ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1967)
ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)
П. Kołakowski (Warszawa)
Qualitative study of a system of three polynomial differential equations
1. Introduction. In the present paper we shall study the following system of three nonlinear ordinary differential equations:
П
(1) w
= 2= -P .M =
pI(x) +pI+'(x) + ...+ p ” и ,
i+j+k=l
s = 1, 2, 3, where als?k = const, x — (aq, x 2, x 3), l is a fixed integer such that 2 < l < n, and Р “ (ж) (w& = h Z+l, • • •, n) are homogeneous polynomials of order m. We shall study the projections of integrals of these equations from the space x, t on the space x, or the so-called trajectories of the system (1). We shall assume that the point x = 0 is an isolated singular point, that is, there exists a neighborhood U of the origin such that for every О Ф x e l l we have P s(x) Ф 0, P s(0) = 0 (s = 1, 2, 3). We shall dis
cuss in detail two methods of studying system (1): one method is based on the study of the behavior of the trajectory as q{x, 0) -> oo {q(x, 0)
= V'х \ф х \ф х \) for the purpose of the study of point x — 0, the second method is the “splitting” method. Quite frequently these two methods enable us to learn about the behavior of the trajectory in the whole space x. The detailed description of both methods for the two-dimensional ease (x — (хг, x 2)) can be found in paper [2]. We shall try to generalize some of the results of paper [2] and give several examples.
2. The study of the behavior of trajectories for q(x, 0) -> oo. Homo
geneous equations. At first we shall transform system (1) with several successive transformations T 1} T 2, T 3. The first transformation will be a linear orthogonal transformation Т г of the euclidean space P 3. The transformation T 2 will be defined by the formulas (see [3]):
The transformation T 3 corresponds to introducing new time coordinate:
tx = T 3(t). The aim of this transformation is to obtain equations with right-hand sides being polynomials in ux, u 2, u3. In onr case T 3 can be defined as dtx = u1~ndt (for the sake of convenience in the sequel we shall write t instead of tx). Clearly, for even n, the motion along the tra
jectory for u x < 0 changes its direction. To avoid new notations we shall assume that the transformation T x leads to equations (1) with x changed to y. Applying transformation T 3 (T2) to such equations we get for и Ф 0 the following system
(3)
dux dt du2
dt du3
dt
i + f + k = l
ijk n —i —j —k j к i + j + k = lФ-J
a2 Ux if .) I/ ■
i + j + k = l
\ ' ijk n—i—j—к i к i \ ' iy'As .n—i—j—k j k / J a\ Щ
i-\-jĄ-k=l
, J К \ V Ijn, III-
ЩЩ+ > % Щ i+3 —j— /с = 2
^ 3 •
We make the additional convention that the above formulas hold also for u x — 0. It can be seen easily that the plane u x = 0 is an integral surface. Application of various transformations of the type T x, trans
formation T 3 (T2) and extending of definition of system (3) for u x — 0 allows us to treat system (1) as a system of equations in projective space.
Thus, for instance, if T x is the identical transformation, then the exten
sion of definition of system (3) for ux = 0 corresponds to a continuous extension of the definition of (1) to all points of the plane at infinity except for the points of the line at infinity of the plane x x = 0. In prac
tice, in order to extend the definition of (1) to all points of the plane at infinity it is enough to apply only three transformations of the type T x, the first being the identity, and the remaining two consisting of the change of coordinate axes.
Definition. The singular points of system (3) will be called gen
eralized singular points of system (1).
Due to the fact that the transformations Ti (i = 1, 2, 3) do not change the topological structure of the partition of space E 3 into tra
jectories we have
Theorem 1. Every singular point of system (1) is a generalized sin
gular point of this system.
The generalized singular points with coordinates и = (0, a, /3) = (a, /?) correspond to the singularities of the behavior of trajectories for
q(x, 0) ->oo.
Study of a system of three polynomial equations 1 5 9
In the sequel, in order to obtain clearer geometrical illustration of trajectories of system (1) we shall also apply transformation T4:
oc —-- — . —x V i +q4oo, 0)
which transforms the space E 3 into the unit ball. In view of Theorem 1 the singular points will be transformed into points from the interior of the unit ball, and the generalized singular points with coordinates и = (a, /5) will be transformed into the points of the unit sphere; to each generalized singular point there correspond two points of this sphere.
Let the point u0 = (a, /?) be a singular point of the system (3). Intro
ducing the variable n\
U = U -j- Uq
we obtain the equations П
- £ a. f u P ' - i- t- k(u2 + a f ( u s+ f ) f , 1 к = ^
n
n
+ £ « Р +
iĄ-jĄ-k—l
n
ctikuf г J' k (и2-\-аУ (w3+ P)k + i+}+k=l
n
+ a3ku f г 7 k (u zf-a f (m3+ fl)k, i+1+k=l
dux dt du2
dt
(4)
du3 dt
where the free terms
A(a, ft) Л £ ”( 1, a, ft)-aP 1( 1, a, ft), В (a, ft) й Р Ц 1, a, a, ft) are the polynomials in a and /5 of the order at most n-\-1.
Clearly, the real roots of the system of equations
(5)
А(щ fi) = 0, B(<x, P) = 0 are the coordinates of the singular points.
The roots of the characteristic equation of system (4) can be written in the following way
P),
Ką = Л а-\-Ва + ( - l ) g / x + p ; \ 2 a ' a;
1 2 / 1 B'a Ą q — 2, 3.
R em ark. As we have already pointed out, for practical purposes it is sufficient to apply only three transformations of type T x, each of them consisting of the change of the coordinate axes. Thus, for the de
termining of the coordinates of all singular points which possess corre
sponding points on the unit sphere in the space x, it is enough to consider only three systems of equations
As{aS) Pa) = 0)
B s(as, P s ) = 0, s = I, 2 ,3 , where
Ax{ax, Pi) = ^-(a, P), P i K , Pi) = В {a, /5);
A 2{a2, p2) = Рз(/52, 1, a2) — a 2P%{P2,1, a 2),
Р г ( а2? Pf) = В i {Pzi 1 , a2) P2 P 2 (p2, 1 , a 2) 5 ( а з? Рз) = В i ( a 3 , P3} 1 ) a 3 P 3 ( a 3 , /?3 , 1 ) , Р з ( « з > ^ з ) == B 2(a3, рз , 1 ) p3P 3(a3, рз, 1 ) .
The polynomials A 8, B s (s — 1, 2, 3) will be called characteristics of infin
ities (we preserve the terminology of [2]).
It is well-known that to each system (1) there corresponds a parti
tion of the space x , or — which amounts to the same if we consider trans
formation 1 \ — a partition of the open unit ball into trajectories. More
over, due to the extension of the definition of system (1) for points at infinity, to each system (1) there corresponds a partition into trajectories of the closed unit ball. The unit ball filled with trajectories correspond
ing to system (1) will be called the Poincare image of system (1).
Let us consider an isolated singular point ( a 0 , /30) such that лх ( a 0, po) Ф 0.
Besides, let us assume that
1) if imA2 = —imA3 Ф 0, then reA2 = reA3 Ф 0, 2) if imA2 = im A3 = 0, then 12 Ф 0, or A3 Ф 0.
Thus, the point ( a 0 , po) can be (see [4]) a node, focus, saddle, saddle- focus, saddle-node, “saddle-focus’’-focus, or a singular point of type “C”.
The behavior of trajectories in a sufficiently small neighborhood of each of the above-mentioned singular point is, by definition, the following (see [3], [4]):
Study of a system of three polynomial equations 1 6 1
N ode. All trajectories enter the point (a0, /30) as t -> -f- oo (t — oo) with well-defined tangents.
F ocu s. All trajectories enter the point (a0, /30) as t -> + oo (t -> — oo).
Two of them enter with a well-defined common tangent, and all other trajectories are spiral lines.
Sad dle. A certain surface, called the separating surface, crosses the point (a0, (30). All trajectories situated on this surface enter the point (a0, /?„) as t -> + oo (£-?- — oo) with well-defined tangents. Two trajec
tories situated on two opposite sides of this surface (the so called sepa- ratrices) enter the point (a0, /?0) as t -> — oo (t -> + oo) with the common tangent, and all the remaining trajectories do not enter the point (a0, /?0) at all.
S a d d le -fo c u s. A certain surface, called the separating surface, crosses the point (a0, j30). All trajectories situated on this surface enter the point (a0, /?0) for t + oo (t -> — oo) as spiral lines. Two trajecto
ries, situated on two opposite sides of this surface (the so called separa- trices) enter the point (a0, /30) as t-> — oo (t -> + oo) with a well-defined common tangent, while the remaining trajectories lie at a positive distant from the point (a0, /30).
S a d d le-n o d e. An integral surface F crosses the point (a0, /?„). The surface F has the property that all trajectories situated on it enter the point (a0, /?0) as t + oo (t -> — oo). All trajectories on one side of F enter (a0, /?0) as t -> + oo (t -> — oo). One trajectory on the other side of F enters the point (a0, /?0) as t -» — oo (^ -> + oo) while all the re
maining trajectories lie at a positive distance from (a0, /30). In other words, on one side of F we have a node domain, while on the other side we have a saddle domain.
“S a d d le - f o c u s ” -fo c u s. An integral surface crosses the point (a0, /?„). This surface is such that all trajectories on this surface are spiral lines, while on one side of this surface we have a node domain, and on the other side we have a saddle domain.
S in g u la r p o in t of th e ty p e “C”. There are two integral half
surfaces such that the trajectories lying on one of them enter the point (a0, /?0) as t -> + oo while the trajectories on the other half-surface enter (a0, /50) for t -> — oo. All remaining trajectories lie at a finite distance from (a0, /30).
Definition. A node, focus, saddle, saddle-focus, saddle-node, “saddle- focus’’-focus, or a singular point of the type “C” with coordinates (a0, /30) will be called A +-limiting if a point moving along a trajectory у in the PoincaiA image approaches the point (a0, /?0) as t -> + oo. If it approaches (a0, /?0) as t — oo, the point (a0, /30) will be called A~-limiting.
Naturally, such a trajectory у always exists.
11 — Prace Matematyczne XI (1967)
Theorem 2. A singular point (a0, /30) is A +-limiting (A~-limiting) if the surface P™{ 1, a, /5) lies above {below) the plane a, (5 in the neighborhood of the point (a0, /30). (See fig. 1.)
P roof. We notice that the right-hand side of the first equation (4) starts with the term Xx u\ all other terms are of the higher order. Thus, for sufficiently small й we have
sgn^i = sgn — ,du
therefore for > 0 the curve у enters (from the side и > 0) the point (a0, j80) as t — oo, while for Хг < 0 it enters this point as t -> + oo, which proves Theorem 2.
We shall study system of differential equations of the type (6) — - = P “(a?), cloc s = 1 ,2 ,3 , x = {xu x 2, xf)
dt
for which the point x = 0 is, by definition, the only singular point. Ob
viously the rays
(7) — = «o, /у» d/j — =/у»d/j are the solutions of this system.
Let us consider a surface of Poincare image of system (6) which can be spht into qualitatively identical cells, each of them filled with trajec
tories bounded by the sides of the curvilinear triangle Ox 0 2 0 3 that is trajectories joining the singular points — vertices of this triangle (fig. 2).
Such a partition can be performed, for instance if (see [1]):
1) For the generalized singular point (a0, |80) we have A2 and A3 real and non-vanishing.
Study of a system of three polynomial equations 1 6 3
2) The surface of Ротсагё image does not contain any limiting cycles.
Let В denote the tetrahedron with base Ox 0 2 0 3 (fig. 2) and the vertex at the point x — 0, and let Op, 0 Q be two of the vertices Ox, 0 2, 0 3.
We assume that each edge of the tetrahedron В is the image of one of the rays (7), and each of its sides is the image of an integral surface of system (6).
De f in it io n. The interior of tetrahedron D filled with trajectories will be called an elliptical region if all trajectories from interior of В enter the point x = 0 for t oo and for t -> — oo; it will be called a hyper
bolical region if all trajectories from the interior of В enter the point Op as t + oo (t -> — oo) and enter the point Oq as t -> — oo (t -f oo).
Finally, it will be called a parabolical region if all trajectories from the interior of В enter the point x = 0 as t -> -f oo (t -> — oo) and enter the point Op as t —^ — oo (t -> + oo).
Theorem 3. I f the points Ox, 0 2, 0 3 are saddle points, then the in
terior of the tetrahedron В is an elliptical region. I f the point Ox is a saddle and points 0 2, 0 3 are nodes, then the interior of В is a hyperbolical region.
Finally, if Ox, 0 2 are saddle points and 0 3 is a node then the interior of В is a parabolical region.
The proof follows directly from the analogous theorem for two- dimensional case (see [2], Theorem 2) and from invariance of homo
geneous equations under homothetic with center at the origin.
In our case we use only the topological structure of the partition into trajectories of the neighborhood of the cell 0 X 0 2 0 3. Thus, if the replacing of points Ox, 0 2, 0 3 by any three of nodes, saddles, saddle- nodes, or singular points of type “C” does not lead to the change in topological structure of the partition of the neighborhood of the cell Oi 0 2 0 3 into trajectories, then the partition of the tetrahedron В will remain the same.
3. Applications. We shall now consider several examples.
Ex a m ple 1. Let us build the Poincard image for following system
(8)
dxx
~ d f dx 2 dt dx a
dt
a x\f-hx\x2,
bx \ f - k x \ x x,
cxl, where a < 0, b > 0, c < 0.
The characteristics of infinities are:
А г(cq, р г ) = Ъа\ — а а г ,
• ® i ( a i? Pi) = GPi a Px ^ a i Pi1
- ^2( ^2? PY) == ^® 2 ^ ^ 2 k(l2P2^
-B2(ct2> ^2) = ^ 2 ^2 5
^ з(аз? ^з)
^з(«з, /^з)
^Щ-\~каврз ^аз?
Ъ(31+1сагр1—срз.
The coordinates of all generalized singular points which have the corresponding points in the sphere of Poincare image are given by real solutions of the system
I CGq ^^2 kd2P2 — 0 ? I 0t3 = 0 7
0; /J8 = 0; U = 0.
i&a?—acq = 0, cfi\ — aPx—kaxPx
The table below gives the values of roots of particular equations (9) together with the roots of the corresponding characteristic equations.
t a b l e 1
Roots of equations (9)
(«1» Pi) = (a2 ’ P%) — («3 > Pa) = (0, 0) (0, Va/c) (0, —Yajc) (0, 0) (0, 0)
h — a — a — a - b — c
A2 — a — a — a - b — c
A3 — a 2a 2 a - b — c
The following surfaces: the integral surface aq = 0, x 2 = 0, x 3 — 0, the separating surface of the saddle (cq, fix) = ’(0,]/ajc) which crosses
“end-points” of the lines
х г = 0, [ x 2 = 0, x 3 = 0, ( x 3 = Va/cxx
and the separating surface of the saddle (a1,/51) = (0, —Va/c) which crosses “end-points” of the lines
IXx — 0, a?3 = 0,
split the space x into 16 parts. To each of these parts there corresponds (inside the unit ball) a tetrahedron with two vertices in nodes, one vertex in a saddle and one at the point x — 0. Thus, by Theorem 3 we have a hyperbolical region, therefore the point x = 0 represents a saddle point
x 2 = 0, x 3 = —VajcXx
Study of a system of three ‘polynomial equations 1 6 5
The separatrices of the saddle x = 0 are the rays of the #2-axis, for they enter x = 0 as t -> — oo. The image of the integral surface x 2 — 0 is the following: The “end-points” of the axis x 3 and x x are plane A “-nodes (that is A“ -limiting nodes), while the “end-points” of lines
x 3 = ]/ ajcx1, x 3 = —1/ a fcx-y
are plane A -saddles. Thus the point (0, 0) of the plane x 2 = 0 is a node (figs. 3, 4).
A+-node
Exam ple 2. Let as consider the system
(10)
dxx
dt ax\—bxxx 3, dx2
dt —ах1х2-\-Ьх2х1, dx a
dt cx\Ą-bx337 where a < 0, b > 0, c > 0.
The characteristics of infinities are
•^•i(ai?/^i) — 2ax(d bf$ i), -Bi(ai> &) = ca?-j-2&0j—
-^•2(^2? /^2) = c-\-da2^27 B 2(a2, fi2) = 2(afi2 ba2fi2);
-^з(аз? /^3) = «a3 2&a3 Ca.3fi3f B 3(a3, p 3) = (шА($3 ^ з'
The coordinates of all generalized singular points which have corre
sponding points in the sphere of Poincare image are given by real solu
tions of the systems
j —2 ax(a — b(5\) = 0, I CGtj-j~2&y5j— x — 0j
*3
I c — 0, a3 = 0, /^2 — 0; /53 = 0.
Using the criteria which allow us to determine the type of singular point (see [4]) we easily obtain that the point (a8, 03) = (0, 0) is a saddle- node. The point {ax, ftj) is a node. The figures below illustrate the images of integral surfaces x x = 0 (fig. 5) and = 0 (fig. 6).
Study of a system of three ‘polynomial equations 1 6 7
The corresponding table is
T A B L E 2
Solutions of systems (11)
(<h>Pi) = (°з> /^3) ~
(0, 0) (0, 0)
h — a - b
^2 — 2 a - 2 b
^3 — a 0
The planes х г = 0 and x 2 — 0 divide the space x into four parts.
In the unit ball to each of these parts there correspond a tetrahedron with vertices in one node, two saddle-nodes and at x = 0. The trajecto
ries filling the interior of an arbitrary tetrahedron are such as if vertices of this tetrahedron lying on the surface of unit ball were two nodes and one saddle. In view of Theorem 3 we see that the point x = 0 is simply a saddle (fig. 7).
A+-saddle-node
The separating surface of this saddle is the plane = 0 (fig. 5) and the separatrices are the rays of the avaxis. The separatrices enter the point x = 0 as t -> — oo (fig. 6).
4. The method of “ splitting” . Let the point x = 0 be an isolated singular point of the system (1). To study this point we may proceed as follows: By applying the inversion $0:
X — ---и
e
2(o ,
u)we transform the neighborhood of the point x = 0 into the neighborhood of the plane at infinity. As in section 2, it will be convenient to introduce the time coordinate t1 = $ х(<) defined by the formula
dt1 = [{?2(0, u)]~1ldt
(for the sake of convenience we shall write t instead of tx). The next step will consist of the study of topological structure of the partition into trajectories of the surface and its neighborhood of the Poincare image of the system obtained from (1) after applying transformation $ i($ 0).
The third step will consist of symmetric reflection of the neighborhood of the above mentioned Poincare image with respect to its surface. Then the limiting passage of this surface to x = 0 will give us the qualitative picture of the trajectories of system (1) in the neighborhood of point x = 0. The method described above is called “splitting” method (see [2]).
The advantage of this method lies in the fact that it allows us to replace the study of generally complicated singular point x = 0 by the study of simpler generalized singular points determined by directions of well- defined tangents of trajectories entering the point x = 0.
Thus, let us consider the system (1). The transformation $ x($0) yields :
dux 2 2 „
TT~ = ( ^2”Ь^З ^ l ) $ l 21U1U2Q2
at
(12) ^ = —2*^ 20! + (m; +m5—^ ) 0 2—2w2ws0 3, dt
du, 9 9 9
—— = 2u1u 3Qi ~ 2 u 2u 3Q2~\~ —Щ)Яз>
dt where
Q„ = P \(u)(u\+ ul + u\)n- l+ P ,*1(u)(u\+ul + u \ f - ,~l + ... + P ”(m), Let
8 = 1, 2, 3.
A*(as, ($s) > B t(as, s = 1, 2, 3)
be the characteristics of infinities of the system (12) and let As(as, /?e), B s{as, ps) (в = 1, 2, 3) be the characteristics of the infinities of the system
= P ls(x) {8 = 1 ,2 ,3 ).
(13) dt
Study of a system of three polynomial equations 1 6 9
Let us denote by A* (As) (s = 1, 2, 3) the roots of characteristic equa
tions of system (12) ((13)) with respect to those generalized points 0*t {Om) of the system (12) ((13)) which have corresponding points on the surface of Poincare image of this system.
Definitio n. The characteristic equations of the system (12) with respect to the singular points 0*n will be called the characteristic equations of (1) with respect to point x — 0 (see [2]).
We notice easily that the roots of the characteristic equations of system (1) with respect to x = 0, he. the numbers A*, satisfy the rela
tions
К = - ( i + c i + p J f-'+ U ,, (14)
К = ( i + o i + $ r ' +4 S = 2, 3.
The analogous formulas for the characteristics of infinities are the following
A t(as,/3S) = (1 + as+/5s)n 1+1-As(aSi (3S) i (15)
B t ( a „
ft) = (l+ai+ft;f-!+IĄ,(a8, ft),
s = 1 , 2 , 3 .Thus, we have a simple method of finding the coordinates of points Om and corresponding numbers A*. It is worth mentioning, that the above definition is very helpful in the case when the right-hand sides of (1) do not contain the linear terms, since in that case the roots of char
acteristic equations
Р щ ( 0 )-A
Р щ ( 0 )
0)
P ' i x 2 (0)
> i* (0 -A )
■Pir2(0)
Р щ ( 0)
Ргж3(0) Рщ { 0)-A
= 0
are equal to zero, hence their discussion yields nothing.
Prom the description of the splitting method and from equalities (14) we get
Theorem 4. To every A +-limiting {A~-limiting) generalized singular point of the system (1) there corresponds an A~-limiting (A +-limiting) gen
eralized singular point of the system (12).
Obviously, the assertion of Theorem 4 does not require any assump
tions about the polynomials P s(x). In the sequal we shall study some examples.
5. Applications of the “ splitting” method.
Example 3. Suppose we want to study the trajectories of the following system of equations
dx
——- = ax\Jr k x,x2-\-x1{P\{x)JrP \(x)Jr ... + P ” ^ж)), (16) clcc■ ■■ = bx2-\-JcX1xl~\-X2^Pl(x) ~{-Р2(х) H“ ••• ~\~P%. 1 (ж)) j
at
= схз~1г00з{Р1(х )р Р з ( х )~1г ••• +P ? 1 (л?))?
where a < 0, b > 0, c < 0, in the neighborhood of the point x = 0.
The following table gives the data concerning this system.
T A B L E 3 Coordinates
of points 0*i
(«1» h) =
(0,0) (0 , ] / a/ c) (0, —Vajc)
(«2» ft2) =
(0, 0)
(a3> Рз) —
(0, 0)
Af Я2*
Я3*
a 11 + — c
2a 1 +
(■ +7)' a 1 +
2a 1 +
b
~ b - b
Thus, all points Ofn are saddle points. Simple geometrical considera
tions show that the point x = 0 is a saddle point. The particular steps of the “splitting” method are applied only to the neighborhood of the point (0, 0) of the integral surface x 2 = 0; they are presented on fig. 8.
The separating surface of the saddle x = 0 is the plane x 2 = 0 with the node at the point (0, 0). The'separatrices of the saddle x — 0 are the rays of the a?2-axis, and they enter the point x — 0 as t -> — 00.
Example 4. We shall study the behavior of the trajectories of the system
(Hoc — ax\-\-x1{P\(x)JrP\{x)-\r ... 4-Pi 1(ж))?
(17) --A = bai+x2(PHx)+Pl(w)+ ... + Р Г Ч *)), cloc
= CXl P X3[Pl{X) Jr P l ( X) Jr ••• + P 3 1{x ))l
where a > 0, b > 0, c > 0 in the neighborhood of the point x = 0.
Coordinates of pointsО
Study of a system of three polynomial equations 171
The corresponding table is of the form
t a b l e t
*§
(«1» 0i) =
( 0 , 0 ) (ajb, ajc) (0, o/o)
h*
2 \ я - 1
e ' 1+ P + *
, 2 л .2 \ м- 1
“|1 + б5+ ^ / n2 а2\п~ 1
(1 +
6i +W
2\ W- 1
а\ 1 + 2\П-1 -O l f
, a‘
йц + -2 \ П— 1
(ajb, 0 ) ( 0 , 0 )
(a2» —
a 1 +
a2\n-i W
a 2 \ » - l
'*'I + W
a2\“-l
»|1 +'p) - 6
(b/c, 0 )
/ b2^ - 1
ni+^)
/ & 2 \ n - l
/ & 2 \ n - l (0,0)
We see easily that all points 0*n except the point (au /Зх) = {a/b, ajc) are saddle points, while the point (cinfix) — (a/b, ajc) is a node point.
The successive steps of the “splitting” method applied to the neighbor-
hood of the point (0, 0) of each of the integral surfaces x x = 0, x 2 = 0, x 3 — 0 give us (fig. 9):
The data in Table 4 and corresponding geometrical considerations lead to the following conclusion: all trajectories in the sufficiently small neighborhood of the point x — 0 from the side x x > 0 , x 2 > 0, x 3 > 0 (xx < 0 , x 2 < 0, x 3 < 0) enter this point as t -> — oo (t, -> -f- oo). All other trajectories, which do not lie on the integral planes x 1 = 0, x 2 = 0, x 3 = 0 do not enter the point x — 0 at all.
Example 5. Let us consider the behavior of the trajectories of the system
^ — — a1x\-\-a2x 1x \ JrX1{P\{x)-\-I>\{x)Jr ... + P i( # ) ) ? dx
(18) —— = Ъхх\агЪ2х\х2-\-х2{Р\{х)-\-Р\{х)-\- ... + P 2 (ж)), at
-ТГ = c1x l JrC2x lx 3+c3x2doc 1x 3+ x 3(Pl(x) + P t { x ) ... +Рз(я?)), at
whereu1 > 0 , Ъг > 0, cx> 0 , с2 — Ъх> 0, а2—Ъх< 0, b2—ax> 0, c3—ax> 0, in the neighborhood of the point x = 0.
Study of a system of three polynomial equations 1 7 3
The table, analogous to the previous tables, is:
TABLE 5
Coordinates
of points От к , ft) = (0, 0) (а2, &) = (0, 0) (а3,|?з) = (0, 0)
Ai* аг h ci
А2* \ — c2- b i ~ С1
А? cz~~ai а2 ~ С1
From this table, and from the conditions imposed on the parametres a-i, a2, 61, &2, Ci, c2, c3 it can be seen that the points (ax, /ЗД = (0, 0), (a2,/?2) = (0,0), (a3, / y = (0,0) are respectively: node, saddle, saddle.
Thus the point x = 0 is a node, and the trajectories entering the point x = 0 enter as t — oo.
I wish to express my thanks to Dr J. Bratkowski for the valuable help in preparing this paper.
R eferences
[1] А. А. А н д р о н о в , А. А. Витт и G. Э. Х а й к и н , Теория колебаний, Москва 1959.
[2] Ю. Г. Б р а т к о в с к и й , Некоторые вопросы качественной теории алгеб
раических дифференциальных систем в малом и в большом, Учен. зап. Рязанск.
гос. пед. ин-та. 15 (1958), рр. 88-108.
[3] Р. М. Минц, Исследование траекторий системы трех дифференциальных уравнений в бесконечности, Сборник памяти академика А. А. Андронова, Москва 1955, рр. 499-534.
[4] — О характере состояния равновесия системы трех дифференциальных уравнений в случае, когда один из корней характеристического уравнения равен нулю, ДАН СССР 111 (3) (1956), рр. 535-537.
