The global flow in the Chapman mechanism

Nonlinear Analysis 71 (2009) 88–95

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The global flow in the Chapman mechanism E. Pérez-Chavela a,∗ , F.J. Uribe b , R.M. Velasco b a

Department of Mathematics, Universidad Autónoma Metropolitana - Iztapalapa, México D. F. 09340, Mexico

b

Department of Physics, Universidad Autónoma Metropolitana - Iztapalapa, México D. F. 09340, Mexico

article

info

Article history: Received 20 December 2007 Accepted 10 October 2008 Keywords: Stratospheric ozone dynamics Poincaré compactification Chapman’s mechanism Qualitative and quantitative analysis

a b s t r a c t The Chapman mechanism is the simplest photochemical mechanism to understand the formation of ozone in stratospheric conditions. In this paper we apply the Poincaré compactification to the dynamical system generated by the above mechanism. We analyze the critical points both in the finite part and at infinity, and find the heteroclinic connections among them, getting in this way a complete qualitative analysis of the global flow. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction The photochemical mechanisms play an important role in formation and destruction of some chemical species in the atmosphere. In particular the ozone cycle in the stratosphere is one of the most relevant ones because its influence as a filter to the ultraviolet radiation arriving to the planet surface. In 1930 Chapman proposed a photochemical mechanism to study the ozone dynamics in the stratosphere, which though it is the simplest one, allows us to understand some relevant characteristics of the problem [1]. In this paper we are interested in the study of the mathematical properties concerning the dynamical system describing such a mechanism. The differential equations in a chemical mechanism represent the balance of concentrations according to the rates of reactions. In the Chapman mechanism the solar radiation, as well as the atmospheric temperature and pressure play a role in the calculation of the photolytic and reaction rates. The set of differential equations define a polynomial vector field for which we will study the global flow using the Poincaré compactification. In Section 2 we give the equations of motion together with the known characteristics concerning the steady states in the finite region of the space. Section 3 will be devoted to the Poincaré compactification and we will compute the critical points at infinity. In Section 4 we describe the global flow and in Section 5 we give some concluding remarks. 2. The Chapman mechanism This mechanism considers molecular oxygen (O2 ), atomic oxygen (O), ozone (O3 ), and a third body or buffer (M). The differential equations describing the mechanism are balance equations for the concentrations of atomic oxygen nO , molecular oxygen nO2 and ozone nO3 which depend on time, they are given by: d dt d dt

∗

nO (t ) = 2 jO2 nO2 (t ) + jO3 nO3 (t ) − k4.2 nO (t ) nO2 (t ) nM − k4.4 nO3 (t ) nO (t ), nO2 (t ) = jO3 nO3 (t ) + 2 k4.4 nO3 (t ) nO (t ) − jO2 nO2 (t ) − k4.2 nO (t ) nO2 (t ) nM ,

Corresponding author. Tel.: +52 55 5804 4654; fax: +52 55 5804 4653. E-mail addresses: [email protected] (E. Pérez-Chavela), [email protected] (F.J. Uribe), [email protected] (R.M. Velasco).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2008.10.061

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Table 1 Approximate values for the rates to be used as an example in our calculations. The values of the rates k4.2 and k4.4 were rounded off, see [4] for more detailed information.The values of the photolytic rates were taken from [5] and correspond to calculated values at mid latitudes under equinox conditions at noon at a height of 25 km. The value of nM is taken from the standard atmosphere at mid latitudes in the month of March and its value is typical of an altitude of about 27.5 km [6]. These values were chosen so that the interested reader can make quick calculations. Rate constants k4.2 = 10−33 k4.4 =

cm6

≡a

mol .2 s

cm3 10−15 mol .s

≡c

Photolytic rates

Concentrations

jO2 = 3 × 10−12 s−1 ≡ d

nM =

jO3 = 5.5 × 10−4 s−1 ≡ b

. f = 0.24 × 1018 mol cm3

3 5

. ×1018 mol ≡e cm3

Table 2 Specific values of the critical points corresponding to the dynamical system given by Eq. (3), see Eq. (4), and the corresponding eigenvalues (µ1,2 ) of the Jacobian matrix at each critical point. xc (108

mol. cm3

)

zc (1012

+0.5244 −0.5244 +0.24 × 109

d dt

mol. cm3

)

µ1 (1/s)

+0.6864 −0.6864

−2.1 × 10 +2.1 × 10−7 +75.25 −7

0

nO3 (t ) = k4.2 nO (t ) nO2 (t ) nM − jO3 nO3 (t ) − k4.4 nO3 (t ) nO (t ).

µ2 (1/s)

Notation

−72.00 −72.00 −459.25

P+ P− P0

(1)

The above system satisfies a conservation law (first integral) which represents the atoms conservation in chemical reactions, nO (t ) + 2 nO2 (t ) + 3 nO3 (t ) = f ,

(2)

where f is a constant of motion, notice that this relation is valid only when the system is closed. This means that the mass transport as well as the emission rates are not considered. The quantities jO2 and jO3 in Eq. (1) are the photolytic rates, their values depend on the geographical location, the spectral distribution of the radiation coming from the sun and some optical properties of the atmosphere. On the other hand, k4.2 and k4.4 represent the rates for reaction between the respective oxygens which depend on the temperature, pressure and some characteristics of the molecules involved. Finally the quantity nM denotes the concentration of the third body or buffer, air in our case. In this work their values are taken from the literature and they are constants with values given in Table 1. In [2,3] the reader can find a complete description of the variables and the constants involved in the Chapman mechanism, as well as a discussion of the two different time scales which appear in the equations. It is obvious that the Chapman mechanism omits a big number of interesting phenomena, nevertheless it permits to go deeper into its mathematical analysis. People interested in these topics can review References [7,8] among others, where more complex systems are explored. Using the first integral (2) we can reduce the number of independent variables in (1), getting a two-dimensional system given by, dx dt dz dt

= df − 3dz − dx + bz − =

aef 2

x−

3ae 2

xz −

ae 2

aef 2

x+

3ae 2

xz +

ae 2

x2 − cxz , (3)

x2 − bz − cxz ,

where x(t ) ≡ nO (t ) and z (t ) ≡ nO3 (t ) are the oxygen and ozone concentrations, a ≡ k4.2 , b ≡ jO3 , c ≡ k4.4 , d ≡ jO2 , and e ≡ nM . The equilibrium points of (3) correspond to critical concentrations or the steady states, and their specific values for the data in Table 1 are given in Table 2. It is easy to check by straightforward computations that the equilibrium points are given ± by P0 = (f , 0) and P ± = (x± c , zc ), where

√ ±

xc =

d2 c 2 + 4aecdb

dc ±

2aec axc e(f − x± c )

zc± =

, (4)

±

±.

3aex± c + 2b + 2cxc

The coordinates of the point P0 indicate that the molecular oxygen and ozone are completely destroyed and we will have only atomic oxygen. On the other hand, though the point P − is not physical because the concentrations are negative we notice that its coordinates are finite, so in a mathematical sense these points are in a finite region of the phase space (x, z ). Lastly, the point P + is the most interesting one from the physical viewpoint since the concentrations are positive, finite, and is the one that roughly corresponds to the experimental data.

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Fig. 1. Here we have the unit sphere. At the north pole the plane Π is represented by the shaded region. By means of the central projection any point in

Π corresponds to two points over the unit sphere.

By computing the eigenvalues of the respective critical points, see Table 2, one infers that P0 and P − are saddle points whereas P + is a stable node. All three equilibrium points are in a finite region, however, it is interesting to study how these points are reached, when coming from extreme initial conditions corresponding to the extension of the phase space to infinity. Also, it is interesting to know if initial positive concentrations can lead to negative concentrations when integrated according to the flow provided by the Chapman mechanism (3). This information will allow us to separate the phase space in well limited regions, in such a way that if we have the initial conditions in a certain region we will know if the physical steady state can be reached or not. 3. Global analysis: Poincaré Compactification We start our analysis noticing that the differential equations (3) define a polynomial vector field in a two-dimensional space, so in order to study the orbits coming back from (going to) infinity we apply the Poincaré compactification, which we describe here in general, for additional information see [9–11]. Given X = (P 1 , P 2 ) a polynomial vector field in R2 , we identify R2 with the hyperplane Π = {y = (y1 , y2 , y3 ) ∈ R3 |y3 = 1} tangent to the sphere S 2 = {y = (y1 , y2 , y3 ) ∈ R3 |y21 + y22 + y23 = 1} at the north pole. After that, we take the central projection from the sphere S 2 to the hyperplane Π , that is, for each point in Π we draw the straight line through this point and the origin of R3 , obtaining in this way two antipodal points in S 2 , one in the open northern hemisphere H + of S 2 and the other in the open southern hemisphere H − of S 2 (See Fig. 1). More concretely, this construction defines the following two diffeomorphisms

φ + : R2 −→ H + and φ − : R2 −→ H − given by

φ + (x) =

1

∆(x)

φ − (x) = −

(x1 , x2 , 1)

1

∆(x)

(x1 , x2 , 1)
1/2

(5)

(x1 , x2 ) ∈ R2 . In this form X induces a vector field Xˆ on H + ∪ H − defined by  (Dφ + )x X (x) if y = φ + (x) Xˆ (y) = (Dφ − )x X (x) if y = φ − (x).

where ∆(x) = 1 + x21 + x22

and

(6)

The expression for Xˆ (y) in H + ∪ H − is



1 − y21 ˆX (y) = y3  −y2 y1 −y3 y1

 −y1 y2  ˆ 1  P 1 − y22  ˆ 2 P −y3 y2

(7)

where Pˆ i (y1 , y2 , y3 ) = P i (y1 /y3 , y2 /y3 ). The equator S 1 = {y ∈ S 2 |y3 = 0} of the Poincaré sphere corresponds to the infinity of R2 , and the key point of the Poincaré compactification is the possibility to extend the flow given by Xˆ defined on S 2 \ S 1 to S 2 . In this way we will be able to study the orbits of X going to, or coming from the infinity in R2 . This extension is possible due to the polynomial character of X . Thus, the vector field



1 − y21 −1 ˆ ˜X (y) = ym  X (y) = −y2 y1 3 −y 3 y 1

 −y1 y2  ˜ 1  P 1 − y22  ˜ 2 P −y3 y2

(8)

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91

Table 3 The critical points at infinite (y3 = 0) in units of 108

particles cm3

particles cm3

for oxygen and 1013

∞

for ozone. This means using α = 108 and β = 1013 in Eq. (14).

∞

y 1 = xc

y2 = zc

Notation

0 0

1

A+ A− C+ B+ C− B−

−1 −0.0000013552873 +0.0002213552818 +0.0000013552873 −0.0002213552818

+0.999999999999082 +0.999999975500919 −0.999999999999082 −0.999999975500919

is analytic on the whole S 2 . Here m = max{degree(P 1 ), degree(P 2 )} is the degree of X and k P˜ k (y1 , y2 , y3 ) = ym 3 P (y1 /y3 , y2 /y3 ).

(9)

Notice that P˜ k are homogeneous polynomials of degree m. The vector field X˜ is called ‘‘The Poincaré Compactification of X ’’. Coming back to the analysis of the Chapman mechanism, in order to apply the Poincaré compactification to system (3), we will consider the plane where we have the concentrations of atomic oxygen (x) and ozone (z ), as the tangent plane Π in the above construction, by all the above (in this case m = 2) the Poincaré compactification of the Chapman mechanism takes the form: dy1 dt dy2 dt dy3 dt

= (1 − y21 ) P˜1 − y1 y2 P˜2 , = −y1 y2 P˜1 + (1 − y22 ) P˜2 , = −y1 y3 P˜1 − y2 y3 P˜2 ,

(10)

where, P˜ 1 = d f y23 − P˜ 2 =

aef 2

 d+

aef



2

y1 y3 + (b − 3 d) y2 y3 +

y1 y3 − b y2 y3 −

ae 2

y21 −



3ae 2

 +c

ae 2

y1 y2 ,

y21 +



3ae 2

 −c

y1 y2 , (11)

where a, b, c , d and e have the values given in Table 2 without units, this point is explained with more detail in the next section. Let us observe that the equator of the Poincaré sphere (y3 ≡ 0), corresponds to the infinity and that this set is invariant by the flow of system (10). The finite critical points have been already computed in (4), then to compute the infinite critical points, those with y3 = 0, we analyze the first two equations in (10), using that y21 + y22 = 1 by straightforward computations we obtain six critical points at infinity given by: A+ = (0,

s +

B

=

C

=

0),

∆2+ 1 + ∆2+

s +

1,

∆2− 1 + ∆2−

1

,

s

∆+ 1

,

∆−

∆2+ 1 + ∆2+

s

∆2− 1 + ∆2−

! ,

0 ,

! ,

0 .

(12)

where A− = −A+ , B− = −B+ , C − = −C + , and

√ ∆+ =

−2ae − c + √

∆− =

−2ae − c −

a2 e2 + 6aec + c 2 ae a2 e2 + 6aec + c 2 ae

, (13)

.

Using the values of the constants given in Table 1 we obtain the values given in Table 3. Since the Poincaré sphere S 2 is a two-dimensional differentiable manifold, the above analysis can also be done in the respective local charts, when the dimension of the space is ≥2, this is the usual technique; however in our case we preferred to do the analysis in the whole system because it is more direct.

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Table 4 The critical points for the Poincaré compactification given by Eqs. (10) and (15) for α = 2.4 × 1017 and β = 5 × 1018 . The values for the chemical and photolytic rates, and the concentrations are taken from Tables 1 and 2 (without units). yc1

yc2

2.1851 × 10 −2.1849 × 10−10

+0.13728 × 10 −0.13732 × 10−5

−10

−5

√1

2

0 0

Notation

0.9999999999991 0.9999999999991

P+ P− P0

0

√1

1

0 0 0 0 0 0

2

−1 −0.6505241 × 10−2 +0.7282017 +0.6505241 × 10−2 −0.7282017

+0.9999788 +0.6853629 −0.9999788 −0.6853629

yc3

A+ A− C+ B+ C− B−

4. The Global flow In this section we study the global flow of the Poincaré compactification of the Chapman mechanism. Since it is necessary to give specific values for the coefficients that appear in Chapman’s mechanism we will use those given in Table 1. There is one point that we would like to stress and that can easily be seen from Table 3, it is difficult to see in one graph the detailed behavior of the flow for all the critical points due to the different scales. In particular the critical points on the Poincaré circle are very close together; so the first step in our analysis is to bring them apart by using certain units in the following way; using nO (t ) = α x(t ) particles/cm3 , nO3 (t ) = β z (t ) particles/cm3 , t = τ s, and with the values given in Table 1 we can write the dynamical system given by Eq. (3) as, dx

d? f ?

3 d? z β

−

− d? x +

b? z β

a? e ? f ?

3 a? e? β

α a? e?

x2 − c ? x z β, α α α 2 2 2 dz α a? e ? f ? 3 α a? e? α 2 a? e? 2 = x− xz − x − b? z − α c ? x z , (14) dτ 2β 2 2β where the values of a? , . . . , e? are the same as the corresponding non-starred quantities given in Table 1 without units. For simplicity, in the following we will take out the star for these quantities. Choosing the values of α and β we can make the dτ

=

−

x+

xy +

Poincaré compactification of the dynamical system given by Eq. (14) to give critical points at infinity that are separated. However, one must check that the chosen values do not change the nature of the finite critical points, this means that we must have two saddles and one stable node. Explicitly, the Poincaré compactification of the dynamical system given by Eq. (14) is obtained directly from Eq. (10) where now,

    aef β ae 2 3ae − d+ y1 y3 + (b − 3 d) y2 y3 + y1 α + − c y1 y2 β, α 2 α 2 2   2 aef α ae 2 α 3ae P˜ 2 = y1 y3 − b y2 y3 − y1 − + c y1 y2 α. 2 β 2 β 2 P˜ 1 =

d f y23

(15)

In Table 4 we show the critical points of the Poincaré compactification for one election of α and β , recall that while the Poincaré compactification may be defined for (y1 , y2 , y3 ) ∈ R3 , we are interested only in the flow on S 2 . Now, we will give some additional qualitative features of the global flow. First, from the conservation law given in (2), we have that 2 nO2 (t ) = f − x(t ) − 3z (t ),

(16)

since nO2 (t ) must be a non-negative quantity for the physical phenomena to be studied, we can define the physical region Ω as the region in the x–z bounded by the straight lines: x = 0, z = 0 plane and L = f − x − 3z = 0, that is

Ω = {(x, z ) ∈ R2 |x ≥ 0, z ≥ 0 and L = f − x − 3z ≥ 0}.

(17)

Using Eq. (3), it is easy to check that the vector field on the vertical axis with 0 < z < f /3 points into region Ω , the same happens on the horizontal axis for 0 < x < f , and also on L (see [3] for a complete description of this fact). All the above shows that solutions with initial positive concentrations in the physical region Ω can not lead into regions with negative concentrations. In other words, the physical region Ω is a positively invariant set (see Fig. 4 for a qualitative description of these facts). Finally, in order to have a complete idea of the global flow, we will use numerical methods, in particular we have used Adams’ method to integrate the equations. There are other ways to obtain a solution, as the Backward Differentiation Formula (BDF) or the Runge–Kutta methods, however Adams’ method performs better in terms of the accuracy needed to obtain the numerical solution. Besides the results coming from all the three methods are almost identical. There are two possible ways to proceed in order to obtain the global flow; the first one is to solve numerically the differential system in the plane and then use the central projection to obtain the curves in the unit sphere, the second one

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93

Fig. 2. Projections of the Poincaré compactification in the y1 –y2 plane (y3 = 0). The solid circles are the projections of the initial conditions in the unit sphere, the solid lines are the projected numerical solution constructed by integrating in forward time while the dashed lines are the projections of the numerical solution constructed by integrating in backward time. The diamonds are the critical points at infinity. There are two different time scales, the solid and dashed lines were constructed by taking time increments ∆τ that varied from 10−3 to 10−1 (∆t = ∆τ s). For more information see Reference [3].

is to use the field provided by the Poincaré compactification, see Eqs. (10) and (15), and solve the corresponding differential system in three dimensions. We decided to use the latter approach to have an alternative test for the numerical methods and also because the calculated points should lie on the unit sphere and this is easy to see from the numeric output. Given an initial condition, numerically we follow the corresponding orbit in both senses, that is, in the direction pointed by the vector field (forward time) and in the opposite direction (backward time). In Fig. 2 the projection on the y1 –y2 plane of the numerical solutions is given. There are nine critical points for the Poincaré compactification, see Table 4, three of them correspond to the finite part and are represented as open circles. The points P + and P − are very close to the origin due to the scale in Fig. 2, both of them are inside the small circle around the origin in Fig. 2, whereas the point P0 is on the y1 -axis. The critical points at infinity are represented by diamonds on the Poincaré circle. With respect to the heteroclinic connections it is simpler to start with the invariant Poincaré circle. The heteroclinic orbits apparent from Fig. 2 are: B+ −→ A+ , B+ −→ C + , A− −→ C + , A− −→ B− , C − −→ B− , and C − −→ A+ . If we now consider the whole region several conclusions can be made. For example the heteroclinic connection B+ −→ C + seems to be unique but there are infinite many heteroclinic orbits of this type, all of them corresponding to initial conditions in the interior of region D1 (each region Di is limited by one or two of the coordinate axis y1 , y2 , and the curves marked in Fig. 2). The dynamics inside the small circle around the origin is not possible to be described in this scale. Furthermore, an infinite number of heteroclinic orbits B+ −→ P + are also possible when the initial conditions are arbitrarily chosen in region D2. Similar conclusion holds for the apparently unique heteroclinic connection A− −→ B− and the infinitely many heteroclinic orbits C − −→ B− . In Table 5 we provide a more complete description of our numerical findings. There are however a couple of points that are worth to mention. Firstly, in the first and second quadrants the orbits come together near the y2 -axis—this is just a broad statement since on the scale shown, the details are hidden and below we will be more precise—what really happens to the solutions is that once they are near the axis they move slower in the y2 -axis and tend to the stable node P + . For points near the stable node it can be shown that the orbits actually move along the line joining the stable node and the non-physical saddle P − [3]. Of course, as one is near the stable node the orbits will move slower. Similar behavior is observed in the third and four quadrants but in this case this behavior in the orbits is observed when the integration is performed backward in time. Notice should be made that the physical region Ω , in which the field points to the stable point is contained in the region D2, see [3]. In fact, the physical region corresponds to a small region, how small it is can be seen by noting that the point (0, f /3) which is in the boundary of Ω goes under the central projection, with the scale factors α = 2.4 × 1017 and β = 5 × 1018 , to the point (0, 0.016, 0.9999). In Fig. 3 the physical region is expanded. Finally, we would like to mention that in a small neighborhood around the origin Fig. 2 does not give detailed information of the flow, the interested reader may take a look at Reference [3]. In particular we have not shown in Fig. 2 the connection between P − and B− .

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Table 5 The α and ω limit sets for the different points on the unit sphere. Here, x denotes the projection of a point in the unit sphere on the y1 –y2 plane.

x x x x x x x

∈ int(D1) ∈ int(D2) ∈ int(D3) ∈ int(D4) ∈ int(D5) ∈ int(D6) ∈ int(D7)

α limit set

ω limit set

B+ B+ C− C− A− A− A−

C+ P+ P+ B− B− P+ C+

Fig. 3. Projections of the Poincaré compactification on the y1 –y2 plane, using the scales α = 2.4 × 1017 and β = 1013 . The solid circles are the projections of the initial conditions on the unit sphere. The solid lines are the projections of the numerical solutions by integrating forward in time while the dashed lines correspond to the projections constructed by integration backward in time. The diamonds are the critical points. The dot-dashed line is the projection of the image under the central projection of the line L (see Eq. (17)). Since when using α = 2.4 × 1017 and β = 1013 , there are situations in which the time to generate the solution with BDF is about twenty thousand times faster than with Adams’ method, this figure was constructed using BDF.

Fig. 4. The separatists connections on the Poincaré disk.

In order to get an idea of the behavior of the dynamical system near the three finite critical points it is convenient to use a different scale: α = 2.4 × 1017 and β = 1013 (in Fig. 2, the physical region is hidden along the y2 -axis). The results are given in Fig. 3 where a heteroclinic connection between P − and P + is shown. Notice that the six critical points at infinite now appear as two. At the north pole we have together the critical points A+ = (0, 1, 0), B+ and C − , while at the south pole we have the critical points A− , B− and C + . Referring to Fig. 2, we see that on the left part of the Poincaré circle in Fig. 3 we

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have the region D4, regions D3 and D5 are hidden. Thus, the heteroclinic orbit shown is between B− and C − . For the initial condition shown on the right side of the Poincaré circle, its ω limit set is P + while its α limit is C + . There is also a heteroclinic orbit between P − and B− that is not shown in Fig. 2. The line joining A+ with P + corresponds roughly to a set in which the orbits move slowly (slow manifold). To get a sensible idea of the meaning slow it should be mention that starting from the initial condition it takes ∆τ ≈ 10−4 to reach the y2 -axis, and from there it takes about ∆τ ≈ 108 to reach P + . The orbit from P − to P + shown in Fig. 3, was generated with ∆τ ≈ 108 . As the lector have noticed, due to the two different scales involved in Chapman mechanism, graphically is impossible to have a unique picture which shows all the above information. So, in order to facilitate the lecture of this paper in Fig. 4 we give a purely qualitative picture of all heteroclinic connections. 5. Conclusions Using the Poincaré compactification we have analyzed the global flow of the Chapman mechanism. The global flow is seen on the unitary open disk whose boundary, the circle S 1 represents the infinite part. On the finite part there are three critical points and six critical points at infinity; numerically we have found all the heteroclinic connections among them. From Fig. 2, we can see that there are heteroclinic orbits in the finite part of the phase space which separate it in disjoint invariant regions. The most important from the physical viewpoint is region D2, where the physical region is contained, hence for each point P in the physical region, the ω-limit of the orbit by z is the critical point P + [3]. In Fig. 4 we give a qualitative picture of the global flow by showing all heteroclinic connections among the critical points, in the finite part and at infinity. Acknowledgments We express our deep thanks to J. Llibre for helpful discussions and to the anonymous referees for their valuable comments. E.Pérez-Chavela was partially supported by CONACYT México, grants No. 47768. Work financed in part by PROMEP under grant Nos. UAM-I-CA-45 and UAM-I-CA-55. References [1] S. Chapman, Mem. Roy. Met. Soc. III 26 (1930). [2] E. Pérez-Chavela, F.J. Uribe, R.M. Velasco, in: L.S. García Colín, J. R. Varela (Eds.), Contaminación Atmosférica V, El Colegio Nacional (2006) (in spanish), a copy of the work is available upon request ([email protected]). [3] R.M. Velasco, F.J. Uribe, E. Pérez-Chavela, Stratospheric ozone dynamics according to the Chapman mechanism, J. Math. Chem. 44 (2) (2008) 529–539. [4] H. Seinfeld, S.N. Pandis, Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Wiley, 1998. [5] D.G. Andrews, An Introduction to Atmospheric Physics, Cambridge, 2000. [6] J.T. Houghton, The Physics of Atmospheres, Cambridge University Press, Cambridge, 1991. [7] B.J. Finlayson-Pitts, J.N. Pitts, Chemistry of the Upper and Lower Atmosphere, Academic Press, 2000. [8] A. Dessler, The Chemistry and Physics of Stratospheric Ozone, Academic Press, 2000. [9] A. Cima, J. Llibre, Bounded polynomial vector fields, Trans. Amer. Math. Soc. 318 (1990) 557–579. [10] J. Delgado, E.A. Lacomba, J. Llibre, E. Pérez, Poincaré compactification of polynomial hamiltonian vector fields, in: Hamiltonian Dynamical Systems, History, Theory and Applications, in: The IMA Volumes in Mathematics and its Applications, vol. 63, Springer-Verlag, 1994, pp. 99–115. [11] A. García, E. Pérez-Chavela, A. Susín, A generalization of the Poincaré compactification, Arch. Ration. Mech. and Anal. 179 (2006) 285–302.