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# . are, in fact, polynomial functions of ‘previous’ coefficients

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As for the third step, if the desired particular solution of the cut system (11) exists, we can always complete the above solution up to a particular solution of the entire system (1) in a form of series. To completely construct series (10), we should perform the following recurrent procedure. We consider only the case with the sign ‘’. Let us first do the exponential change of time logt. Then by substituting series (10) into system of equations (1), we obtain an infinite chain of linear differential equations with constant coefficients and polynomial right-hand sides.
 (15)

Here k, k1,2,... are, in fact, polynomial functions of ‘previous’ coefficients x0,...,xk-1, and KkkIK, where K is the so-called Kovalevsky matrix
 (16)

System (15) always has a polynomial particular solution. Hence, series (10) can be completely constructed.
As for convergence of series (10), in general, we can only affirm that there is always a particular solution of (1), infinitely smooth on an interval [T,), for which (10) is an asymptotic expansion [3,4,6]. The proof of the above fact is based on a version of the abstract implicit function theorem. Nevertheless, if the vector field v(x) is analytic and if 1 is the only eigen value of the Kovalevsky matrix of the type k,k1,2,... the above series converges on an interval [T,) and we can explicitly construct an infinitely sheeted Riemann surface on which the corresponding particular solution of (1) is holomorphic [6]. Precisely speaking, that means that the desired solution can be obtained in the form of series
 , (17)

which converges on a small complex disk , where ss(t) is a function inverse to the function , is a certain real parameter.
It is worth noticing that 1 is always an eigen value of the Kovalevsky matrix K in accordance with the lemma from [7].
So, the following result holds.
Theorem I. If the cut system (11) has a particular solution (9), then the entire system (1) has a particular solution x(t)0 as t or t for which (10) is an asymptotic expansion.
Let us consider now several examples.
Example V. We can partially prove the hypothesis mentioned in Section 1.
Theorem II. Let the dimension of the phase space n of system (1) be odd and v(x) be an analytic vector field with a nilpotent linear part possessing an invariant measure with a smooth density. If there exists a quasi-homogeneous structure such that the origin x0 is the only critical point of the cut vector field vm(x), the equilibrium position x0 of the entire system (1) is unstable both ‘in the future’ and ‘in the past’.
Using lemma I, we can simply prove the above theorem. Indeed, according to the lemma I, there exists either ‘positive’ or ‘negative’ eigen vector of problem (12). Consequently, there is a particular solution x(t) of the entire system (1) going to the origin x0 either as t or as t. That means that x0 is unstable either ‘in the future’ or ‘in the past’. But since system (1) possesses an invariant measure, instability ‘in the future’ results in instability ‘in the past’ and vice versa instability ‘in the past’ results in instability ‘in the future’.
Example VI. Let us now consider a gradient system (5) with a harmonic potential. The following statement holds.
Theorem III. The equilibrium position x0 of every gradient system with a harmonic non-constant potential (x) is unstable both ‘in the future’ and ‘in the past’.
Proof. Let us consider the Maclaurin expansion of the potential (x)2(x).... If that expansion starts at a second order form, the statement can be proved only by means of the linear approximation, as was shown in Section 1. That is why we confine ourselves only to the case when the Maclaurin expansion starts at a form of order m1,m2. Then the above expansion reads (x)m+1(x).... As we have already shown, the cut system (13) possesses two rectilinear solutions going to zero as t and as t which can be completed to particular solutions of the entire system (5) with the same asymptotic properties which leads to both instabilities.
Example VII. A particular case of the inversion of the Lagrange-Dirichlet theorem on stability of an equilibrium position of a mechanical system and the Earnshaw theorem on instability of a point charge in an electrostatic field.
Let us consider a mechanical system described by a Hamiltonian system of differential equations
 , (18)

where H(q,p) is an analytic function of the kind . Here K(q) is a positive definite symmetric matrix of coefficients of the kinetic energy of the system (without loss of generality we can assume that K(0)I), U(q) is the potential energy of the system. If q0 is a critical point of the potential energy, qp0 is an equilibrium position of the system. The Lagrange-Dirichlet theorem [1] states that the above equilibrium position is stable if q0 provides U(q) with a strict minimum. The following question arises. Is the trivial equilibrium unstable if U(q) does not have a minimum at the point q0? The positive answer was quite recently given by V.Palamodov [8] who completely proved the above statement. We give a simple proof of a weaker theorem.
Theorem IV. Let us consider the Maclaurin expansion of the potential energy at the equilibrium position U(q)Um+1(q)...m1. Then if the first non-trivial form Um+1(q) does not have a minimum, the equilibrium position qp0 is unstable [4,9].
It is worthy to notice that via the time-reversibility of equations (18) the above instability is ‘two-sided’.
Proof. The case m1 can be simply studied by means of the linear approximation. Let the Maclaurin expansion of U(q) start at order greater than two. Then introducing a quasi-homogeneous structure by means of the following positive definite diagonal matrix , we can obtain a cut system
 , (19)

for the entire system (18).
If conditions of theorem IV hold, the quasi-homogeneous cut system (19) admits a particular solution , where the vector is parallel to a unit vector e providing Um+1(q) with a minimum a,a0 on the unit sphere Sn-1. The length of the vector c can be calculated as . Therefore, the entire system (18) has a particular solution (q(t),p(t))(0,0) as t. This leads to instability ‘in the past’ and consequently ‘in the future’.
The equations of motion of a point charge in an electrostatic field also have the form of (18) where n3,K(q)I, and U(q) is a harmonic function. All the forms in the expansion of the potential U(q) into the Maclaurin series are alternating. Hence, to prove instability (the Earnshaw theorem), we can refer to theorem IV.
Example VIII. At the end of this Section we consider an extension of the Lyapunov criterion on instability of a trivial equilibrium position of a system of differential equations with a nilpotent linear part [3,4]. Lyapunov studied only the case n2. We consider the case of a higher co-dimension when n2.
Let us write the system of equations for that critical case as follows
 , (20)

where the dots present non-linear terms for which the monomial in the last equation is singled out.
Those dots show the manner how system (20) can be truncated. In fact, the cut system is quasi-homogeneous with respect to the diagonal matrix . That cut system has a particular solution if . Here . That solution can be completed to a particular solution of the entire system (20) entering the equilibrium position as t. This means instability ‘in the past’. Analogously, a solution of (20) which enters the equilibrium position as t can also be constructed. And instability ‘in the future’ also takes place. This holds if n2.
If, nevertheless, a0, the initial system of equations should be rewritten as follows
 (21)

If we neglect all the non-linear terms denoted by dots, we obtain a quasi-homogeneous system of equations associated with the matrix (here we assume that n3). That system has a particular solution if , , . As previously, that solution can be built up to a particular solution of the entire system (21) which goes to the origin as t. In this case the origin is unstable ‘in the past’. On the other hand, it is also possible to construct a particular solution of system (21) entering the origin as t which results in instability ‘in the future’.

3o. Let us pass to the case when the characteristic equation (2) has non-zero roots. We mainly confine ourselves to the case of purely imaginary roots of equation (2). The matrix  of the linear approximation to system (1) can be expressed as a sum DJ, where D is diagonalizable and J is nilpotent. Then (1) reads
 , (22)

where u(x)Jx..., dots represent all the non-linear terms.
The next step is to transform system (22) into a normal form. Let us carry out a formal power transformation , where Yp(y) are homogeneous vector forms of order . After that system (21) gets the form
 , (23)

where w(y)Jy... .
Let us remind of the following definition. System (23) is said to be written in Poincare’s normal form if (see, for example, [10]). Thus, we can use the following construction. After the following linear non-autonomous bounded invertible transformation system (23) becomes
 , (24)

where formal vector field w(z) has a nilpotent linear part.
Thus, we find ourselves in the situation of the previous section and can perform the whole scheme described above. But though, in general, the eigen vector problem (12) is solvable for diagonal matrices G which the Newton polyhedron method provides us with, in concrete critical cases it is not so and we have to use a more refined technique.
Let there exist n0n independent linear semi-simple fields of symmetry for system (24) Djz,j1,...,n0 with diagonalizable matrices Dj. Let also G be a diagonal matrix defining a quasi-homogeneous structure obtained by means the Newton polyhedron technique such that it commutes with all matrices of linear fields of symmetry (GDjDjG,j1,...,n0). Let the system of equations
 , (25)

be a quasi-homogeneous cut system for (24).
The following statement holds.
Lemma III [4]. The set of matrices , where is a set of arbitrary real parameters, defines a quasi-homogeneous structure under which (25) is quasi-homogeneous and (24) is semiquasi-homogeneous and (25) is the corresponding cut system.
For instance, we can use the matrix as a matrix of the desired structure. Using that approach, we obtain not only solutions of the cut system of the ‘ray’ type but also solutions looking like curled rays.
Opposite to (9) particular solutions of (25) may have the following form in general
 (26)

The further construction is the same. Hence, the desired formal particular solution of (23) can be obtained in a form of series

 , (27)

However, we should bear in mind that in general a normalizing transformation diverges and since the normalized system (23) is only a formal system of differential equations, series (27) are only formal series. But using a partial normalizing transformation and a kind of the implicit function theorem technique we can prove that a partial sum of (27) of high order approximates a real smooth solution of a partially normalized system which goes to the equilibrium position as t or t.

The following general result takes place.
Theorem V. If the cut system (25) has a particular solution (26), then the entire system (22) possesses a particular solution x(t)0 as t or t.
Example IX. Non-exponential asymptotic solutions of general systems of differential equations with additional 1:1 frequency resonance and non-simple elementary divisors.
Let us consider a 4D system of differential equations for which the characteristic equation has purely imaginary roots with a non-diagonalizable matrix of the first approximation.
Stability of the trivial equilibrium position of the system under consideration was investigated in [11].
Let y(y1,y2,y3,y4) be a phase vector of the normalized system. The diagonalizable part of the matrix of the linear approximation may be represented as
 (28)

The cut system analogous to (25) reads [11]
 , , (29)

a,b are real parameters of the system. Here the diagonal matrix is used to perform the necessary quasi-homogeneous truncation.
Obviously, the matrix G commutes with the matrix D and we can apply the procedure described above. The mentioned algorithm makes us to search for particular solutions of system (29) in the following form
 (30)

Those solutions (30) form a one-parameter family , .
Here is the parameter of the above family and the magnitudes , have to be determined. They satisfy the following algebraic system of equations
 (31)

Conditions of solvability of system (31) can be presented in a complex form which coincides with conditions to instability found in [11]
 (32)

If inequality (32) holds, the initial system of equations possesses a one-parameter family of solutions which goes to the equilibrium position as t. We can also prove the existence of solutions going to the equilibrium position as t.

It is worth pointing out some obstacles we meet while investigating the case when some of roots of the characteristic equations (2) do not lie on the imaginary axis. In this case we can reduce the system under consideration onto a center manifold. After that we can apply the procedure described above. But the problem is that the center manifold may have only finite order of smoothness. That is why, series like (27) remain only formal series and the question of persistence of non-exponential asymptotic solutions requires a more careful analysis.
4o. Now we stop at the situation which can be characterized as singular. This means that we can construct series like (10) but they diverge in most cases even for analytic right-hand sides of the system under consideration. A classical example has been already given by Euler.
Example X. Let us consider the following 2D system of differential equations.

 (33)

System (33) has a particular solution as which means, of course, instability. The function x1(t) can be developed into power series with respect to inverse powers of t. That series takes the form and, of course, diverges.

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