A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle Текст научной статьи по специальности «Математика»

УДК 517.9

A Class of Quintic Kolmogorov Systems with Explicit Non-algebraic Limit Cycle

Ahmed Bendjeddou

Department of Mathematics, Faculty of sciences University of Setif 1, 19000 Algeria

Mohamed Grazem*

Department of Mathematics, Faculty of sciences University of Boumerdes, 35000 Algeria

Received 26.11.2018, received in revised form 29.01.2019, accepted 06.02.2019 Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Subsequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form

x = x P4 (x, y) , y = y Q4 (x,y) ,

where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant {(x,y) € R2; x > 0, y > 0}. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result.

Keywords: Kolmogorov systems, First integral, Periodic orbits, algebraic and non-algebraic limit cycle. DOI: 10.17516/1997-1397-2019-12-3-285-297.

1. Introduction and preliminaries

The so-called Kolmogorov systems on the plane are differential equations of the form

x = — = xP (x, y),

ft (1) ■ dy n< \

y = ~dt = yQ (x,y),

where P and Q are two coprime polynomials of R [x, y] and the derivatives are performed with respect to the time variable. By definition, the degree of the system (1) is the maximum of the degrees of the polynomials P and Q. These systems arise in great variety of applications, for example, ecology and population dynamics [20,22,25], chemical reaction and plasma physics [19], hydrodynamics [10], economics, etc ...

System (1) is said to be integrable on an open set Q of R2 if there exists a non constant continuously differentiable function H : Q -—> R called a first integral of this system on Q which

* med_grazem@univ-boumerdes.dz © Siberian Federal University. All rights reserved

is constant on the trajectories of the polynomial system (1) contained in Q i.e., if dH dH dH

~dt (x,y) = ~3x (x,y) xP (x> y) + ~3y (x,y) y Q (x, y) = 0 in the points of Q.

Moreover, H = h is the general solution of the above equation, where h is an arbitrary constant. It is well know that for planar differential system, the existence of a first integral determines its phase portrait, see [11].

In the qualitative theory of planar polynomial differential systems [12], one of the most important topics is related to the second part of the unsolved Hilbert 16th problem concerned essentially by the number H (n) of limit cycles of (1) and their positions in the phase space. There is an extensive literature on that subject, most of it deals essentially with detection, number and stability of limit cycles.

We recall that in the phase plane, a limit cycle of system (1) is an isolated periodic solution in the set of all its periodic solutions. If limit cycle contained in the zero set of invariant algebraic curve of the plane, then we say that it is algebraic, otherwise it is called non-algebraic.

In the literature, we can find also another interesting but even more difficult problem is to give an explicit expression of a limit cycle. The limit cycles previously known in an explicit way were algebraic see [3,4,15].

After the Odani's work [23], where it has been proved that the limit cycle appearing in the Van der Pol equation is not algebraic without giving an explicit expression, several articles have been published presenting differential systems polynomials for which non-algebraic limit cycles exist and are explicitly determined see [1,6,9,14,16].

Concerning the Kolmogorov systems, most of the studies were limited to study the existence of limit cycles for classes of these systems see [17,20,21,25-27]. To our knowledge, the exact analytic expressions of the limit cycles for a given Kolmogorov system is still unknown except in algebraic case. For instance, Bendjeddou, Cheurfa and Berbache in [2] showed that the quartic system admits the circle as an invariant curve which corresponds of course to the limit cycle. In the same context, Benyoucef and Bendjeddou studied in [7, 8] two polynomial systems of any degrees. They showed in the first paper that the considered system can admit up to four algebraic limit cycles in the plane and in the second one the system can admit a unique algebraic limit cycle in the first quadrant.

In this paper, we are interested on the quintic Kolmogorov systems of the form

{x = x P4 (x, y),

iy>1 (2) y = y Q4 (x,y) ,

where

P4 (x, y) = 4A - 4 (3 + 2A) x + 2 (23 - 3A) y + 2 (33 + 4A) x2 + 2 (4A - 3) xy + 2 (A - 23) y2 -

- 4 (3 + A) x3 - (3 + 5A) x2y + (6/ - 2A - 1) xy2 + (A - 3) y3+

+ (3 + A) x4 + (3 + A) x3y + (1 - 23) x2y2 + (A - 3) xy3 + (3 - A) y4,

Q4 (x,y) = -4A + 2(23 + A) x + 4 (2A - 3) y + 2Ax2 - 63xy + 2(33 - 4A) y2 -

- 3 (3 + A) x3 + (23 - 2A - 1) x2y + (53 - A) xy2 + 4 (A - 3) y3+

+ (3 + A) x4 + (3 + A) x3y + (1 - 23) x2y2 + (A - 3) xy3 + (3 - A) y4,

and A, 3 are reals constants. Within this class, we study the existence of a limit cycle in the realistic quadrant {(x,y) G R2; x > 0, y > 0} and show under appropriate conditions that this cycle is non-algebraic giving its explicit form.

For presenting our main result, the coordinates are translated by a vector (1,1), which trasforms our system (2) to the following,

„2) if a _i_ \ ^ J2 i t\ _ R\„.2\ <„(„, < n2

(3)

where

x =(x + 1) ((x - 2y + x2 + xy - y2) ((3 + A) x2 + (A - 3) y2) + x (y + 1)2 (x + 1)) , y = (y +1) ((2x + y + x2 + xy - y2) ((3 + A) x2 + (A - 3) y2) + y (x + 1)2 (y + 1)) . We can write the system (3) in polar coordinates (r, 9) through x = r cos 9, y = r sin 9, as

r =8r (fi (9) r4 + f2 (9) r3 + f3 (9) r2 + f4 (9) r + 8) , 9 = r2 (A + 3 cos 29) (2 + r (cos 9 + sin 9)),

f1 (9) = 1 + 43 + 4A (2 cos 29 + sin 29) + (43 - 1) cos 49 + 23 sin 49, f2 (9) = 4 ((23 + 4A +1) cos 9 + sin 9 + (23 - 1) cos 39 + sin 39), f3 (9) = 8 (1 + A + 3 cos 29 + 2 sin 29), f4 (9) = 16 (cos 9 + sin 9). Since we are dealing with solutions of system (2) in the first quadrant, we have r cos 9 > -1 and r sin 9 > -1 hence (2 + r (cos 9 + sin 9)) > 0. If A + \3\ < —, then (A + 3 cos 29) is strictly

■ d9

negative and as a result 9 = — is negative for all t. This signifies that (1,1) is the unique equilibrium point of system (2) in the first quadrant and the orbits (r (t) ,9 (t)) of system (4) have opposite orientation with respect to (x (t) ,y (t)) of system (2).

2. The main result

Our main result on the limit cycles of the quintic Kolmogorov system defined by (2) is as follows

Theorem 2.1. Consider the polynomial system (2), then the following statements hold

1) If 3 = 0 and A + \3\ < , the system (2) has non-algebraic, stable and hyperbolic limit cycle

explicilty given in polar coordinates (r, 9) by

. . A (9) (cos 9 + sin 9) + J A2 (9) + 4A (9) - A2 (9) sin 29 r (9,ro) =-2 - A (9) sin 29-,

where A (9) = exp (9) (+ 0 ^PLiLds) and w Vro + 1 0 A + 3cos2s J

/

In

e2n f-ds , . .

0 A + 3 cos 2s / / r2n _e-s / r2n „-s

r0 = --^-r-\\ e2n[ --^-ds +< e2n [ --^-ds + 4(e2n- 1)

0 2 (e2n - 1) \Y J0 A + 3 cos 2s y J0 A + 3 cos2s v y

2) If 3 = 0 and A < —, the system (2) has algebraic, stable and hyperbolic limit cycle explicilty

, , , ns , /n s (cos 9 + sin 9) + a/1 - 4A - sin 29

given in polar coordinates (r, 9) by r (9, r0) =-—-:——-, and in Cartesian

- (2A + sin 29)

coordinates by A (x - 1)2 + A (y - 1)2 + xy = 0.

For the demonstration of Theorem (2.1), we need the following lemmas Lemma 2.2. The system of the form

d H

r = F (0) H (r, 0) - — (r, 0) + G (0),

h dH , 0 = ^

d0............(5)

possess a first integral expressed as

f9 f9 ps

L (r, 0) = H (r,0)exp(- / F (s) ds) - G (s)exp(-/ F (w) dw)ds. (6)

J 0 J 0 J 0

, e . e , s .

Proof. Let set A (r, 0) = H (r, 0) exp -J F (s) ds) and B (0) = / G (s) exp -J F (w) dw Ids,

^ 0 ' 0 ^ 0 ' then the derivatives of A and B with respect to 0 are

dA (r, 0) = (dH (r, 0) - F (0) H (r, 0))exp( - f' F (s) ds),

d-B (0)= G (0) exp ^ - J' F (s) ds)

By replacing the expression of derivatives of A and B with respect to 0 in the expression of L, dL ~dB

By the chain rule, the derivative of L with respect to t is given by following expression, dL( . . „ . . x dL , dr dL . d0

dt(r (t) ,0 (t)) = (7 (r,0) dtt + He (r,0) dt =

e

dT i dH \ f e \

it follows that — (r,0) = ^^ (r, 0) - F (0) H (r,0) - G (0) exp -J F (s) ds).

d0 y d0 J 0 '

d-H (r, 0) exp ( - 0 F (s) ds^ (f (0) H (r, 0) - (r, 0) + G (0)) + (( 1H (r,0) - F (0) H (r, 0) - G (0)) exp ( - J° F (s) dsfj

0

+ ( ( (r, 0) - F (0) H (r, 0) - G (0) ) exp ( - f F (s) ds) ) (r,0) = 0.

So L (r, 0) is a first integral of system □

Lemma 2.3. Let X, 3 £ R such that X + \3\ < --, then the following statements hold

e2n 2 n —e~s

1) 0 < - f -ds < 2.

y e2n - 1 0 X + 3 cos 2s

e —e~s

2) The function g defined on [0, 2^] by g (0) = 2 exp (-0) + J x + 3-2"ds is strictly decreasing.

e2n 2 n —e~s

Furthermore g (0) > - f ----ds.

w e2n - 1 0 X + 3 cos 2s

( e?n 2 n —e-s e e-s \

3) 0 < A (0) = exp (0) -- f ----— ds + f----— ds \ < 2.

7 w KW\e2n - 1 0 X + 3cos2s 0 X + 3cos2s J

Proof of statement 1) of Lemma 2.3 We have X + 3 cos 2s < X + \3\ < — which implies 0 <

_e-s e2n 2n _e-s 2e2n 2 n

----— < 2e s, consequently 0 < -- f ----— ds < -- f e s ds, whence

X + 3 cos 2s J e2n - 1 0 X + 3 cos 2s e2n - 1 0

e2^ f2n —e-s

0 < ^- -ds < 2.

e2n - 1J0 A + 3 cos 2s

□

Proof of statement 2) of Lemma 2.3. Over the interval [0, 2n], the function g is differentiable and

g' (6) = —2 exp (—6) + - eXP ( = - exp (—6) f 2+---1-- .

yy> FV A + P cos 26 V A + P cos 26 J

Since A + P cos 26 < A + \P\, then g' (6) < — exp (—6) ^2 + < 0- Therefore g is

strictly decreasing function. On the other hand, from the statement 1) of Lemma 2.3, we have 1 2n -e—s -t i -;-^-^ ds < 2e-2n, which implies

e2n — 1 0 A + P cos 2s

e2n f2n —e-s f2n -e-s 2n 1 ds — ----— ds < 2e-

e2n — 1 J0 A + P cos 2s J0 A + P cos 2s

because —^ = ^ —^ — 1j , consequently

2n

f e2n

1= \e2^ —

„2n

e [ .

:2n _ 10

ds < 2e-2n + / -ds = g (2n),

J0 A + P cos 2s yV ''

e2n 2n —e~s

as g is strictly decreasing function, then -- f ----— ds < g (6). □

J 6 e2n — 1 0 A + P cos 2s w

Proof of statement 3) of Lemma 2.3. Let us first show that A is strictly positive. From the relationship of Chasles

A (6) = exp (6) | f2n | I -——-ds +i -——-ds) —i -——-ds) ,

w y e2n — J0 A + P cos 2s J9 A + P cos2s J J0 A + P cos2s y'

which implies

( (?n C2n -e-s ( e2n \ (e -e-s \ A (6) = exp (6) I ^- -ds + -n--1 / -ds)

y e2n — 1je A + P cos2s Ve2n — 1 J J0 A + P cos2s J

e-

Since --1 > 0 and ---- > 0, then A (6) > 0.

e2n — 1 A + P cos 2s w

Let us now show that A (6) < 2. From the statement 2) of Lemma 2.3, we have

e2n r2n _e s rd _e s

ds < 2exp— 6)+ ----— ds,

e2n — 1 J0 A + P cos 2s v ' J0 A + P cos 2s

( e2n 2 n —e-s 9 e-s \

which implies -- f ----— ds + f----— ds \ < 2 exp (—6), therefore

\e2n — 1 0 A + P cos 2s 0 A + P cos2s / v '

( e2n i2n —e-s C9 e-s \

A (6) = exp (6) ^- -ds + -ds) < 2,

FV ' ye2n — 1 J0 A + P cos 2s J0 A + P cos2s J '

whence 0 < A (6) < 2. □

Proof of Theorem 2.1. We assume that A + \P\ < —1. In the new independent variable 6, the differential system (4) becomes

dr = 1 fi (6) r4 + f2 (6) r3 + f3 (6) r2 + f4 (6) r + 8

d6 8r (A + Pcos 26) (2 + r (cos 6 + sin 6)) , ( )

which can be expressed as

dH

dr F (e) H M) - — M) + G (e)

de

_de

dH .

(8)

where H (r, e) =

F (0) = 1 and G (0) =

1

(r cos e + 1)(r sin e + 1)' " A + ß cos 2e'

By Lemma 2.2, the solutions of the equation (7) are expressed as

r2 ( f es \

, —-- - exp (e) k + -ds

(rcose + 1)(rsine + 1) Jo A + ßcos2s J

0, where k £ R.

(9)

In the region 2 - A (0) sin 20 = 0 and A2 (0) + 4A (0) - A2 (0) sin 20 > 0 the equation (9) has two

solutions _

A (0) (cos 0 + sin 0) ± v7A2 (0) + 4A (0) - A2 (0) sin 20

r1 '2 (0) = -Ô-A(a\ ^Ofl-, (10)

2 - A (e) sin2e

ds .

with a (0)=exp(0^ *+0 ^^

Notice that system (3) has a periodic solution if and only if equation (7) has a strictly positive 2n-periodic solution. For 0 = 0, we have

ri (0) = 1 (h + Vk (k + 4)) and r2 (0) = 1 (k - y/k (k + 4)) ,

ri'2 (0) are defined if only if k g ]-œ, -4[ U ]0, +œ[. Over the interval ]-œ, -4[, r1 (0) and r2 (0) are negative, and in ]0, +œ[, r1 (0) is positive but r2 (0) is negative. Consequently, the admitted solution of equation (9) is

r (e) = n (e)

a (e) (cos e + sin e) + v7a2 (e) + 4A (e) - a2 (e) sin 2e

where A (e)=exp(eW k + 0 a + ß cos 2s

ds and k

2 - A (e)sin2e

r2 (0)

(11)

> 0.

r (0) + 1

The solution of the equation (9) starting at r (0, r0) = r0 > 0 is given by

r ( e, ro )

a (e) (cos e + sin e) + Va2 (e) + 4A (e) - a2 (e) sin 2e

where A (e) = exp (e) + / -—|-—

w I r0 + 1 0 A + ß cos 2s

2 - A (e) sin2e

ds \ and r0 = r (0).

The condition of the periodic solution with 2n-periodic starting at r (0, r0)

r (0, r0) = r (2n, r0). For 0 = 2n, we obtain

r0 > 0 is

(2n, ro) = 2 (A (2n) + VA (2n) (A (2n)+4))

where A (2n) = e

2n

' r2 2n e-s

+ 2 -e-

ro + 1 o A + ß cos 2s

ds

2

r

— s

s

The resolution of equation r (0, r0) = r (2n, r0) gives

/

2 n _e s

e f X+ 3 cos 2s ds ( I —p-s I T2* -e-s \

Ve2M -ds + </e2W -ds + 4 (e2^ - 1)1 .

iV J0 X+ 3 cos 2s Y J 0 X + 3 cos 2s V '

r0 = —_0_ | i e2n _e_ds + 1 e2n _—

0 2 (e2n - 1) IV J0 X+ 3 cos 2s V J0 X + 3 cos2s

r2 e2n 2n —e-s

By some simplifications, we obtain —= —- f ----— ds. Finally, the general so-

r0 + 1 e — 1 0 X + 3 cos 2s

lution of (4) is given explicitly by

„ . A (0) (cos 0 + sin 0) + VA2 (0) + 4A (0) - A2 (0) sin 20 ^) =-2 - A(0)sin20-, (12)

e2n 2 n —e-s

with a (0) =exp (0^0 ds + 0 ds)and r0 =r (0). Periodicity of r (0, r0). Let us now show that A (0) is 2n-periodic function. We have

e- f2n -e-s , fe+2n e-s \

A (0 + 2n) = exp (0 + 2n)| - I ---ds +/ ---ds

K ' FV ' ye2n - 1 J0 X + 3 cos 2s J0 X + 3 cos 2s ;

it follows

i.e.

1 f2n —p-s fe+2n e-s

A (0 + 2n) = eee2n[-n- ----ds + / ----ds

K ' 1 e2n - 1 J0 X + 3 cos 2s J2n X + 3 cos 2s

e+2n e-s e e-s

e - -2n " e

f e2n f2n -e-s ds f2n e-s ^ fe+2n e-s \

1 e2n - 1 J0 X + 3 cos 2s s + J0 X + 3 cos 2s s + J2n X + 3 cos2s

(e2-1

by the change of variable u = s — 2n, we obtain f --^-ds = e-2n f --^-ds,

J 5 ' ¿n X + 3 cos 2s 0 X + 3 cos 2s '

then

A (0 + 2n) = ee ( -4— /2"-—-ds + e2n e-2n f"-—-ds) ,

v 7 y e2n - 1 J0 X + 3 cos 2s J0 X + 3 cos 2s y'

therefore A (0 + 2n) = A (0). Furthermore, as 0 -—> sin 0, 0 -—> cos 0 and 0 -—> A (0) are 2n-periodic functions, then r (0, r0) is also.

Strict positivity of r (0, r0). By the statement 3) of Lemma 2.3, we have 0 < A (0) < 2, then the denominator of r (0,r0) is strictly positive. Two cases are distinguished

i) If (cos 0 + sin 0) >0, the numerator of r (0, r0) is strictly positive, consequently r (0, r0) is also.

ii) If (cos 0 + sin 0) < 0, we have 4A (0) - 2A2 (0) sin 20 > 0, which implies

A2 (0) + 4A (0) - A2 (0) sin 20 > A (0) + A2 (0) sin 20,

i.e.

A2 (0) + 4A (0) - A2 (0) sin 20 > (-A (0) (cos 0 + sin 0))2,

then

■\JA2 (0) + 4A (0) - A2 (0) sin 20 > -A (0) (cos 0 + sin 0),

r

hence

A (6) (cos 6 + sin 6) + VA2 (6) + 4A (6) — A2 (6) sin 26 > 0.

Therefore r (6,r0) is stictly positive. Finally r (6,r0) defines through (4) is a periodic solution. Let us show that this peiodic solution is a limit cycle. For this aim, we introduce the Poincare return map

Y ^ n (y) = r (2n, y) = 2 (A (2n) + vA2 (2n) + 4A (2n)) ,

( Y2 2 n e-s \

where A (2n) = exp (2n)--h f ----ds ) and show that the function of Poincare

V 7 V ^V Y +1 0 A + P cos 2s J

dn (y)

first return verify

dY

= 1 see [12]. We remark that

Y=ro

VA2 (2n) + 4A (2n) = e vi(Y2+DY+D)je2nJY2+DY+D)T4(YT1j, (Y + 1)

with

i'2n e-s

D = -ds.

J0 A + P cos 2s

We have

d (A (2n)) = e2n,

&Y (Y +1)2

and

(^H) V(Y2 + Dy + D) (e2n (y2 + Dy + D) + 4 (y + 1)))

BY

Y (Y + 2) en (e

(y + 1)2 vWTDYTDYJenJYTTDYTD)T4{YT1)'

consequently

d /1

dY (j (A (2n) + VA2 (2n) + 4A (2n)

_ 1 Y (Y + 2) en ( (e

2 (y + 1)2 VvWTDYTDjjeniY2TDYTDy+4(Y+1T)

then

,

Y

dn (y) _ 1 Y (Y + 2) en ( (e-

2W„ ,2

22

Y=ro

+ en

2 (y + 1)2 \ V(Y2 + Dy + D) (e2n (y2 + Dy + D) + 4 (y + 1))

= 1 r0 (r0 + 2) en ( (e2n (r2 + Dr0 + D) + 2 (r0 + 1))

2 (r0 + 1)2 V V(r2 + Dr0 + D) (e2n (r2 + Dr0 + D) + 4 (r0 + 1))

Since A + \P\ < —1, then 0 < (r2 + Dr0 + D) < r2 because D < 0 and A> 0. Therefore

(e2n (r2 + Dr0 + D) +4 (r0 + 1)) < (e2nr2 + 4 (^ + 1)) < e2n (r2 + 2r0 + 1) = e2n (r0 + 1)2 because e2n > 4. It follows that

-2 + Dr0 + D) (e2n (r2 + Dr0 + D) + 4 (r0 + 1)) < e2n (r0 + 1)2 = enr0 (r0 + 1),

i.e.

>

V(r2 + Dr0 + D) (e2n (r2 + Dr0 + D) + 4 (r0 + 1)) r0 (r0 + 1)

-, which implies

(e2n (r2 + Dr0 + D) +2 (r0 + 1))

>

2(r0 + 1)

vVo + Dro + D) (e2- (r2 + Dro + D) + 4 (ro + 1)) e-ro (ro + 1) e-ro '

because (e2- (r2 + Dro + D) +2 (ro + 1)) > 2 (ro + 1). Consequently

1 ro (ro + 2) e- ( 2

dn(Y )

dy

> -

Y=r о

(4-+«') =

\en r0 У

2 (r0 + 1)2

(r0 + 2) +1 r0 (r0 + 2) «2п

+ ^ о «

(r0 + 1)2 2 (r0 + 1)2

(r0 + 2) ( е2п ) / е2п л -( 1 + — r0 j > ( because — > 1J

>

Hence

(r0 + 1)' (r0 + 2) (r0 + 1)2 (r0 + 2) > . (r0 + 1)

dn (7)

(1 + r0) =

dy

>1.

Y=ro

Therefore the solution of differential equation (4) is unstable and hyperbolic limit cycle see [12], consequently, it is a stable and hyperbolic limit cycle for the system (2).

1) If 3 = 0, this limit cycle is non-algebraic, due to the expression of A (0).

)2 , 1)2 0 arctan /y - 1'

yx — 1/

J- ( 11 ¡S -y— W, UlllO 111111U V, V lO null ClilS^V^l^l ClilV., 'Ill' UW lillVi IjApiUJOlUll W1 ^ A IV I .

More precisely, in Cartesian coordinates^r2 = (x — 1)2 + (y — 1)2, 0 = arctan ^-—, the

22

curve defined by this limit cycle is f (x, y) =

'y — 1

(x — 1)2 + (y — 1)2

B (x, y) = exp arctan

xy

—e

— B (x, y) = 0, with

(arctan(x-!)) («¿-r(l лг-^) +i

y-1

x — 1

Л+ в cos 2s

-ds \ .

S J

dnf dn f

There is no integer n for which both —— and —— vanish identically. To be convinced by this

dxn dyn

df

fact, one has compute for example ——, that is

dy

df , , — x2 +2x + y2 — 2 x — 1

7T(x,y) = -2--7-772-;-772 B (x, y) +

dy xy2 (x — 1)2 + (y — 1)2

A+ P cos ^2 arctan ^J

Since B (x, y) appears again, it will remains in any order of derivation, therefore the curve f (x, y) = 0 is non-algebraic and the limit cycle of the system (2) will also be non-algebraic. This complete the proof of statement 1) of Theorem 2.1.

2) If в = 0, we have / exp—)ds = 1 (1 — e-) and e2n / -«0 Л Л 0 Л

ds = 1 (1 — e2n) , by Л

— 1 _ r2

simplification we obtain r0 = — (л/1 — 4Л + 1) and 0

2Л

values of ro and A (0) in (12), the solution of (4) becomes

r° = —1 = A (0). By substituting the

ro + 1 A

1

1

2

s

e

%T0) =

Л (cos 0 + sin 0) + - Л - sin 20

T

Л2 Л Л2

2 + -т sin 20

Л

(cos 0 + sin 0) + Vi - 4Л - sin 20 - (2Л + sin 20) '

In Cartesian coordinates, the curve defined by this limit cycle is X (x - 1)2 + X (y - 1)2 + xy = 0 which is algebraic. This complete the proof of statement 2) of Theorem 2.1. □

3. Applications

In this section, we present some examples to illustrate the applicability of the our main result. In addition, a plot of phase portraits in the Poincare disc for each example were performed showing a limit cycle in the first quadrant.

Eexample 3.1. In the system (2), we take X = -2 and 3 =1 f X + \3\ = -1 < -— j, we obtain

-8 + i2x + 16y - 10x2 - 18xy - 8y2 + 4x3 + 9x2y

+9xy2 - 3y3 - x4 - x3y - x2y2 - 3xy3 + 3y

У = y

8 - 20y - 4x2 - 6xy + 22y2 + 3x3 + 5x2y +7xy2 - 12y3 - x4 - x3y - x2y2 - 3xy3 + 3y

(13)

which has a non-algebraic, stable and hyperbolic limit cycle whose expression in polar coordinates (r, 0) is

A (0) (cos 0 + sin 0) + VA2 (0) + 4A (0) - A2 (0) sin 20

r(0,ro) =

2 - A (0) sin 20

where A (0) = exp (0)

2

' 0 + r exP(-s) r0 + 1 0 -2 + cos 2s

ds\ and r0 ~ 1'1877 (Fig. 1).

Fig. 1. The phase portrait on the Poincare disc of the system (13), showing a limit cycles in the first quadrant

r

4

4

Eexample 3.2. In the system (2), we take A = -10 and ¡3 = 0 A < — , we obtain

t) ■

-40 + 80x + 60y - 80x - 80xy - 20y2 + 40x3 + 50x2y

y = y

+19xy2 - 10y3 - 10x4 - 10x3y + x2y2 - 10xy3 + 10y

40 - 20x - 80y - 20x2 + 80y2 + 30x3 + 19x2 y + 10xy2 -40y3 - 10x4 - 10x3y + x2y2 - 10xy3 + 10y4

,

(14)

which has an algebraic, stable and hyperbolic limit cycle given by the expression (Fig. 2).

-10 (x - 1)2 - 10 (y - 1)2 + xy = 0.

4

Fig. 2. The phase portrait in the Poincare disc of the system (14), showing a limit cycle in the first quadrant

Conclusion

In this paper, a quintic Kolmogorov system with two parameters X and 3 having (1,1) as positive equilibrium point was investigated. By translation the coordinates of vector (1, 1) and rewritten the system in polar coordinates, we mainly shown that there is a sufficient condition for the existence of a limit cycle. Moreover, this limit cycle is non-algebraic in the case 3 = 0.

Finally, It is of interest to extend this study by answering to the following question: Is there a quartic or quintic Kolmogorov system that exhibit more than one non-algebraic limit cycle? This is left as a topic for future research.

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Класс квинтических колмогоровских систем с явным неалгебраическим предельным циклом

Ахмед Бенджедду

Кафедра математики, факультет наук Университет Сетиф 1, 19000 Алжир

Мохамед Гразем

Кафедра математики, факультет наук Университет Бумердес, 35000 Алжир

Различные физические, экологические, экономические и т.д. явления перекрываются планарными дифференциальными системами. Впоследствии некоторые исследования привлекут внимание к изучению предельных циклов из-за их интереса к пониманию этих систем. Целью данной работы является исследование одного класса квинтических колмогоровских систем, а именно систем вида

х = х Р4 (х, у) , У = у ( (х,у) ,

где Р4 и (4 — квартичные полиномы. В этом классе наше внимание ограничено изучением предельного цикла в реалистическом квадранте {(х,у) € К2; х > 0, у > 0}. Согласно гипотезам доказано существование алгебраического или неалгебраического предельного цикла. Кроме того, этот предельный цикл явно задан в полярных координатах. Некоторые примеры представлены для того, чтобы проиллюстрировать возможности применения нашего результата.

Ключевые слова: колмогоровские системы, первый интеграл, периодические орбиты, алгебраический и неалгебраический предельные циклы.