3.1 Linear autonomous systems

As we already mentioned in Chapter 2, we consider dynamical processes of a finite dimension n which are differentiate and deterministic.

In the simplest case, the system of differential equations (2.3.3) is linear with constant coefficients

 

 

(3.1.1)

where L denotes the constant, non-singular (nxn)-matrix of the coefficients. Using this example, we now demonstrate which solutions can occur and which patterns the relevant trajectories form in the corresponding phase space.

We introduce the ansatz

 

(3.1.2)

into equ. (3.1.1), thus obtaining a homogeneous linear system of equations

 

 

 

 

(3.1.3)

which only possesses non-trivial solutions if the system determinant vanishes

 

 

 

Fig. 3.1.1 Survey of the eigenvalues of singular points

 

 

 

 

P( A) is a polynomial of the nth degree in A and is called characteristic or secular equation. The zeroes of P(A) are the eigenvalues of L. A non-vanishing vector y which satisfies equ. (3.1.3) is called the eigenvector of L appertaining to the eigenvalue A. If A and y satisfy equ. (3.1.3), equ. (3.1.2) is a solution of equ. (3.1.1). For each pair Ai,j/i, we obtain in accordance with equ. (3.1.2) a solution of the form

 

 

If all n eigenvalues Ai are different, n linearly independent eigenvectors y, exist and the general solution of equ. (3.1.1) can be expressed by the linear combination

 

with n integration constants C; which are determined by the initial conditions a;(to) (Arnol'd, 1980; Braun, 1979).

From a physical point of view, finding all the states of equilibrium xs is of interest. These are stationary states of the system in which the dynamical process does not undergo any change; they are thus characterised by xs = o. If we observe the corresponding point in the phase space, the vector F(xs) which defines the change of the trajectory vanishes here. For this reason, xs is thus also denoted singular point. The decisive factor in characterising the stationary state is the behaviour of the trajectories in the neighbourhood of the singularity. We speak of stable, unstable or neutral equilibrium. If all the trajectories within certain neighbourhoods of xs are captured, the singular point is asymptotically stable (sink); if, one the other hand, all the trajectories that come close enough to xs are repelled, xs is asymptotically unstable (source). On the basis of a system of two differential equations, we now demonstrate the classification of singularities

 

 

 

 

 with the origin xs — o as the singular point. The eigenvalues of the correspond­ing equ. (3.1.3) serve as a basis to make the following six distinctions, illustrated diagrammatically in fig. 3.1.1; here, we ignore the degenerate case of a zero eigen­value.

 

 

Initially, we assume that L possesses different eigenvalues X\ / A2 and denote the corresponding eigenvectors as 3/1,3/2 which we can presume to be normalised without loss of generality.

 

Due to

We now introduce a new system of coordinates

 

 

 

which is defined by the transformation

 

 

Thus, the eigenvectors 3/1,3/2 are the basis vectors of the new coordinate system. Substitution of the transformation equ. (3.1.9) into the given system equ. (3.1.7) yields

 

 

(3.1.11)

and is thus reduced to diagonal form. The transformed system equivalent to equ. (3.1.7) now becomes

(3.1.12)

Here, we can distinguish for Ai ^ A2 a total of four possibilities (see fig. 3.1.1). If the eigenvalues are real, the solution of equ. (3.1.12) according to equ. (3.1.6) is

 

 

or, after eliminating the time parameter t,

 

 

We next differentiate between two cases having different signs (see fig. 3.1.1):

 

 

Fig. 3.1.2

D can be simplified as follows

 

 

 

Case Al: Stable node (Ai, A2 real; A2 < Ai < 0)

 

If the eigenvalues have the same sign, equ. (3.1.14) describes a family of parabolae of the order A2/A1 which have a common tangent at their origin (see fig. 3.1.2). We then speak of a node. If the eigenvalues are negative, the node is stable, as can be seen in the parameter representation of equ. (3.1.13) since all trajectories tend towards the singular point for t —> <x>.

 

 

If Ai and A2 possess different signs, the solution is

 

 

i.e. the trajectories are hyperbolae. The corresponding singular point is then called a saddle point (see fig. 3.1.3).

Fig. 3.1.3

Case A2: Saddle point (Ai, A2 real; Ai < 0 < A2)

If Ai,A2 are conjugate complex, L can always be reduced by applying a (real) linear coordinate transformation T to the simple form

 

 

The eigenvectors y 1 and y2 appertaining to the eigenvalues Ai,2 = a±iuj are also conjugate complex

 

 

 

and reduce in accordance with equ. (3.1.9) the transformation of L to a diagonal form; in accordance with standard usage, y* denotes the conjugate complex vector of y\. Thus, points x of the real phase space are transformed into points x of a two-dimensional complex phase space C2. The solution of equ. (3.1.12) can be expressed immediately as

 

 

where the complex constants C\ and C2 coincide with the initial values 5i(0) and z2(0). It is now easy to see that C1 and C2 are conjugate complex. From the inverse transformation of equ. (3.1.9) for an initial vector x(0)

 

 

we obtain the relation i2(0) = ^i(O), i.e. C2 = C*. Thus, the solutions according to equ. (3.1.16a) are also conjugate complex, i.e. x2(t) — x*(t) applies. The inverse transformation of the initial system hence yields a real solution

(3.1.16b)

We observe that the initial vector x(0) is rotated about an angle u>t, and stretched with the factor eQt.

 

002

We once again distinguish between two cases (see fig. 3.1.1):

 

 

 

 

 

 

 

This formulation describes a

Equation (3.1.16b) can also be interpreted as a representation in polar coordinates for which we set

iamily ot logarithmic spirals. It the real part ol A; is negative, i.e. Re (A;) = o < 0. we have a stable focus; on the other hand, for a > 0, the singular point is unstable. Figure 3.1.4 shows the spirals in the original xi,x2 coordinate system. They are affinely distorted by the inverse transformation.

 

 

In the special case of a = 0, the phase curves in the x^,x2 coordinate system become concentric circles, the centre of which is the singular point. We then speak of a centre or vortex point. Figure 3.1.5 shows the trajectories in the original Xi,X2 coordinate system in which L possesses a general form. Here, we observe concentric ellipses.

 

 

 

 

 

We now move on to a discussion of the case of identical eigenvalues, Ai = A2 = A. Here, we distinguish between two cases (fig. 3.1.1):

(Bl) There are two linearly independent eigenvectors y \ . iyj for A.

In analogy to equs. (3.1.10), (3.1.11), the system x = Lx can be transformed into a diagonal form, x; = Ax;(i = 1,2), with the solutions

 

 

or

 

 

 

 

 

Fig. 3.1.6

Case Bl: stable dicritical node (Ai = A2 = A < 0)

 

(B2) There is only one linearly independent eigenvector y for A.

In this case, a sufficient number of eigenvectors does not exist to span the whole phase space; thus, L can no longer be transformed to diagonal form. However, in matrix theory, it is shown than any non-diagonalisable (nxn) matrix can always be transformed uniquely to the so-called Jordan normal form where each Jordan block is associated with an eigenvector (Zurmiihl, 1964).

 

 

The corresponding coordinate transformation is

For n = 2, there is thus a distinguished X\, x2 coordinate system in which the set of differential equations (3.1.7) can be transformed to the Jordan normal form

(3.1.19)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Having treated row sequences and their relatives, we turn to diagonal sequences, ray sequences, and their relatives. As an introduction, we prove Theorem 6.5.3, which is a simple theorem, and subsequently we generalize it substantially. This theorem was originally proved by Nuttall [1970b], using Szego's theorem, but the proof given by Zinn-Justin [1971], based on Hermite's formula, is somewhat simpler.

Theorem 6.5.3. Let f(z) be a meromorphic function. Suppose that e, 8 are given positive numbers. Then M0 exists such that any [M/M] Pade approxi- mant satisfies

 

 

for all M > M0 on any compact set of the z-plane except for a set &M of measure less than 8.

 

 

(5.12)

 

Proof. We will set the scale of the z-plane by proving convergence for |z|< 1 except on a set &M of measure less than 8. Define t] = j^8/tt . We may assume 0<tj< 1 without loss of generality. Define

 

 

For some A, no matter how small, in the range 0<A< 1, R exists satisfying

 

poles located at

 

 

 

 

 

Next we need Hermite's interpolation formula using the polynomial

 

 

 

 

and consequently for

 

 

 

 

We have chosen to consider \z\<\, R>Rmin>2, and so

 

 

The denominator of the right-hand side of (5.13) is a polynomial with leading coefficient unity, and is bounded by

 

 

 

except for z in a set &M of measure irt}2 = 8. Assembling (5.12), (5.13), and (5.14),

 

 

 

 

 

 

 

 

Provided M>2m, and recalling that

 

 

for any M>(some M0), except on the set &M of measure less than 8.

As already stated, Theorem 6.5.3 is a weak form of both what is known to be true and what is expected to be true about convergence in measure of ray sequences. Nonetheless, it provides a basis for further development.

First, the diagonal sequence may be replaced by the sequence \Lk/Mk\. k= 1.2     nrovided that, for anv \ in the ranse 0<\< 1. however small.

 

 

Equation (5.15) confines the Pade approximants to a fan-shaped region of the Pade table as shown in Figure 1. Provided {5.15) holds and Lk +Mk —»oc. this weaker constraint is sufficient to allow convergence in measure.

From (5.6), with some M'<:M, we proved that

Second, /(z) need not be meromorphic, but may also have a countable number of isolated essential singularities. This means that exp[ —(1 — z)-1] and exp[zT(z)] are allowable functions, but not functions whose singulari­ties have a limit point in the finite z-plane.

 

Figure 1. The allowed domain of the Pade table for the approximants in Theorem 6.5.4.

 

 

 

Theorem 6.5.4 [Pommerenke, 1973]. Let f{z) be a function which is analytic at the origin and analytic in the entire z-plane except for a countable number of isolated poles and essential singularities. Suppose e>0 and S>0 are given. Then Mn exists such that any \L/M] Pade approximant of the ray

 

sequence with

 

satisfies

 

 

for any M Ss M(). on any compact set of the z-plane except for a set &M of measure less than S.

 

and

 

Proof. As in Theorem 6.5.3, we take

 

assume

We choose

 

(5.16)

 

and some

 

such that

 

Define the polynomial

 

(5.17)

 

 

where p=p(M) is defined so that the ratio p=p/M satisfies

 

Hermite's formula, with

 

 

 

 

 

 

Hence, for

 

The purpose of this is to be able subsequently to let M—> oo and >oc simultaneously but keep RM{z) of sufficiently low degree. At any rate. Rm(z) has leading coefficient unity and degree less than M. We use

where C is a closed contour containing the origin and no singularities of QM(z)f(z). By enlarging the contour so as to enclose the essential singulari­ties, we find

where

 

 

Equation (5.20) is a contour integral round a small circle of radius 8k enclosing the essential singularity at z = wk. To bound Ik(z), we require that \z-wA>28„ Using the maximum-modulus theorem for the polynomial

We now specify the radii Sk of the small circles by defining

 

 

 

(5.22)

Equation (5.16) ensures that the regions \z — wk\<2Sk surrounding the essential singularities in |z| < 1 have arbitrarily small total measure. From (5.22),

 

 

We define

 

 

where Kk is independent of M, and then

 

 

 

where K' is independent of M and k. Assembling (5.19) and (5.23), we find

 

 

where K" is also independent of M.

Again, when QM(t) has M' zeros within |/|<2/?, we note that M'<M and RM(t) is a polynomial of degree m+pa, so that (5.6) and (5.7) yield provided  a set of measure less than wtj2. Since L=\M and |z|< 1,

(5.18), (5.24), and (5.25) yield

 

 

where K"' is also independent of M. From (5.18), pn<j\M, and then

(5.26) gives

<e except on the set &M and the small circles enclosing the essential singularities of/(z) in |z|< 1.

Corollary 1. This theorem can also be generalized to treat arbitrary sequences in the region of the Pade table shown Figure 1.

 

Let

 

 

Corollary 2 [Zinn-Justin, 19711. Let f(z) be meromorphic in

the number of zeros of

 

Then if

 

 

 

 

 

the [M/M] Padé approximants of f(z) converge in measure in

Corollary 3 [Zinn-Justin, 1971]. If f(z) is an analytic function of exponential order less than 2/X, then the sequence of[\M/M] approximants converges on any compact set of the z-plane except on a set of arbitrarily small measure.

We present these corollaries without proof. The second and third are interesting because they show that further restrictions on the class of functions considered lead to stronger convergence results. However, no theorem yet proved gives convergence in measure of diagonal Pade ap­proximants in \z\<R for functions known only to be meromorphic in |z|</?. We refer to Edrei [1975b] for a result which allows the essential singularities to be limit points of pole sequences, rather than isolated essential singularities as in Pommerenke's theorem.