrated. As follows, on the open region on the surface , denoted by . On this region a typical reference point is depicted. Also on the open region on the surface , denoted by . This region contains the typical reference point, denoted by .

The bifurcation diagrams of system (1.1) at the points and are illustrated in Figures 3 and 4, respectively. The theorem is proved.

*Remark 4.5. *There are numerical evidences that the sign of the third Lyapunov coefficient is always positive along the curve given by the intersection of the surfaces and (see Figure 2(b)). For see the article in [5]. For see Theorem 4.6 in what follows. Figure 5 shows the behavior of the third Lyapunov coefficient as a function of , for on the curve . So the bifurcation diagrams depicted in Figures 7 and 8 are essentially the same for all values of .

All the calculations presented here are illustrated in Figure 6 for the cases (a) and (b). In this figure are depicted the surfaces , the solid curves where , the regions and where and , respectively, the dashed curves where and the arcs and where and , respectively. The bifurcation diagrams for typical points and are depicted in Figure 3 while the bifurcation diagrams for typical points and are shown in Figure 4. In the next theorem we analyze the sign of the third Lyapunov coefficient at the point , that is, for the case or equivalently for . For the case the calculations are very similar.

**(a)**

**(b)**

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Theorem 4.6. *Consider in system (1.1). Then there is a unique point where the surfaces and intersect transversally on the Hopf hypersurface . For the parameter values at the point , system (1.1) has a transversal Hopf point of codimension 3 at which is unstable since . The bifurcation diagram of system (1.1) at the point is illustrated in Figure 7. In Figure 8 the bifurcation diagram of system (1.1) at a typical point of Figure 7 is drawn.*

*Computer Assisted Proof*

The point is the intersection of the surfaces and on the Hopf hypersurface taking into account . It is defined and obtained as the solution of the equations (see Theorem 4.3) and (see Theorem 4.4). The point has coordinates
The existence and uniqueness of with the previous coordinates has been established numerically with the software MATHEMATICA 5 [18] (see also [17]) and can be checked by the Newton-Kantorovich Theorem [19].

For the point take five decimal round-off coordinates , , and for simplicity. For these values of the parameters one has
From (3.9), (3.10), (3.11), and (4.27) one has
The previous calculations have also been corroborated with 20 decimals round-off precision performed using the software MATHEMATICA 5 [18]; see [17].

The gradients of the functions , given in (4.20), and , given in (4.23), at the point are, respectively,
The transversality condition at is equivalent to the nonvanishing of the determinant of the matrix whose columns are the above gradient vectors, which was evaluated and has furnished the value . The transversality condition being satisfied, the bifurcation diagrams in Figures 7 and 8 follow from the work of Takens [16], taking into account the orientation and signs.

Define . The open region in the parameter space where system (1.1) has three small periodic orbits bifurcating from the equilibria can be described by , , and with . See Figure 7. The phase portraits of system (1.1) for the flow restricted to the center manifold and its continuation are shown in Figure 8(b). For parameters at the equilibrium is unstable; for parameters at the equilibrium is an weak unstable focus (Hopf point with positive ); for parameters at the equilibrium is stable and an unstable limit cycle appears from the Hopf bifurcation; for parameters at the equilibrium is an weak stable focus (Hopf point with negative ) and there is an unstable limit cycle; for parameters at the equilibrium is unstable and a stable limit cycle appears from the Hopf bifurcation, so there are two limit cycles encircling the equilibrium; for parameters at the equilibrium is unstable and two cycles collide corresponding to a nondegenerate fold bifurcation of the cycles; for parameters at the equilibrium is stable and two cycles collide corresponding to a nondegenerate fold bifurcation of the cycles; for parameters at the equilibrium is stable and two cycles collide corresponding to a nondegenerate fold bifurcation of the cycles; for parameters at the equilibrium is stable and there are three limit cycles encircling the equilibrium.

#### 5. Numerical Simulations

In this section we present some numerical simulations of system (1.1) for several values of the parameters , , , and . The main purpose is to illustrate the creation of stable and unstable limit cycles through the Hopf bifurcations at the equilibria and , proved to occur in the previous sections. The simulations were developed using the Software MAPLE 8, under the Runge-Kutta method of fourth order with several different stepsizes. We also perform a numerical study of the continuation of periodic orbits which arise from the Hopf bifurcations at the point . Throughout this analysis we have found period doubling and homoclinic bifurcations which seems to lead to the creation of strange attractors for (1.1), for some parameter values. For the study of homoclinic bifurcations in three-dimensional systems see [20, 21].

##### 5.1. Bifurcations at

For , system (1.1) has the origin as its unique equilibrium point. According to Theorem 4.1, the system undergoes a Hopf bifurcation when the parameter crosses the critical value with . This type of bifurcation is illustrated in Figures 9 and 10. To draw these figures we have taken , , and have varied the parameter , whose values are presented in the captions of the figures. Observe that for this parameter values we have . One can observe that when the parameter tends to the critical value from above the spiralling behavior of the solution becomes slower as expected, corroborating the correctness of the calculations presented in the previous sections (see Figures 9(a) and 9(b)); the limit cycle which arises when the parameter crosses is very small (observe the range of the state variables in Figure 10). The numerical analysis performed suggests that such small limit cycle which birth in the Hopf bifurcation for dies for ; that is, the cycle exists for . One can observe that as the parameter stay away from the critical value the solutions tend faster to the limit cycle (see Figure 11). The shape and behavior of the limit cycles in Figure 11 resemble the one of the Van der Pol system, see [22, page 267].

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##### 5.2. Bifurcations at

For , system (1.1) has the origin and also the symmetric equilibria . According to Theorem 4.3, the system undergoes a Hopf bifurcation when the parameter crosses the critical value , for . This type of bifurcation is illustrated in Figures 12 and 13, where we have taken , , and have varied the parameter , whose values are presented in the captions of the figures. Observe that for these values of the parameters , , and we have . In the figures are shown the -dimensional unstable manifolds of , which spiral to the equilibria for . Observe that when the parameter tends to the critical value the spiralling behavior of the solutions becomes slower (Figure 12(b)); for the unstable manifolds spiral to the symmetric limit cycles arise in the Hopf bifurcation for (Figure 13).

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