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Bifurcation analysis of the Watt governor system

Abstract

This paper pursues the study carried out by the authors in Stability and Hopf bifurcation in the Watt governor system [14], focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium.

centrifugal governor; Hopf bifurcations; periodic orbit


Bifurcation analysis of the Watt governor system

Jorge SotomayorI; Luis Fernando MelloII; Denis de Carvalho BragaIII

IInstituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brazil

IIInstituto de Ciências Exatas, Universidade Federal de Itajubá Avenida BPS 1303, Pinheirinho, 37500-903 Itajubá, MG, Brazil

IIIInstituto de Sistemas Elétricos e Energia, Universidade Federal de Itajubá Avenida BPS 1303, Pinheirinho, 37500-903 Itajubá, MG, Brazil

E-mails: sotp@ime.usp.br/ lfmelo@unifei.edu.br/ braga_denis@yahoo.com.br

ABSTRACT

This paper pursues the study carried out by the authors in Stability and Hopf bifurcation in the Watt governor system [14], focusing on the codimension one Hopf bifurcations in the centrifugal Watt governor differential system, as presented in Pontryagin's book Ordinary Differential Equations, [13]. Here are studied the codimension two and three Hopf bifurcations and the pertinent Lyapunov stability coefficients and bifurcation diagrams, illustrating the number, types and positions of bifurcating small amplitude periodic orbits, are determined. As a consequence it is found a region in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium.

Mathematical subject classification: 70K50, 70K20.

Key words: centrifugal governor, Hopf bifurcations, periodic orbit.

1 Introduction

The Watt centrifugal governor is a device that automatically controls the speed of an engine. Dating to 1788, it can be taken as the starting point for the theory of automatic control (see MacFarlane [10] and references therein). In this paper the system coupling the Watt-centrifugal-governor and the steam-engine will be called simply the Watt Governor System (WGS). See Section 2 for a description and illustration, in Fig. 1, of this system.


Landmarks for the study of the local stability analysis of the WGS are the works of Maxwell [11] and Vyshnegradskii [16]. A simplified version of the WGS local stability based on the work of Vyshnegradskii is presented by Pontryagin [13]. A local stability study generalized to a more general Watt governor design was carried out by Denny [4] and pursued by the authors in [14].

Enlightening historical comments about the Watt governor local mathematical stability and oscillatory analysis can be found in MacFarlane [10] and Denny [4]. There, as well as in [13], we learn that toward the mid XIX century, improvements in the engineering design led to less reliable operations in the WGS, leading to fluctuations and oscillations instead of the ideal stable constant speed output requirement. The first mathematical analysis of the stability conditions and subsequent indication of the modification in the design to avoid the problem was carried out by Maxwell [11] and, in a user friendly style likely to be better understood by engineers, by Vyshnegradskii [16].

From the mathematical point of view, the oscillatory, small amplitude, behavior in the WGS can be associated to a periodic orbit that appears from a Hopf bifurcation. This was established by Hassard et al. in [5] and Al-Humadi and Kazarinoff in [1]. Another procedure, based in the method of harmonic balance, has been suggested by Denny [4] to detect large amplitude oscillations.

In [14] we characterized the surface of Hopf bifurcations in a WGS, which is more general than that presented by Pontryagin [13], Al-Humadi and Kazarinoff [1] and Denny [4]. See Theorem 4.1 and Fig. 3 for a review of the critical curve on the surface where the first Lyapunov coefficient vanishes.



In the present paper, restricting ourselves to Pontryagin's system, we go deeper investigating the stability of the equilibrium along the above mentioned critical curve. To this end the second Lyapunov coefficient is calculated (Theorem 4.4) and it is established that it vanishes at a unique point (see Fig. 4 and 5). The third Lyapunov coefficient is calculated at this point (Theorem 4.5) and found to be positive. The pertinent bifurcation diagrams are established. See Fig. 6, 7 and 9 . A conclusion derived from these diagrams, concerning the region - a solid ''tongue'' - in the space of parameters where an attracting periodic orbit coexists with an attracting equilibrium, is specifically commented in Section 5.






The extensive calculations involved in Theorems 4.4 and 4.5 have been corroborated with the software MATHEMATICA 5 [18] and the main steps have been posted in the site [17].

This paper is organized as follows. In Section 2 we introduce the WGS and review the Pontryagin differential equations [13]. The stability of the equilibrium points is also analyzed. This section is essentially a review of [13, 5, 1, 14]. The Hopf bifurcations in the WGS differential equations are studied in Sections 3 and 4. Expressions for the second and third Lyapunov coefficients, which fully clarify their sign, are obtained, pushing forward the method found in the works of Kuznetsov [8, 9]. With this data, the bifurcation diagrams are established. Concluding comments, synthesizing and interpreting the results achieved here, are presented in Section 5.

2 The Watt centrifugal governor system

2.1 Differential equations for the Watt governor system

According to Pontryagin [13], p. 217, the differential equations of the WGS illustrated in Fig. 1 are

where j Î (0, ) is the angle of deviation of the arms of the centrifugal governor from its vertical axis S1, W Î [0, ¥) is the angular velocity of the rotation of the flywheel D, q is the angular velocity of S1, l is the length of the arms, m is the mass of each ball, H is a sleeve which supports the arms and slides along S1, T is a set of transmission gears, V is the valve that determines the supply of steam to the engine, t is the time, y = d j/d t, g is the standard acceleration of gravity, q = c W, c > 0 is a constant transmission ratio, b > 0 is a constant of the frictional force of the system, I is the moment of inertia of the flywheel, F is an equivalent torque of the load and µ > 0 is a proportionality constant. The reader is referred to Pontryagin [13] for the derivation of (1) from Newton's Second Law of Motion.

After the following change in the coordinates and time

the differential equations (1) can be written as

where a > 0, 0 < b < 1 and e > 0 , given by

are the normalized variable parameters. Thus the differential equations (3) are in fact a three-parameter family of differential equations which can be rewritten as x¢ = f(x, µ), where

and

2.2 Stability analysis of the equilibrium points

The differential equations (3) have one admissible equilibrium point

The Jacobian matrix of f at P0 has the form

and its characteristic polynomial is given by p(l), with

Theorem 2.1. For all

the WGS differential equations (3) have an asymptotically stable equilibrium point at P0. If 0 < e < 2 a b3/2 then P0 is unstable.

The proof of this theorem can essentially be found in Pontryagin [13]; it has also been established in a more general setting in [14].

The surface of critical parameters µ0 = (b, a, ec) such that ec = e(b, a) = 2 a b3/2 is illustrated in Fig. 2. In the Section 4 we will analyze the stability of P0 as ec = 2 a b3/2. The change in the stability at the equilibrium P0 as the parameters cross the critical surface produces a Hopf bifurcation in the WGS, whose analysis has been carried out by [1], [5] and, in a more general setting, by [14].

From (4), e represents the friction coefficient of the system. The case e = 0 maybe of theoretical interest due to its connection with conservative systems. However, as made explicit in Vyshnegradskii's Rules, friction is an essential ingredient to attain stability. This point is neatly presented in Pontryagin [13], of which Figure 2 is a geometric, dimensionless, synthesis.

3 Lyapunov coefficients

The beginning of this section is a review of the method found in [8], pp. 177-181, and in [9] for the calculation of the first and second Lyapunov coefficients. The calculation of the third Lyapunov coefficient has not been found by the authors in the current literature. The extensive calculations and the long expressions for these coefficients have been corroborated with the software MATHEMATICA 5 [18].

Consider the differential equations

where x Î n and µ Î m are respectively vectors representing phase variables and control parameters. Assume that f is of class C¥ in n × m. Suppose (11) has an equilibrium point x = x0 at µ = µ0 and, denoting the variable x - x0 also by x, write

as

where A = fx(0, µ0) and

for i = 1, ..., n.

Suppose (x0, µ0) is an equilibrium point of (11) where the Jacobian matrix A has a pair of purely imaginary eigenvalues l2,3 = ±i w0, w0 > 0, and admits no other eigenvalue with zero real part. Let Tc be the generalized eigenspace of A corresponding to l2,3. By this is meant that it is the largest subspace invariant by A on which the eigenvalues are l2,3.

Let p, q Î n be vectors such that

where AT is the transposed matrix. Any vector y Î Tc can be represented as y = wq + , where w = áp, yñ Î . The two dimensional center manifold can be parametrized by w, , by means of an immersion of the form x = H(w, ), where H: 2® n has a Taylor expansion of the form

with hjk Î n and hjk = kj. Substituting this expression into (11) we obtain the following differential equation

where F is given by (12).

The complex vectors hij are to be determined so that system (22), on the chart w for a central manifold, writes as follows

with Gjk Î .

Solving for the vectors hij the system of linear equations defined by the coefficients of the quadratic terms of (22), taking into account the coefficients of F in the expressions (13) and (14), one has

where In is the unit n × n matrix. Pursuing the calculation to cubic terms, from the coefficients of the terms w3 in (22) follows that

From the coefficients of the terms w2 in (22) one obtains a singular system for h21

which has a solution if and only if

áp, C(q, q, ) + B(, h20) + 2B(q, h11) – G21qñ = 0.

Therefore

The first Lyapunov coefficient l1 is defined by

The complex vector h21 can be found by solving the nonsingular (n + 1)-dimensional system

with the condition áp, h21ñ = 0.

For the sake of completeness, in Remark 3.1 we prove that the system (29) is nonsingular and that if (v,s) is a solution of (29) with the condition áp, vñ = 0 then v is a solution of (26).

Remark 3.1. Write n = Tc Å Tsu, where Tc and Tsu are invariant by A. It can be proved that y Î Tsu if and only if áp, yñ = 0. Define

a = C(q, q, ) + B(, h20) + 2B(q, h11) – G21q

Let (v, s) be a solution of the homogeneous equation obtained from (29). Equivalently

From the second equation of (30), it follows that v Î Tsu, and thus (i w0In - A)v Î Tsu. Therefore áp, (i w0In - A)vñ = 0. Taking the inner product of p with the first equation of (30) one has áp, (i w0In - A)v + sqñ = 0, which can be written as áp, (i w0In - A)vñ + sáp, qñ = 0. Since áp, qñ = 1 and áp, (i w0In - A)vñ = 0 it follows that s = 0. Substituting s = 0 into the first equation of (30) one has (i w0In - A)v = 0 . This implies that

But 0 = áp , vñ = áp , aqñ = aáp, qñ = a. Substituting a = 0 into (31) it follows that v = 0. Therefore (v, s) = (0,0).

Let (v, s) be a solution of (29). Equivalently

From the second equation of (32), it follows that v Î Tsu and thus (i w0In - A)v Î Tsu. Therefore áp , (i w0In - A)vñ = 0. Taking the inner product of p with the first equation of (32) one has áp, (i w0In - A)v + sqñ = áp, añ, which can be written as

áp, (i w0In - A)vñ + sáp, qñ = áp, añ.

As áp, añ = 0, áp, añ = 1 and áp, (i w0In - A)vñ = 0 it follows that s = 0. Substituting s = 0 into the first equation of (32) results (i w0In - A)v = a. Therefore v is a solution of (26).

The procedure above will be adapted below in connection with the determination of h32 and h43.

From the coefficients of the terms w4, w3 and w2

2 in (22), one has respectively

where the term -2h11(G21 + 21) has been omitted in the last equation, since G21 + 21 = 0 as l1 = 0.

Defining

32 as

and from the coefficients of the terms w3

2 in (22), one has a singular system for h32

which has solution if and only if

where the terms -6G21h21 - 321h21 in the last line of (36) actually does not enter in last equation, since áp, h21ñ = 0.

The second Lyapunov coefficient is defined by

where, from (38), G32 = áp, 32ñ.

The complex vector h32 can be found solving the nonsingular (n + 1)-dimensional system

with the condition áp, h32ñ = 0.

From the coefficients of the terms w4, w4

2 and w3
3 in (22), one has respectively

Defining

43 as

and from the coefficients of the terms w4

3, one has a singular system for h43

which has solution if and only if

where the terms - 6 (2G32h21 + 32h21 + 3G21h32 + 221h32) appearing in the last line of equation (44) actually do not enter in the last equation, since áp, h21ñ = 0 and áp, h32ñ = 0.

The third Lyapunov coefficient is defined by

where, from (46), G43 = áp, 43ñ.

The expressions for the vectors h50, h60, h51, h70, h61, h52 have been omitted since they are not important here.

Remark 3.2. Other equivalent definitions and algorithmic procedures to write the expressions for the Lyapunov coefficients lj , j = 1, 2, 3, for two dimensional systems can be found in Andronov et al. [2] and Gasull et al. [6], among others. These procedures apply also to the three dimensional systems of this work, if properly restricted to the center manifold. The authors found, however, that the method outlined above, due to Kuznetsov [8, 9], requiring no explicit formal evaluation of the center manifold, is better adapted to the needs of this work.

A Hopf point (x0, µ0) is an equilibrium point of (11) where the Jacobian matrix A = fx(x0, µ0) has a pair of purely imaginary eigenvalues l2,3 = ± i w0, w0 > 0, and admits no other critical eigenvalues - i.e. located on the imaginary axis. At a Hopf point a two dimensional center manifold is well-defined, it is invariant under the flow generated by (11) and can be continued with arbitrary high class of differentiability to nearby parameter values. In fact, what is well defined is the ¥-jet - or infinite Taylor series - of the center manifold, as well as that of its continuation, any two of them having contact in the arbitrary high order of their differentiability class.

A Hopf point is called transversal if the parameter dependent complex eigenvalues cross the imaginary axis with non-zero derivative. In a neighborhood of a transversal Hopf point - H1 point, for concision - with l1¹ 0 the dynamic behavior of the system (11), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to the following complex normal form

w' = (h + iw)w + l1w|w|2

w Î , h, w and l1 are real functions having derivatives of arbitrary high order, which are continuations of 0, w0 and the first Lyapunov coefficient at the H1 point. See [8]. As l1 < 0 (l1 > 0) one family of stable (unstable) periodic orbits can be found on this family of manifolds, shrinking to an equilibrium point at the H1 point.

A Hopf point of codimension 2 is a Hopf point where l1 vanishes. It is called transversal if h = 0 and l1 = 0 have transversal intersections, where h = h(µ) is the real part of the critical eigenvalues. In a neighborhood of a transversal Hopf point of codimension 2 - H2 point, for concision - with l2¹ 0 the dynamic behavior of the system (11), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to

w' = (h + iw0)w + tw|w2 + l2w|w|4,

where h and t are unfolding parameters. See [8]. The bifurcation diagrams for l2¹ 0 can be found in [8], p. 313, and in [15].

A Hopf point of codimension 3 is a Hopf point of codimension 2 where l2 vanishes. A Hopf point of codimension 3 is called transversal if h = 0, l1 = 0 and l2 = 0 have transversal intersections. In a neighborhood of a transversal Hopf point of codimension 3 - H3 point, for concision - with l3¹ 0 the dynamic behavior of the system (11), reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to

w¢ = (h + iw)w + tw|w2 + nw|w|4 + l3w|w|6

where h, t and n are unfolding parameters. The bifurcation diagram for l3¹ 0 can be found in Takens [15].

Theorem 3.3. Suppose that the system

x¢ = f(x, µ), x = (x, y, z), µ = (b, a, e)

has the equilibriumx = 0 for µ = 0 with eigenvalues

l2, 3(µ) = h(µ) ± iw(µ),

where w(0) = w0 > 0. For µ = 0 the following conditions hold

h(0) = 0, l1(0) = 0, l2(0) =0,

where l1(µ) and l2(µ) are the first and second Lyapunov coefficients, respectively. Assume that the following genericity conditions are satisfied

1. l3 (0) ¹ 0, where l3 (0) is the third Lyapunov coefficient;

2. the map m ® (h(µ), l1(µ), l2(µ)) is regular at µ = 0.

Then, by the introduction of a complex variable, the above system reduced to the family of parameter-dependent continuations of the center manifold, is orbitally topologically equivalent to

w¢ = (h + iw0)w + tw|w|2 + nw|w|4 + l3w|w|6

where h, t and n are unfolding parameters.

Remark 3.4. The proof of this theorem given by Takens for C¥ families of vector fields, using the Malgrange-Mather Preparation Theorem [7], is also valid in the present case of arbitrarily high, but finite, class of differentiability, using the appropriate extensions of the Preparation Theorem. See Bakhtin [3] and Milman [12],among others.

4 Hopf bifurcations

The stability of the equilibrium point P0 given in (7) as ec = e(b, a) = 2 a b3/2 is analyzed here. According to (13) and the subsequent expressions (14), (15), (16), (17), (18) and (19), for Bi to Li, one has

where

and referring to the expressions in equations (11) and (12)

where

From equations (13), (14), (15), (16), (17), (18), (19) and (50) one has

where

where

where

where

where

where

The eigenvalues of A (equation (48)) are

and from (20) one has

Theorem 4.1. Consider the three-parameter family of differential equations (3). The first Lyapunov coefficient is given by

If

is different from zero then the three-parameter family of differential equations (3) has a transversal Hopf point at P0 for ec = e(b, a) = 2 a b3/2.

If (b, a, ec) Î S ÈU (see Fig. 3) then the three-parameter family of differential equations (3) has a H1 point at P0. If (b, a, ec) Î S then the H1 point at P0 is asymptotically stable and for each e < ec, but close to ec, there exists a stable periodic orbit near the unstable equilibrium point P0. If (b, a, ec) Î U then the H1 point at P0 is unstable and for each e > ec, but close to ec, there exists an unstable periodic orbit near the asymptotically stable equilibrium point P0.

This theorem summarizes Proposition 3.2 and Theorems 3.5, 3.6 and 3.7 established in [14]. Equation (61) gives a simple expression for the sign of the first Lyapunov coefficient (60). Its graph is illustrated in Fig. 3, where the signs of the first Lyapunov coefficient are also represented. The curve l1 = 0 divides the surface of critical parameters into two connected components denoted by S and U where l1 < 0 and l1 > 0 respectively.

We have the following theorem.

Theorem 4.2. Consider the three-parameter family of differential equations (3) restricted to e = ec . The second Lyapunov coefficient is given by

where

Proof. Define the following functions

From (39) one has

The theorem follows by expanding the expressions in definition of the second Lyapunov coefficient (39). It relies on extensive calculation involving the vector q (58), the vector p (59), the functions B, C, D and E, listed equations (51), (52), (53) and (54), respectively, the long complex vectors h11, i i 20, h30, h21, h31 and h22, and the above functions T1 to T15.

The calculations in this proof, corroborated by Computer Algebra, have been posted in [17]. Here, the complex vectors h11, h20, h30, h21, h31 and h22 have particularly long expressions. They have listed in MATHEMATICA 5 files.

Theorem 4.3.For the system (3) there is unique point Q = (b, a, ec), with coordinates

b = 0.86828033997971281542..., a = 0.85050048430685017856...

and

ec = 1.37624106484659953171...

where the curves l1 = 0 and l2 = 0 on the critical surface intersect and there do it transversally.

Computer assisted proof. The point Q is the intersection of the curves l1 = 0 and l2 = 0 on the Hopf critical surface. It is defined and obtained by the solution of the equations

g(b, a) =0,

given in (61), and

where h(b, a, ec) is given by (62). The existence and uniqueness of Q with the above coordinates has been established numerically with the software MATHEMATICA 5.

Figure 4 presents a geometric synthesis interpreting the long calculations involved in this proof. The sign of h(b, a) gives the sign of the second Lyapunov coefficient (62). The graph of h(b, a) = 0, where the signs of the first and second Lyapunov coefficients are also illustrated. As follows, l2 < 0 on the open arc of the curve l1 = 0, denoted by C1. On this arc a typical reference point R is depicted. Also l2 > 0 on the open arc of the curve l1 = 0, denoted by C2. This arc contains the typical reference point, denoted by T. See also Fig. 5.

The bifurcation diagrams of the system (3) at the points T and R are illustrated in Fig. 6 and 7, as a consequence of [8] and [15].

The main steps of the calculations that provide the numerical evidence for this theorem have been posted in [17].

Theorem 4.4. If (b, a, ec) Î C1ÈC2 then the three-parameter family of differential equations (3) has a transversal Hopf point of codimension 2 at P0. If (b, a, ec) Î C2 then the H2 point at P0 is unstable and the bifurcation diagram is drawn in Fig. 6. If (b, a, ec) Î C1 then the H2 point at P0 is asymptotically stable and the bifurcation diagram is illustrated in Fig. 7.

This theorem is a synthesis of the discussion in the last part in the proof of Theorem 4.3.

Theorem 4.5. For the parameter values at the point Q determined in Theorem 4.3, the three-parameter family of differential equations (3) has a tranversal Hopf point of codimension 3 at P0 which is asymptotically unstable since l3(Q) > 0. The bifurcation diagram of system (3) at the point Q is illustrated in Figs. 8 and 9 .

Computer assisted proof. For the point Q take five decimal round-off coordinates b = 0.86828, a = 0.85050 and ec = 1.37624. For these values of the parameters one has

From (28), (39), (47), (64), (65) and (66) one has

l1(Q) = 0, l2(Q) = 0, l3(Q) = Re G43 = 0.39050.

The calculations above have also been corroborated with 20 decimals round-off precision performed using the software MATHEMATICA 5 [18]. See [17].

The gradients of the functions l1 , given in (60), and l2 , given in (62), at the point Q are, respectively

(0.80095, -0.31847), (-0.38861, -0.85118).

The transversality condition at Q is equivalent to the non-vanishing of the determinant of the matrix whose columns are the above gradient vectors, which is evaluated gives -0.80552. The transversality condition being satisfied, the bifurcation diagrams in Figs. 8 and 9 , follow from the work of Takens [15], taking into consideration the orientation and signs established in Theorems 4.3 and 4.4.

5 Concluding comments

The historical relevance of the Watt governor study as well as its importance for present day theoretical and technological aspects of Automatic Control has been widely discussed by Denny [4] and others. See also [10, 14].

This paper starts reviewing the stability analysis due to Maxwell and Vyshnegradskii, which accounts for the characterization, in the space of parameters, of the structural as well as Lyapunov stability of the equilibrium of the Watt Centrifugal Governor System, WGS. It continues with recounting the extension of the analysis to the first order, codimension one stable points, happening on the complement of a curve in the critical surface where the eigenvalue criterium of Lyapunov holds, as studied in [5], [1] and by the authors [14], based on the calculation of the first Lyapunov coefficient. Here the bifurcation analysis at the equilibrium point of the WGS is pushed forward to the calculation of the second and third Lyapunov coefficients which make possible the determination of the Lyapunov as well as higher order structural stability at the equilibrium point. See also [8, 9], [6] and [2].

The calculations of these coefficients, being extensive, rely on Computer Algebra and Numerical evaluations carried out with the software MATHEMATICA 5 [18]. In the site [17] have been posted the main steps of the calculations in the form of notebooks for MATHEMATICA 5.

With the analytic and numeric data provided in the analysis performed here, the bifurcation diagrams are established along the points of the curve where the first Lyapunov coefficient vanishes. Pictures 8 and 9 provide a qualitative synthesis of the dynamical conclusions achieved here at the parameter values where the WGS achieves most complex equilibrium point. A reformulation of these conclusions follow:

There is a ''solid tongue'' where two stable regimes coexist: one is an equilibrium and the other is a small amplitude periodic orbit, i.e. an oscillation.

For parameters inside the ''tongue'', this conclusion suggests, a hysteresis explanation for the phenomenon of ''hunting'' observed in the performance of WGS in an early stage of the research on its stability conditions. Which attractor represents the actual state of the system will depend on the path along which the parameters evolve to reach their actual values of the parameters under consideration. See Denny [4] for historical comments, where he refers to the term ''hunting'' to mean an oscillation around an equilibrium going near but not reaching it.

Finally, we would like to stress that although this work ultimately focuses the specific three dimensional, three parameter system of differential equations given by (1), the method of analysis and calculations explained in Section 3 can be adapted to the study of other systems with three or more phase variables and depending on three or more parameters.

Acknowledgement. The first and second authors developed this work under the project CNPq Grant 473824/04-3. The first author is fellow of CNPq and takes part in the project CNPq PADCT 620029/2004-8. This work was finished while he visited Brown University, supported by FAPESP, Grant 05/56740-6.

The authors are grateful to C. Chicone and R. de la Llave for helpful comments.

Received: 20/IV/06. Accepted: 25/XI/06.

#663/06.

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  • Publication Dates

    • Publication in this collection
      10 May 2007
    • Date of issue
      2007

    History

    • Accepted
      25 Nov 2006
    • Received
      20 Apr 2006
    Sociedade Brasileira de Matemática Aplicada e Computacional Sociedade Brasileira de Matemática Aplicada e Computacional - SBMAC, Rua Maestro João Seppe, nº. 900 , 16º. andar - Sala 163, 13561-120 São Carlos - SP Brasil, Tel./Fax: 55 16 3412-9752 - São Carlos - SP - Brazil
    E-mail: sbmac@sbmac.org.br