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ical point, one obtains an implicit ordinary differential equation

v,(z, z) = g(X,(r,r),z.z)

for the radius :{t) and the wave speed :(t).

4. Statement of Main Results. The main result of this paper, taken in

conjunction with Bukiet [5, 6] is that the velocity of the expanding detonation

is equal to the plane wave velocity plus a correction which is to lowest order

linear in the shock curvature k = 1/z. Moreover the constant of propor-

tionality is determined by the separatrix solution of (3.7). This statement

follows on a numerical level from our equations together with their numerical

solution and the comparison to the numerical solution of the full Euler equa-

tions by Bukiet. One consequence of this result is that standard methods of

computation of detonation waves [13] which use the experimentaj values of

the planar detonation velocity can be improved in accuracy by these

18 -

corrections. Moreover since the correction can be computed from the chem-

istry, we believe that the correction can be predicted from some phenomeno-

logical equation of state and rate law. at least after the latter have been recali-

brated to reflect the new requirement that they reproduce both planar speeds

and leading order curvature corrections. Such a predictive capability would

minimize the amount of experimental calibration necessary to use this new

theory in numerical computations.

We begin our analysis of the model (3.7) with the statement of our main

theorem. We will see in subsequent sections that the linear part of the vector

field at the plane wave sonic critical point is singular, and is consequently a

bifurcation point for the system. Theorem 4.1 identifies the effect of curva-

ture in (3.7) as a perturbation of the sonic bifurcation point.

THEOREM 4.1 Assume that there is a radius :â– , such that T^. < T when-

ever < ,v and r, < z. Then there is a k^^.^ > and a neighborhood

(^mm, -^max) of Zq SUCh that

i) The sonic bifurcation point v^ = (v^y + ^JL"(i" - ^o))^ ", ^b ^ ^ ^^ ^^e

vector field (3.7) bifurcates into a saddle point as k = â€” is increased from

zero, for alii 6 (imin> ^max)-

ii) For K^,^ sufficiently small, and f 6 (inun, ima.x)- ^^e saddle point in i)

is the unique sonic critical point of (3.7).

iii) The location of the saddle point in the phase plane is a C function of

K 6 (0, K^^J and : i (z^,,, z^^J.

iv) The restriction of the vector field (3.3) to the stable separatrix of the

saddle point is continuous, uniformly in r â‚¬ (imm. ^max)-

19-

v) The unique smooth transonic solution of (3.3) is given by the stable

separatrix of the saddle point.

In [3, 5] Bukiet solved the shooting problem numerically for the func-

tional dependence of the wave speed i on the curvature k. The separatrix

solution in Theorem 4.1 is computed simultaneously, employing the wave

speed determined by the shooting problem. The computations by Bukiet

indicate that the wave speed depends linearly on the curvature to leading

order.

We now present some of the tools which this analysis will require.

Those familiar with bifurcation theory for vector fields may wish to skip to

Section 5.

By the phase flow ^l{yi) of a vector field â€”â€” - f(w) we mean the gen-

dy

eral solution of the differential equation interpreted as a map taking an initial

point w at v = to its image at some later value of v. Thus

^ such that all C^ perturbations of size e of f at w^ are C^

equivalent to f at Wq. By bifurcation point we mean any point at which f is

not structurally stable.

In general, bifurcation refers to a change in the topological equivalence

class of the vector field under a perturbation. In practice, the perturbation is

controlled by a set of bifurcation parameters , and the different equivalence

classes for the perturbed vector field correspond to different regions of the

parameter space. An n parameter family of perturbations of a bifurcation

point is called an n parameter unfolding of the bifurcation. We will show in

Section 6 that k is a bifurcation parameter for the plane wave sonic bifurca-

tion point, so that (3.7) is a one parameter unfolding of the bifurcation.

Local equivalence is usually defined by means of the elegant machinery of

jets [1]. We have chosen a more prosaic definition which is adequate for our

purposes. We will also make use of the following restricted form of struc-

tural stability.

DEnNITION 4.8 Let K denote a subset of the set of all C^ perturba-

tions at Wq. a vector field f is said to be C* stable to class K perturbations at

Wq if all perturbations in A' of f at Wq are C* equivalent to f at Wq.

PROPOSITION 4.9 Let f(w) be a continuous vector field in a domain of

R" with no critical points. Then f is locally structurally stable.

PROPOSITION 4.10 (Hartman-Grobman Theorem). Let w^ be an iso-

lated critical point of a smooth vector field f(w). If none of the eigenvalues

23-

of the Jacobian derivative Df(w^.) of f at at Wq have a zero real part, then f is

topologically equivEilent to the linear system â€” â€” = Df(w^.)(w - w^,) at w^.

Proposition 4.9 follows easily from the Rectification Theorem for con-

tinuous vector fields, which states that the vector field is C' equivalent to a

constant field in a neighborhood of a noncritical point. (The Rectification

Theorem may be found, for example, in Arnold [2].) Any sufficiently small

perturbation of a constant field is noncritical and therefore rectifiable, but all

nonzero constant fields are topologically equivalent. This implies that a

bifurcation point is necessarily a critical point. A proof of Proposition 4.10 is

presented in Arnold [1]. It is easily verified that the linear system in Proposi-

tion 4.10 is structurally stable at the critical point, and thus the Hartman-

Grobman theorem implies that a bifurcation point must possess at least one

eigenvalue with zero real part. Critical points possessing no eigenvalues with

a zero real part are termed hyperbolic.

The Hartman-Grobman theorem proves structural stability for hyperbolic

critical points by relating them to the critical points of a topologically

equivalent linear system which is easy to analyze. The method we will

employ to analyse the sonic bifurcation point is analogous to what the

Hartman-Grobman theorem accomplishes for hyperbolic critical points,

except that nonlinear terms are required because of the degeneracy of the

bifurcation point. We transform the vector field in a neighborhood of the

bifurcation point by a smooth change of variables to a field which is topologi-

cally equivalent but simpler in form. It turns out that in most cases, the

topological equivalence class of a vector field at a critical point contains

-24-

polynomial fields. In such a case there are fields of smallest degree, and pos-

sessing the fewest number of terms. The study of these minimal fields was

initiated by Poincare, and the fields are referred to as normal forms for the

equivalence class. To find such a field, we will take advantage of the analyti-

city of the vector field by expanding in powers of the deviations v, \ from

the critical values (v^, 1). We then seek a diffeomorphism of the phase space

that preserves the linear part of the vector field while eliminating as many

nonlinear terms as possible. This procedure is customarily carried out order

by order; first quadratic terms are eliminated, then cubic terms and so on. In

the present case, only the second degree terms are required. The next step is

to show that the transformed field may be truncated at some finite order to

obtain a topologically equivalent field. This latter step is usually the hardest,

in part because it is usually necessary to identify the class of perturbations

which preserve the equivalence class, and in part because the construction of

topological equivalence maps is often a highly nontrivial enterprise. The nor-

mal form thus possesses the same local topological structure as the original

field, but is much easier to study. This method may also be applied to an

unfolding of the bifurcation, so we may speak of a normal form for the

unfolding. We will demonstrate the first step of this technique in Section 6.

As indicated in the discussion following Definition 4.5, the second step of

local topological equivalence is factorized, and the subproblem of translation

to a fixed critical point is also solved in Section 6. Excellent introductions to

the theory of normal forms for vector fields are available in Arnold [1] and in

Guckenheimer and Holmes [II].

25 -

Assume that A is the matrix of the linear part of a two dimensional ana-

lytic vector field f(w) with a critical point at the origin, so that f has the form

-^ = f(w) - Aw - f'-i(w) ,

ay

where f '-^ = 0(|w|-). For each integer n ^2, the linear operator A induces a

linear operator L^ on the linear space Ji, n ot' 2-vectors having entries which

are homogeneous /ith degree polynomials in w-^, W2,

(4.1) (Lvh), - i l^(Aw), - (Ah),.

Note that L^h is just the Lie bracket [h, Aw] of h with Aw. If L^ were

nonsingular, all nth degree terms in the Taylor series for f could be elim-

inated by the nonlinear change of variables w = w + L^^h where h denotes

the vector of n\h degree terms in the series. In general, only those nth

degree terms of f which lie in the range of L^ can be eliminated. Those ele-

ments of J2, n which do not lie in the range of L^ cannot be eliminated by a

smooth change of variables and are termed resonant. The resonant terms of f

contain the essential nonlinear contributions to the phase plane structure.

Applying L^ to the standard basis (w-(w2, 0)^ and (0, w-fw^)^, j ^ I - n,

yields a set of vectors which span the range of L;^. A basis for the range may

be chosen from this set. We need never include more than codim(range(Lx))

in the set of resonant vectors. This is because two resonant vectors which

differ only by an element of range(LA) are smoothly equivalent (the element

of range(L^) may be transformed away). Consequently we may identify the

set of resonant vectors with the non-zero vectors of the quotient space

^2. â€ž/range(L;^). Choose any basis for this quotient space. These basis

-26

vectors are equivalence classes of elements of J2. n- Now choose any particu-

lar representatives for these equivalence classes. These representatives define

a maximal set of resonant vectors, i.e. f may be smoothly transformed into a

vector field with second degree terms consisting of a linear combination of

this maximal set of resonant vectors. In practice this maximal set is chosen

from among the standard basis vectors, if possible, in order to produce the

maximum simplification of the vector field. Extensions of these ideas to

higher dimensional vector fields and their associated spaces Jâ€ž â€ž of homo-

geneous vector valued polynomials are obvious.

In Sections 5 and 6 we begin a local analysis of the bifurcation point

using the ideas presented above. In Section 5 we present and study a one

parameter unfolding of a proposed normal form for the plane wave sonic

critical point point w - c,k - 1. This degenerate critical point possesses two

zero eigenvalues and is the primary bifurcation point for the system. The

unfolding parameter represents the effect of shock curvature on the phase

plane structure. For this simplified vector field the plane wave sonic critical

point bifurcates into a unique saddle point as the shock curvature Vz is

increased from zero. In Section 6 we show that the unique saddle structure

observed in Section 5 is preserved under a class of perturbations which

includes (3.7), and that the critical point itself is a smooth function of Vz.

5. The Proposed Normal Form. We propose

(5.1)

V

= V

V

-aX

(X - tik)v

27

as a normal form for the vector field (3.7) at the bifurcation point. Here

K = â€” is the bifurcation parameter. The coefficients v, a, and r\ are posi-

tive. The variables v and X. here denote v and X translated to the transonic

critical point, which remains fixed at the origin as k is varied. In this section

we investigate the properties of this proposed normal form. The results here

will assist in understanding the properties of the transonic critical point, as

well as lay a foundation for an eventual proof of local topological equivalence

with (3.7) at the critical point.

If the first of the equations (5.1) is divided by the second, a separable

ordinary differential equation

vrfv = - dX

k â€” r\K

is obtained for the phase curves v(X). The general solution of this equation

is

(5.2)

V = v: - 2a

X - Xr + TiKln

X â€” T|K

[X, - T1K J

where (v^, X^) denotes any fixed reference point.

For K > 0, (5.1) has a unique critical point at (0, 0). The eigenvalues

and corresponding eigenvectors are

p^ = Â±v{ar\K)^~,

Thus the critical point is a saddle.

V. =

1

:::(TiK/a)^

L'2

-28

The phase plane structure for (5.1) is shown in Fig. 5.1. Note that the

horizontal Hne X = tik is a phase curve for (5.1), as well as a horizontal

asymptote for all nearby phase curves. This line corresponds to the X. = 1

line of the original vector field (3.7) (although the location of the X = 1 line

is perturbed slightly from t|k). The region above the X = t|k line is non-

physical.

Figure 5.1

We may use (5.2) to eliminate v from the X^, equation in (5.1), obtaining

L = sgn(v,)v(X - tik)

v; â€” 2a

X - V + TnÂ»' / 2 + -),

, 3 > , |v,| < V,

, p > , |v,| > V,

,3 =

, 3 <

-32-

where

tanh \-Vr/ vj, |v,| < v,,

coth-i(-v,/v,), |v,|>v.

We end this section with some observations about the double zero bifur-

cation point at the origin of (5.4). There are two choices of resonant terms

for A in terms of standard basis vectors. They are

v"

v"

and

v^

v''~^\

Choosing the latter, we find that the only resonant terms in (3.7) at z = ^o

are (0, Xv)^, and (0, Xv^)^. Only the second degree term is retained in the

proposed normal form (5.4). Note that the resonant terms (0, v")^ are miss-

ing from (3.7) for all /i. As a consequence of this degeneracy the bifurcation

has infinite codimension: the number of distinct topological equivalence

classes which may be obtained by a small perturbation of the plane wave vec-

tor field is infinite. This exceptional degeneracy is exhibited as the line of

critical points. A nice presentation of the nondegenerate case, in which both

second degree resonant terms are present, may be found in Guckenheimer

and Holmes [11]. The case of second order degeneracy occurs in models of

33

chemical reactors [14].

6. The Transonic Critical Point. In this section we apply the Poincare

transformations described in Section 4 to facilitate our study of the phase

plane structure of (3.7) in a neighborhood of the sonic bifurcation point. As

indicated in Section 3 we will ignore the functional relationship between r

and z defined by the shooting problem described in Soaion 3, and consider z

and z to be independent parameters of the system. It will be seen that this is

sufficient to determine the topological structure of the bifurcation point.

In order to carry out the transformations in a way which preserves the

correct dependence on k we employ the standard trick of defining an aug-

mented system which consists of (3.7) together with a third equation

-â€” - 0. We then expand the augmented system in a Taylor series about the

origin (v, X, k) = (0, 0, 0) and perform the simplifying nonlinear transfor-

mations as indicated in Section 4, while allowing only transformations that

leave k invariant.

In what follows we will use O to denote a neighborhood of the origin in

the V, \ plane. It will be necessary at several points to restrict (v, \, k) to

some cylinder ftmax ^ fO- ^rna-J to obtain a desired result. For notational

simplicity we will let k^^^ and 0.^^^ denote the minimum over all such res-

trictions. Let Pâ€žâ€ž denote the class of real analytic functions on U C R'"

which are C)(|w|") for w z U, where U is any neighborhood of the origin.

PROPOSITION 6.1 For each

- 34

p., i \2^i 6 {1,2},

q, 6 P,, :,'â– ^ {1.2},

Gi, Ut, bi > 0,

a, â‚¬ R,; â‚¬ {3, â€¢ â€¢ â€¢ ,6},

^^ 6 R, A: 6 {2, 3,4},

there exist p, 6 ^'3 2,

v,(z, z) = g(X,(r,r),z.z)

for the radius :{t) and the wave speed :(t).

4. Statement of Main Results. The main result of this paper, taken in

conjunction with Bukiet [5, 6] is that the velocity of the expanding detonation

is equal to the plane wave velocity plus a correction which is to lowest order

linear in the shock curvature k = 1/z. Moreover the constant of propor-

tionality is determined by the separatrix solution of (3.7). This statement

follows on a numerical level from our equations together with their numerical

solution and the comparison to the numerical solution of the full Euler equa-

tions by Bukiet. One consequence of this result is that standard methods of

computation of detonation waves [13] which use the experimentaj values of

the planar detonation velocity can be improved in accuracy by these

18 -

corrections. Moreover since the correction can be computed from the chem-

istry, we believe that the correction can be predicted from some phenomeno-

logical equation of state and rate law. at least after the latter have been recali-

brated to reflect the new requirement that they reproduce both planar speeds

and leading order curvature corrections. Such a predictive capability would

minimize the amount of experimental calibration necessary to use this new

theory in numerical computations.

We begin our analysis of the model (3.7) with the statement of our main

theorem. We will see in subsequent sections that the linear part of the vector

field at the plane wave sonic critical point is singular, and is consequently a

bifurcation point for the system. Theorem 4.1 identifies the effect of curva-

ture in (3.7) as a perturbation of the sonic bifurcation point.

THEOREM 4.1 Assume that there is a radius :â– , such that T^. < T when-

ever < ,v and r, < z. Then there is a k^^.^ > and a neighborhood

(^mm, -^max) of Zq SUCh that

i) The sonic bifurcation point v^ = (v^y + ^JL"(i" - ^o))^ ", ^b ^ ^ ^^ ^^e

vector field (3.7) bifurcates into a saddle point as k = â€” is increased from

zero, for alii 6 (imin> ^max)-

ii) For K^,^ sufficiently small, and f 6 (inun, ima.x)- ^^e saddle point in i)

is the unique sonic critical point of (3.7).

iii) The location of the saddle point in the phase plane is a C function of

K 6 (0, K^^J and : i (z^,,, z^^J.

iv) The restriction of the vector field (3.3) to the stable separatrix of the

saddle point is continuous, uniformly in r â‚¬ (imm. ^max)-

19-

v) The unique smooth transonic solution of (3.3) is given by the stable

separatrix of the saddle point.

In [3, 5] Bukiet solved the shooting problem numerically for the func-

tional dependence of the wave speed i on the curvature k. The separatrix

solution in Theorem 4.1 is computed simultaneously, employing the wave

speed determined by the shooting problem. The computations by Bukiet

indicate that the wave speed depends linearly on the curvature to leading

order.

We now present some of the tools which this analysis will require.

Those familiar with bifurcation theory for vector fields may wish to skip to

Section 5.

By the phase flow ^l{yi) of a vector field â€”â€” - f(w) we mean the gen-

dy

eral solution of the differential equation interpreted as a map taking an initial

point w at v = to its image at some later value of v. Thus

^ such that all C^ perturbations of size e of f at w^ are C^

equivalent to f at Wq. By bifurcation point we mean any point at which f is

not structurally stable.

In general, bifurcation refers to a change in the topological equivalence

class of the vector field under a perturbation. In practice, the perturbation is

controlled by a set of bifurcation parameters , and the different equivalence

classes for the perturbed vector field correspond to different regions of the

parameter space. An n parameter family of perturbations of a bifurcation

point is called an n parameter unfolding of the bifurcation. We will show in

Section 6 that k is a bifurcation parameter for the plane wave sonic bifurca-

tion point, so that (3.7) is a one parameter unfolding of the bifurcation.

Local equivalence is usually defined by means of the elegant machinery of

jets [1]. We have chosen a more prosaic definition which is adequate for our

purposes. We will also make use of the following restricted form of struc-

tural stability.

DEnNITION 4.8 Let K denote a subset of the set of all C^ perturba-

tions at Wq. a vector field f is said to be C* stable to class K perturbations at

Wq if all perturbations in A' of f at Wq are C* equivalent to f at Wq.

PROPOSITION 4.9 Let f(w) be a continuous vector field in a domain of

R" with no critical points. Then f is locally structurally stable.

PROPOSITION 4.10 (Hartman-Grobman Theorem). Let w^ be an iso-

lated critical point of a smooth vector field f(w). If none of the eigenvalues

23-

of the Jacobian derivative Df(w^.) of f at at Wq have a zero real part, then f is

topologically equivEilent to the linear system â€” â€” = Df(w^.)(w - w^,) at w^.

Proposition 4.9 follows easily from the Rectification Theorem for con-

tinuous vector fields, which states that the vector field is C' equivalent to a

constant field in a neighborhood of a noncritical point. (The Rectification

Theorem may be found, for example, in Arnold [2].) Any sufficiently small

perturbation of a constant field is noncritical and therefore rectifiable, but all

nonzero constant fields are topologically equivalent. This implies that a

bifurcation point is necessarily a critical point. A proof of Proposition 4.10 is

presented in Arnold [1]. It is easily verified that the linear system in Proposi-

tion 4.10 is structurally stable at the critical point, and thus the Hartman-

Grobman theorem implies that a bifurcation point must possess at least one

eigenvalue with zero real part. Critical points possessing no eigenvalues with

a zero real part are termed hyperbolic.

The Hartman-Grobman theorem proves structural stability for hyperbolic

critical points by relating them to the critical points of a topologically

equivalent linear system which is easy to analyze. The method we will

employ to analyse the sonic bifurcation point is analogous to what the

Hartman-Grobman theorem accomplishes for hyperbolic critical points,

except that nonlinear terms are required because of the degeneracy of the

bifurcation point. We transform the vector field in a neighborhood of the

bifurcation point by a smooth change of variables to a field which is topologi-

cally equivalent but simpler in form. It turns out that in most cases, the

topological equivalence class of a vector field at a critical point contains

-24-

polynomial fields. In such a case there are fields of smallest degree, and pos-

sessing the fewest number of terms. The study of these minimal fields was

initiated by Poincare, and the fields are referred to as normal forms for the

equivalence class. To find such a field, we will take advantage of the analyti-

city of the vector field by expanding in powers of the deviations v, \ from

the critical values (v^, 1). We then seek a diffeomorphism of the phase space

that preserves the linear part of the vector field while eliminating as many

nonlinear terms as possible. This procedure is customarily carried out order

by order; first quadratic terms are eliminated, then cubic terms and so on. In

the present case, only the second degree terms are required. The next step is

to show that the transformed field may be truncated at some finite order to

obtain a topologically equivalent field. This latter step is usually the hardest,

in part because it is usually necessary to identify the class of perturbations

which preserve the equivalence class, and in part because the construction of

topological equivalence maps is often a highly nontrivial enterprise. The nor-

mal form thus possesses the same local topological structure as the original

field, but is much easier to study. This method may also be applied to an

unfolding of the bifurcation, so we may speak of a normal form for the

unfolding. We will demonstrate the first step of this technique in Section 6.

As indicated in the discussion following Definition 4.5, the second step of

local topological equivalence is factorized, and the subproblem of translation

to a fixed critical point is also solved in Section 6. Excellent introductions to

the theory of normal forms for vector fields are available in Arnold [1] and in

Guckenheimer and Holmes [II].

25 -

Assume that A is the matrix of the linear part of a two dimensional ana-

lytic vector field f(w) with a critical point at the origin, so that f has the form

-^ = f(w) - Aw - f'-i(w) ,

ay

where f '-^ = 0(|w|-). For each integer n ^2, the linear operator A induces a

linear operator L^ on the linear space Ji, n ot' 2-vectors having entries which

are homogeneous /ith degree polynomials in w-^, W2,

(4.1) (Lvh), - i l^(Aw), - (Ah),.

Note that L^h is just the Lie bracket [h, Aw] of h with Aw. If L^ were

nonsingular, all nth degree terms in the Taylor series for f could be elim-

inated by the nonlinear change of variables w = w + L^^h where h denotes

the vector of n\h degree terms in the series. In general, only those nth

degree terms of f which lie in the range of L^ can be eliminated. Those ele-

ments of J2, n which do not lie in the range of L^ cannot be eliminated by a

smooth change of variables and are termed resonant. The resonant terms of f

contain the essential nonlinear contributions to the phase plane structure.

Applying L^ to the standard basis (w-(w2, 0)^ and (0, w-fw^)^, j ^ I - n,

yields a set of vectors which span the range of L;^. A basis for the range may

be chosen from this set. We need never include more than codim(range(Lx))

in the set of resonant vectors. This is because two resonant vectors which

differ only by an element of range(LA) are smoothly equivalent (the element

of range(L^) may be transformed away). Consequently we may identify the

set of resonant vectors with the non-zero vectors of the quotient space

^2. â€ž/range(L;^). Choose any basis for this quotient space. These basis

-26

vectors are equivalence classes of elements of J2. n- Now choose any particu-

lar representatives for these equivalence classes. These representatives define

a maximal set of resonant vectors, i.e. f may be smoothly transformed into a

vector field with second degree terms consisting of a linear combination of

this maximal set of resonant vectors. In practice this maximal set is chosen

from among the standard basis vectors, if possible, in order to produce the

maximum simplification of the vector field. Extensions of these ideas to

higher dimensional vector fields and their associated spaces Jâ€ž â€ž of homo-

geneous vector valued polynomials are obvious.

In Sections 5 and 6 we begin a local analysis of the bifurcation point

using the ideas presented above. In Section 5 we present and study a one

parameter unfolding of a proposed normal form for the plane wave sonic

critical point point w - c,k - 1. This degenerate critical point possesses two

zero eigenvalues and is the primary bifurcation point for the system. The

unfolding parameter represents the effect of shock curvature on the phase

plane structure. For this simplified vector field the plane wave sonic critical

point bifurcates into a unique saddle point as the shock curvature Vz is

increased from zero. In Section 6 we show that the unique saddle structure

observed in Section 5 is preserved under a class of perturbations which

includes (3.7), and that the critical point itself is a smooth function of Vz.

5. The Proposed Normal Form. We propose

(5.1)

V

= V

V

-aX

(X - tik)v

27

as a normal form for the vector field (3.7) at the bifurcation point. Here

K = â€” is the bifurcation parameter. The coefficients v, a, and r\ are posi-

tive. The variables v and X. here denote v and X translated to the transonic

critical point, which remains fixed at the origin as k is varied. In this section

we investigate the properties of this proposed normal form. The results here

will assist in understanding the properties of the transonic critical point, as

well as lay a foundation for an eventual proof of local topological equivalence

with (3.7) at the critical point.

If the first of the equations (5.1) is divided by the second, a separable

ordinary differential equation

vrfv = - dX

k â€” r\K

is obtained for the phase curves v(X). The general solution of this equation

is

(5.2)

V = v: - 2a

X - Xr + TiKln

X â€” T|K

[X, - T1K J

where (v^, X^) denotes any fixed reference point.

For K > 0, (5.1) has a unique critical point at (0, 0). The eigenvalues

and corresponding eigenvectors are

p^ = Â±v{ar\K)^~,

Thus the critical point is a saddle.

V. =

1

:::(TiK/a)^

L'2

-28

The phase plane structure for (5.1) is shown in Fig. 5.1. Note that the

horizontal Hne X = tik is a phase curve for (5.1), as well as a horizontal

asymptote for all nearby phase curves. This line corresponds to the X. = 1

line of the original vector field (3.7) (although the location of the X = 1 line

is perturbed slightly from t|k). The region above the X = t|k line is non-

physical.

Figure 5.1

We may use (5.2) to eliminate v from the X^, equation in (5.1), obtaining

L = sgn(v,)v(X - tik)

v; â€” 2a

X - V + TnÂ»' / 2 + -),

, 3 > , |v,| < V,

, p > , |v,| > V,

,3 =

, 3 <

-32-

where

tanh \-Vr/ vj, |v,| < v,,

coth-i(-v,/v,), |v,|>v.

We end this section with some observations about the double zero bifur-

cation point at the origin of (5.4). There are two choices of resonant terms

for A in terms of standard basis vectors. They are

v"

v"

and

v^

v''~^\

Choosing the latter, we find that the only resonant terms in (3.7) at z = ^o

are (0, Xv)^, and (0, Xv^)^. Only the second degree term is retained in the

proposed normal form (5.4). Note that the resonant terms (0, v")^ are miss-

ing from (3.7) for all /i. As a consequence of this degeneracy the bifurcation

has infinite codimension: the number of distinct topological equivalence

classes which may be obtained by a small perturbation of the plane wave vec-

tor field is infinite. This exceptional degeneracy is exhibited as the line of

critical points. A nice presentation of the nondegenerate case, in which both

second degree resonant terms are present, may be found in Guckenheimer

and Holmes [11]. The case of second order degeneracy occurs in models of

33

chemical reactors [14].

6. The Transonic Critical Point. In this section we apply the Poincare

transformations described in Section 4 to facilitate our study of the phase

plane structure of (3.7) in a neighborhood of the sonic bifurcation point. As

indicated in Section 3 we will ignore the functional relationship between r

and z defined by the shooting problem described in Soaion 3, and consider z

and z to be independent parameters of the system. It will be seen that this is

sufficient to determine the topological structure of the bifurcation point.

In order to carry out the transformations in a way which preserves the

correct dependence on k we employ the standard trick of defining an aug-

mented system which consists of (3.7) together with a third equation

-â€” - 0. We then expand the augmented system in a Taylor series about the

origin (v, X, k) = (0, 0, 0) and perform the simplifying nonlinear transfor-

mations as indicated in Section 4, while allowing only transformations that

leave k invariant.

In what follows we will use O to denote a neighborhood of the origin in

the V, \ plane. It will be necessary at several points to restrict (v, \, k) to

some cylinder ftmax ^ fO- ^rna-J to obtain a desired result. For notational

simplicity we will let k^^^ and 0.^^^ denote the minimum over all such res-

trictions. Let Pâ€žâ€ž denote the class of real analytic functions on U C R'"

which are C)(|w|") for w z U, where U is any neighborhood of the origin.

PROPOSITION 6.1 For each

- 34

p., i \2^i 6 {1,2},

q, 6 P,, :,'â– ^ {1.2},

Gi, Ut, bi > 0,

a, â‚¬ R,; â‚¬ {3, â€¢ â€¢ â€¢ ,6},

^^ 6 R, A: 6 {2, 3,4},

there exist p, 6 ^'3 2,

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