1. Introduction
There are well-known procedures for putting a system of differential equations
(where v is a formal power series starting with quadratic terms) into normal form with respect to its linear part A. Our concern in this paper is to describe the normal form of the systemm
, that is the set of all v such that
is in normal form where A is the linear part
from the Stanley decomposition of the ring of invariants. Our main result is a procedure that solves the description problem where N is a nilpotent matrix with coupled n Jordan blocks, provided that the description problem is already solved for each Jordan block of N taken separately. Our method is based on adding one block at a time. This procedure will be illustrated with examples and then be generalized.
The idea of simplification near an equilibrium goes back at least to Poincare (1880), who was among the first to bring forth the theory in a more definite form. Poincare considered the problem of reducing a system of nonlinear differential equations to a system of linear ones. The formal solution of this problem entails finding nearidentity coordinate transformations, which eliminate the analytic expressions of the nonlinear terms.
Cushman et al. [1], using a method called covariant of special equivariant solved the problem of finding Stanley decomposition of
. Their method begins by creating a scalar problem that is larger than the vector problem and their procedures are derived from classical invariant theory thus it was necessary to repeat calculations of classical invariants theory at the levels of equivariants. Malonza [2] solved the same problem by “Groebner” basis methods found in [3] rather than borrowing from classical theory.
Murdock and Sanders [4] developed an algorithm based on the notion of transventants to determine the form of normal form of a vector field with nilpotent linear part, when the normal form is known for each Jordan block of the linear part taken separately. The algorithm is based on the notion of transvectants from the classical invariant theory known as boosting to module of equivariants when the Stanley decomposition for the ring of invariants is known.
Namachchivaya et al. [5], studied a generalized Hopf bifurcation with non-semisimple 1:1 Resonance. The normal form for such a system contains only terms that belong to both the semisimple part of A and the normal form of the nilpotent, which is a coupled TakensBogdanov system with ![](https://www.scirp.org/html/10-7401042\e5258415-4669-4676-8c0f-e9a98bad9298.jpg)
This example illustrates the physical significance of the study of normal forms for systems with nilpotent linear part.
Our results are mainly based on the work found in [4] that is application of transvectant’s method for computing normal form for the module of equivariants of nilpotent systems. In section two and three we put together background knowledge for understanding the content of this work. Section four forms the central part of this paper where we shall compute the module of equivariants.
2. Invariants and Stanley Decompositions
Let
denote the vector space of homogeneous polynomials of degree
on
with coefficients in
, where
denotes the set of real numbers. Let
be the vector space of all such polynomials of any degree and let
be the vector space of formal power series. If
,
becomes the ring of formal power series on
, where
denotes the set of real numbers. For such smooth vectors fields, it is sufficient to work polynomials. For any nilpotent matrix
, we define the Lie operator
![](https://www.scirp.org/html/10-7401042\f021ea8d-17e2-4006-b513-a618019c36e0.jpg)
by
(2.1)
and the differential operator
![](https://www.scirp.org/html/10-7401042\1dd7f671-0278-477c-8760-1b7084e5d0ba.jpg)
by
(2.2)
Then
is a derivation of the ring
, meaning that
(2.3)
In addition,
(2.4)
A function
is called an invariant of
if
or equivalently
Since
![](https://www.scirp.org/html/10-7401042\f8f6e5fe-0989-45c4-8edb-f870c96c696d.jpg)
![](https://www.scirp.org/html/10-7401042\4526a5bf-2544-43f5-9715-27b6c60102a1.jpg)
it follows that if f and
are invariants, so are
amd
; that is
is both a vector space over
and also a subring of
, known as the ring of invariants. Similarly a vector field
is called an equivariants of
, if
that is
![](https://www.scirp.org/html/10-7401042\2f1e8ff5-4687-4d0a-9f77-40066fa10b58.jpg)
There are two normal form styles in common use for nilpotent systems, the inner product normal form and the sl(2) normal form. The inner product normal form is defined by
where
is the conjugate transpose of
. To define the sl(2) normal form, one first sets
and constructs matrices
and
such that
(2.5)
An example of such an
triad
is
![](https://www.scirp.org/html/10-7401042\3f992b43-46cd-4144-82c4-b5a84e83cc07.jpg)
Having obtained the triad
we create two additional triads
and
as follows
(2.6)
(2.7)
The first of these is a triad of differential operators and the second is a triad of Lie operators. Both the operators
and
inherit the triad properties (2.5). Observe that the operators
map each
into itself. It follows from the representation theory
that
(2.8)
Clearly the
ia s subring of
, the ring of invariants and it follows from (2.4) that
is a module over this subring. This is the sl(2) normal form module.
3. Boosting Rings of Invariants to Module of Equivariants
In this section we describe the procedure for obtaining a Stanley decomposition of the module of equivariants (or normal form space
) when the Stanley decomposition of the ring of invariants is known.
The module of all formal power series vector fields on
can be viewed as the tensor product
, and in fact the tensor product can be identified with the ordinary product (of a field times a constant vector) since the ordinary product satisfies the same algebraic rules as a tensor product. Specifically, every formal power series vector field can be written as
![](https://www.scirp.org/html/10-7401042\759f2dd4-1ae2-4531-b8fc-77c630a4a316.jpg)
where the
are the standard basis vectors of
. Next, the Lie derivative
can be expressed as the tensor product of
and
, that is
. Under the identification of
with ordinary product, this means
, where ![](https://www.scirp.org/html/10-7401042\321e967c-7f35-4f83-8f6e-ce7100b42fea.jpg)
and
in agreement with the following calculation, in which
because
is constant.
![](https://www.scirp.org/html/10-7401042\d7938667-c5e0-4375-886d-d17ee5b4749c.jpg)
This kind of calculation also shows that
representation (on vector fields ) with triad
is the tensor product of the representation (on scalar fields)
with triad
and the representation (on ![](https://www.scirp.org/html/10-7401042\1ee45472-d1b4-47ea-a367-e27dfd4befe0.jpg)
with triad
that is
![](https://www.scirp.org/html/10-7401042\a0461411-7917-4e86-96b9-968e22f0f511.jpg)
It follows that a basis from the normal form space
is given by well defined transvectants ![](https://www.scirp.org/html/10-7401042\301fa0cf-4726-4a08-9ac0-1fa41a8365b1.jpg)
as
ranges over a basis for ![](https://www.scirp.org/html/10-7401042\ee230fda-f44a-4a4b-b69a-7e4946d0b1e5.jpg)
and
ranges over a basis for
. The first of these bases is given by the standard monomials of a Stanley decomposition for
. The second is given by the standard basis vectors
such that
is the index of the bottom row of a Jordan block in
. It is useful to note that the weight of such an
is one less than the size of the block. Then we define the transvectant
as
![](https://www.scirp.org/html/10-7401042\a0f16b9d-2676-496e-acca-2f56d0894962.jpg)
From here, the computational procedures of box products are the same as those used in describing rings of invariants from [4], except that infinite iterations never arise.
4. Normal Form for Systems with Linear Part N3(n)
Before generalizing we shall consider the normal form for nonlinear systems with linear part having two and three blocks, that is
and
as examples.
4.1. System with Linear Part N33
The Stanley decomposition for the ring of invariants with linear part
is given by:
![](https://www.scirp.org/html/10-7401042\6a0f94aa-3dd9-4b97-9ea0-33b2c5d13651.jpg)
(see [6]). Since
and
has weight zero, it is convenient to remove them since we do not expand along terms of weight zero by setting
and write
![](https://www.scirp.org/html/10-7401042\1af23872-7521-431c-bfb8-90dbc214ed1a.jpg)
In this case the basis elements are
and
. Therefore we need to compute the box product of the ring
with
which are both of weight 2.
Therefore
. Distributing the box product there are two cases to consider.
Case 1:
.
There are four products namely:
a) ![](https://www.scirp.org/html/10-7401042\28722d35-2690-4506-91e0-3759f0c6e730.jpg)
b) ![](https://www.scirp.org/html/10-7401042\e1bc8be3-468c-437a-aec2-08ebc01873d1.jpg)
c) ![](https://www.scirp.org/html/10-7401042\85d97d79-fe22-43e8-8ca9-8a1274de7be5.jpg)
d) ![](https://www.scirp.org/html/10-7401042\175271b6-0e32-4710-8bd4-199434ae3cc9.jpg)
Recombining terms gives
![](https://www.scirp.org/html/10-7401042\9f15216a-e927-4ef4-a7c0-17b00d511e09.jpg)
Case 2: Similarly we have,
![](https://www.scirp.org/html/10-7401042\a0686a81-e22f-4268-8493-7004f7494636.jpg)
Adding terms in case 1 and 2 we obtain:
![](https://www.scirp.org/html/10-7401042\c8c8a4ad-1fe3-4351-a939-0ed8cdb5fccb.jpg)
Finally, to complete the calculation, it is necessary to compute the transvectants that appear. These are of the form
and
for
where
.
![](https://www.scirp.org/html/10-7401042\cf477524-a43b-4fb0-8bd5-8679c618845c.jpg)
![](https://www.scirp.org/html/10-7401042\59009a7e-2240-48b5-861a-93f423f9401d.jpg)
![](https://www.scirp.org/html/10-7401042\81602794-2be4-46a5-85a2-38546ad5c956.jpg)
We ignore the nonzero constants –1 and –2 because we are concerned with computing basis elements. For the basis
we have:
![](https://www.scirp.org/html/10-7401042\d0d0a92f-d65d-406a-be38-d81782fe5c35.jpg)
Therefore the normal form for system with linear part
is:
![](https://www.scirp.org/html/10-7401042\724dd0a9-83cc-4b1f-bddf-b28823ea45b0.jpg)
![](https://www.scirp.org/html/10-7401042\197f8845-647c-4a26-a4f5-f7f8a7ba96bd.jpg)
4.2. System with Linear Part N333
The Stanley decomposition for ring of invariants of a system with linear part
is given by:
![](https://www.scirp.org/html/10-7401042\30ff844e-1db7-441e-84f4-62d47c741cf5.jpg)
(see [6]).
The basis elements for
are
and
. Therefore we need to compute the box product of the invariants ring
with
. Thus
Let
, then
![](https://www.scirp.org/html/10-7401042\f3967f64-6588-4445-95c0-2c36ab104082.jpg)
There are three cases to consider. Computing and simplifying the cases we obtain the normal form as:
![](https://www.scirp.org/html/10-7401042\556fb963-7900-47a4-8aaa-75b3cb59feb4.jpg)
where
and
such that
,
and ![](https://www.scirp.org/html/10-7401042\7f0829de-c3b3-46cc-99de-b6e426d85aee.jpg)
In general, from the above examples we conclude that the normal forms are obtained by computing the box product
![](https://www.scirp.org/html/10-7401042\453e0451-eec5-4119-8194-fdcfccfac356.jpg)
The basis of the normal form of
are transvectants of the form:
where
is the standard monomials of Stanley decomposition of the ring of invariants,
,
and
.
As an example we find the normal form for a system with linear part
, we first find the ring of invariants
where
using
. By inspection
and
, and this generates the entire ring; that is
(4.1)
To check this, we note that the weight of
is two and
is of weight zero, so the table function of
is
![](https://www.scirp.org/html/10-7401042\609d5409-32ab-4fb8-a2d7-cdcac9db8f4e.jpg)
Hence
![](https://www.scirp.org/html/10-7401042\ad683a85-791a-44c5-8d06-a84da4c61b74.jpg)
this implies (0.1).
The next step is to compute
as a module over
.
contains one Jordan block of size 3 hence the differential operators
![](https://www.scirp.org/html/10-7401042\12daa41b-96b7-4816-b73f-c4257600f7a3.jpg)
![](https://www.scirp.org/html/10-7401042\19ba7d0c-89e7-44a7-87e0-4f2d4d8e391f.jpg)
In this case the basis elements is
which is of weight 2 therefore the normal form is
![](https://www.scirp.org/html/10-7401042\1dcd88d3-fb20-4953-a8c1-97fed3c64e1c.jpg)
![](https://www.scirp.org/html/10-7401042\a6bfe784-ca8f-4e3f-89d8-5a2ae75112d6.jpg)
We compute:
![](https://www.scirp.org/html/10-7401042\32259856-1086-4e08-a059-d48908d2b57c.jpg)
![](https://www.scirp.org/html/10-7401042\da7c3796-c533-4d55-b8be-97d04314f74a.jpg)
![](https://www.scirp.org/html/10-7401042\fc232695-16e6-417b-b36e-cea8574cf703.jpg)
The differential equations in
normal form are:
![](https://www.scirp.org/html/10-7401042\29c03215-2506-4772-82b6-f69eea18ff3a.jpg)
![](https://www.scirp.org/html/10-7401042\0a88d4e5-229f-4ce3-8c2d-183e4665230a.jpg)
![](https://www.scirp.org/html/10-7401042\e7ec6a4f-4862-4cd4-9639-465401c23090.jpg)
The normal form upto quadratic term is:
![](https://www.scirp.org/html/10-7401042\8d8ae21b-a5df-4f4b-b16d-a1af630d4a73.jpg)
Remark: The normal form of a dynamical systems is a powerful tool in the study of stability and bifurcations analysis. From the practical point of view, only the normal form with perturbation (bifurcation) parameters is useful in analyzing physical or engineering problems. In this paper the computation of the normal form has been mainly restricted to systems which do not contain perturbation parameters by setting the parameters to zero to obtain the simplified normal form. Having found the normal form of the reduced system we shall then add unfolding terms to get a parametric normal form for bifurcation analysis.
NOTES