LIMIT CYCLES AND PHASE PORTRAIT FOR A CLASS OF DIFFERENTIAL SYSTEMS Текст научной статьи по специальности «Математика»

Математические заметки СВФУ Апрель—июнь, 2022. Том 29, № 2

UDC 517.925

LIMIT CYCLES AND PHASE PORTRAIT FOR A CLASS OF DIFFERENTIAL SYSTEMS R. Boukoucha

Abstract. We introduce an explicit expression of invariant algebraic curves for a class of polynomial differential systems and an explicit expression for its first integral. Moreover, we determine sufficient conditions for these systems to possess a limit cycle, which can be expressed by an explicit formula. Concrete examples exhibiting the applicability of our results are introduced.

DOI: 10.25587/SVFU.2022.61.28.006

Keywords: Hilbert 16th problem, dynamical system, limit cycle, invariant algebraic curve, first integral.

1. Introduction

We consider an autonomous ordinary differential systems in two real variables of the following form:

x' = = y' = = W

where the dependent variables x and y are real, the independent t (time) is real, and P(x,y) and Q(x,y) are polynomials in x and y with real coefficients. Here, the degree of system (1) is denoted by n = max{deg P, deg Q}. In literature, equivalent mathematical objects to refer to these planar differential systems appear as a vector field

d d

X = P(x,y)TT +Q(x,y)TT-ox ay

In the study of planar differential system (1), it is not always possible to find explicit solutions for such systems, so we resort to qualitative theory to seek information about solutions for non-linear systems to investigate their behavior. In the qualitative theory [1,2] limit cycles, or isolated periodic solutions, were and still remain the most sought solutions when modeling physical systems in the plane. Most of the early examples in the theory of limit cycles in planar differential systems were commonly related to practical problems with mechanical and electronic systems, but periodic behavior appears in all branches of science, both technological and natural. Existence of limit cycles is one of the most difficult subjects in the qualitative theory of planar differential equations, whereas even more difficult is the problem to give an explicit expression for them [3-5]. A large amount of works deals with limit cycles:

© 2022 R. Boukoucha

for instance, the famous Hilbert's 16th problem [6] motivated researchers to enter this domain of research. This was one of the main problems in the qualitative theory of planar differential equations in the 20th century.

Let us recall some useful notions.

• A limit cycle of a planar differential system is an isolated periodic solution in the set of all periodic solutions to the system; it is said to be algebraic if it is contained in the zero level set of a polynomial function [7, 8].

• System (1) is integrable on an open set O c R2 if there exists a non-constant C^function H : O ^ R, called the first integral of the system on O, which is constant on the trajectories of system (1) contained in O, i.e. if

dHi^y} = OH^y} + dH^y} t the pointg of ^

at dx dy

Moreover, H = h is the general solution of this equation, where h is an arbitrary constant.

• A curve U(x, y) = 0, where U(x, y) is a polynomial with real coefficients, is an invariant algebraic curve of system (1) if and only if there exists a polynomial K = K(x, y) of degree at most n — 1 satisfying the equality

y) + y) = K(x, y)U(x, y). (2)

dx dy

• A curve is non-singular if it does not contain equilibrium points of system (1), i. e. those that satisfy the relations

x' = ^ = P{x,y) = 0, y' = ^ = Q(x, y) = 0.

In this paper we give an explicit expression of invariant algebraic curves, then prove that these systems are integrable, and introduce an explicit expression of the first integral for a multi-parameter planar polynomial differential system of the form

^ = -(x- 2y)x2 - {2x - y)y1 + x Ylia.x4 + blX2y2 + c,y4)A-

i= 1

, i=n

<H = -(9.x + ,AX2 <- ' 1 T~IV--4 , , „.MXi

dt

(3)

-j- = ~(2x + y)x2 ~(x + 2y)y2 + y Y[(alX4 + b%x2y2 + clVy

i= 1

where n and Aj are positive integers and ai, bj, and q, i = 1,..., n, are real constants.

Moreover, we determine sufficient conditions for a polynomial differential system to possess a limit cycle, which can be expressed by an explicit formula. Concrete examples exhibiting the applicability of our results are introduced.

2. Main result

Our first result on the critical point and the expression of invariant algebraic curves for system (3) is the following

Theorem. Consider a multi-parameter planar polynomial differential system (3). Then the following statements hold.

(1) The origin 0(0,0) is the unique critical point at finite distance. Moreover, 0(0,0) is a star node.

(2) The curve U(x, y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of system (3) with cofactor

2=n

K(x, y) = -4x2 + 2xy - 8y2 + 4 JJ(a,x4 + 6,x2y2 + c,y4)A-.

i=1

Proof. (1) We say that A(x*, y*) G R2 is a critical point of system (3) if

i=n

-(x* - 2y*)x2 - (2x* - y*)y2 + x* JJ(a^4 + &,x2y2 + c,y4)Ai = 0,

-(2x* + y*)x2 - (x* + 2y*)y2 + y* JJ(a^x4 + &,x2y2 + c,y4)Ai = 0.

i=1

Then 2x4 + y 4 + 3x2y 2 = 0 and x * =0, y * =0 is the unique solution to this equation. Thus, the origin is the unique critical point at finite distance.

(2) We prove that U (x, y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of the differential system (3). We have

dU(x,y^ n , dU(x,y)

-~-P(x, y) H--^-Q(x, y)

dx dy

x-2y)x2- (2x-y)y2 + xTT (a,x4 + bx2 y2 + c,y4)Ai ] (8x3 1 c — 2

= ^-(x - 2y)x2 - (2x - y)y2 + x ^(a,x4 + b,x2y2 + c,y4)AiJ (8x3 + 6xy2) + ^-(2x + y)x2 - (x + 2y)y2 + y jJ(a,x4 + b,x2y2 + c,y4)A^ (4y3 + 6x2y) = (-4x2 + 2xy - 8y2 + 4 JJ(a,x4 + b,x2y2 + c,y4)AM (2x4 + 3x2y2 + y4)

i=1

K (x,y)U (x,y);

therefore, U(x,y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of the polynomial differential systems (3) with the cofactor

2=n

K(x,y) = -4x2 + 2xy - 8y2 + 4jj(a,x4 + b,x2y2 + c,y4)A-. □

Our second result on the existence of a first integral and explicit expressions for two limit cycles of system (3) is the following

Theorem 2. Consider a multi-parameter planar polynomial differential system (3). Then the following statements hold. (1) System (3) has the first integral

H,x'a) = ,x2+y2>'-2"H / -A<S>ds

a: / W \

J lß(w)exp — ^ A(s) ds 1 dw,

where

¿V = Ai + ■ ■ ■ + AH, „(.) = ( 2-4iV)i±fi4l, BW^(4JV-2)F-W

1 + cos2 s '

1 + cos2 w

and

T-. / \ TT /'3ßi + b + 3cj a — Ci o a — b + c.„ rn\w) = Il I ----1---— cos zw H--;- cos4w

8

8

(2) If 3a + bj + 3ci > 4|a — Cj| + — bj + cj| for i = 1,..., n, then system (3) has a non-algebraic limit cycle (r) explicitly given in the polar coordinates (r, 0) by the equation

r(0,r*)

where

/0 (B(w) exp JW — A(s) ds) dw + r exp f0 —A(s) ds

/02n(B(w) exp fW — A(s) ds) dw

2-4AT\ ^

\ — 1 + exp /Qn —A(s) ds y Moreover, this limit cycle is a stable hyperbolic limit cycle.

Proof. In order to prove results (1) and (2) of Theorem 2, we write the polynomial differential system (3) in the polar coordinates (r, 0), defined by x = r cos 0 and y = r sin 0, hence the system takes the form

% = -(1 + sin2 0)r3 + Fn(eyN+\ d4- = -(1 + dt dt

cos2 0)r2.

(4)

Since — (1 + cos2 0) < 0 for 0 e R, 0' is negative for all t € R and the orbits (r(t),0(t)) of system (4) have the opposite orientation with respect to those (x(t),y(t)) of system (3).

Taking 0 as an independent variable, we obtain the Bernoulli equation

d 1 + sin2 e —Fn{6) 4N_1 -r(0) = -—;-— r + --— r

dO' y"' 1 + cos2 0' ' 1+ cos2 0' ' ^

Via the change of variables z = r2-4N, this Bernoulli equation (5) is transformed into the linear equation

d

(6)

-z(e) = A(e)z + B(e),

y

y

A

2-4N

where

v ' v 1 + cos 0 v ' l + COS26>

The general solution to the linear equation (6) is

z(0) = exp ^ A(s) ds^j ^ ^B(w) exp ^ —A(s) ds^j ^ dw + A

where A G R.

Consequently, the general solution to (5) is

= //0>H exp/; -A(s) da) dw + A\ ^ V exp /0 —A(s) ds /

where A G R.

From this solution we obtain the first integral in (x,y) of the form

rctai

J ÎB(w)exp — y A(s) ds 1 dw.

Hence, statement (1) of Theorem 2 is proved.

Notice that system (3) has a periodic orbit if and only if equation (5) has a strictly positive 2n-periodic solution. This, moreover, is equivalent to the existence of a solution to (5) such that r(0, r*) = r (2n, r*) and r(0, r*) > 0 for any 0 in (0, 2n). The solution r(0, r0) of the differential equation (5) such that r(0,ro) = r0 is

^ = ¡' (B(w) exp J™ -A(s) ds) dw +

Q

exp fQ —A(s) ds where r0 = r(0).

A periodic solution to (3) must satisfy the condition r(2n,r0) = r(0,ro), which leads to a unique value r0 = r* given by

f(BH exp -A(s) ds) dw\ ™ — 1 + exp — A(s) ds /

The r* is the intersection of the periodic orbit with the OX+ axis. After repalcing r* with r(0,ro), we obtain

to \ i'fo (B(w) exPjT —A(s) ds) dw + r2

r(6,r*) = ( -g-

exp f0 —A(s) ds

arctan ^

In what follows it is proved that r(0,r„) > 0. Indeed

r (6>, r*) = I -^---e-- ( exP / —ds

\ (— 1+exp fQn —A(s) ds) exp f0 — A(s) ds \ J

2n ,

x J ^B(w)ex^y —A(s) dsj dw + J ^B(w)ex^y — A(s) dsj dw

According to the conditions +bj+3ci > 4|a* — c^ + |ai — bi+c^| for i = 1,..., n, hence B(0) > 0 for all 0 € (0,n), this ensures that r* and r(0,r„) are well defined for all 0 € (0,n); therefore, r* > 0 and r(0, r*) > 0 for all 0 € (0,n) and the limit cycle does not pass through the equilibrium point 0(0,0) of system (3). This is the limit cycle for the differential system (3); we note it by (r).

This limit cycle (r) is not algebraic. More precisely, in the cartesian coordinates r2 = x2 + y2 and 9 = arctan the curve (T) defined by this limit cycle is (T): L(x, y) = 0, where

L(x,y) = (x2+y2y-2N 1

(-1 + exp fQ T -A(s) ds) exp /Qarc an * -A(s) ds

2-ir \ arctan £ / w \

x I exp J —A(s) ds 1 J lß(w)ex^y — A(s) ds 1 dw

0

' arctan -

y I B(w) exp J —A(s) ds 1 dw = 0.

2n \ 0

According to the conditions, we have the non-algebraic expression

arctan ^ arctan ^

/, . . f . 1 + sin2 s

-A(s)ds = exp / UN-2)--ds

J 1 + cos2 s 00

in L(x, y), hence the expression L(x, y) is not algebraic. Consequently, (r): L(x, y) = 0 is non-algebraic and the limit cycle is also non-algebraic.

Therefore, the limit cycle (r) of the differential system (3) is a non-algebraic limit cycle.

This completes the proof of statement (2) of Theorem 2. □

3. Examples

The following examples are given to illustrate our results.

Example 1. If we take n = ai = ci = Ai = 1 and bi = 2, then system (3) takes the form

x' = — (x — 2y)x2 — (2x — y)y2 + x(x4 + 2x2y2 + y4),

y' = —(2x + y)x2 — (x + 2y)y2 + y(x4 + 2x2y2 + y4). (7)

0

w

w

The curve U(x,y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of system (7) with the cofactor K(x, y) = —4x2 + 2xy — 8y2 + 4x4 + 8x2y2 + 4y4. System (7) has the first integral

H (x,y) =

a.rct.a.n — ,

1 - - rw 2+2 sin2 s

X2 + y2

2 exp 0

0 l+cosL s

ds

1 + cos2 W

dw.

System (7) is a cubic system that has a non-algebraic limit cycle whose expression in the polar coordinates (r, 0) is

r(0,r*)

0

0 V 1+cos2 w

f0 2+2 sin2 si

eXP Jo 1+cos2 s aS

where 0 G R, and the intersection of the limit cycle with the OX+ axis is the point with

' ffi 2 exp r 2+2si">dS) du7\ +

J0 v 1+cos2 w ^ J0 1+cos2 s '

2 . .. rw 2+2 sin" s , ->0 1+cos2 s

1 I CXD f2*" 2+2sin.2s0'j 1 -+- exp J0 1+cosi s

~ 1.144.

The non-algebraic limit cycle of system (7) is shown in Fig. 1.

Fig. 1. The phase portrait for system (7).

Fig 2. The phase portrait for system (8).

*

Example 2. If we take n = A1 = a1 = b1 = c1 = 1, then system (3) takes the

form

x = — (x — 2y)x2 — (2x — y)y2 + x(x4 + x2y2 + y4),

2 2 4 2 2 4 (8)

y = —(2x + y)x — (x + 2y)y + y(x + x y + y ).

The curve U(x, y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of system (8) with the cofactor

K (x, y) = —4x2 + 2xy — 8y2 + 4(x4 + x2y2 + y4).

System (8) has the first integral

Îarcta

J

arctan ^

31 2 + 2 sin2 s , ds

1 + cos2 s

0

arctan t: / w

7 + cos 4w f 2 + 2 sin2 s

■ exp / —-5— ds dw.

4 + 4 cos2 w J 1 + cos2 s 0

System (8) is a cubic system that has a non-algebraic limit cycle whose expression in the polar coordinates (r, 0) is

0 =

¿(¿^expfields) dw + r"2 ^ ^

0 2+2 sin2 s

r" 2+2 sin2 s , eXPJo 1+cos2 s aS

where 0 € R, and the intersection of the limit cycle with the OX+ axis is the point with

- CC+ri- exp r 2+2sifs ds) dw\^

J0 V 4+4 cos2 w 1 J0 1+cos2 s / \

¡■2-it ( 7+cos 4w rw 2+2 sin2 s

1+4 cos2-ш Jo 1+cos2 s

-1 + ехп f27r 2+2sin,2s ds 1 -+- exp j0 1+cosis us

-1

~ 1.2208.

The non-algebraic limit cycle of system (8) is shown in Fig. 2. Example 3. If we take n = A2 =, b1 = c2 = 2, a1 = c1 = A1 = 1, a2 = 5, and b2 = 3, then system (3) takes the form

x' = —(x — 2y)x2 — (2x — y)y2 + x(x4 + 2x2y2 + y4)(5x4 + 3x2y2 + 2y4)2, y' = —(2x + y)x2 — (x + 2y)y2 + y(x4 + 2x2y2 + y4)(5x4 + 3x2 y2 + 2y4)2. (9)

The curve U(x, y) = 2x4 + 3x2y2 + y4 = 0 is an invariant algebraic curve of (9) with the cofactor

K (x, y) = —4x2 + 2xy — 8y2 + 4(x4 + 2x2y2 + y4)(5x4 + 3x2y2 + 2y4)2. System (9) has the first integral

'ctan ^

„, , , 2 2,_5 , f 10+10sin2 s H(x,y) = (x2+y2) exp | J 1+C0S2S ds

arctan —

w

10F2(w) /"10+ 10 sin2 s , . ,

exp / —--5-ds dw,

1 i 9^1 1 9

1 + cos9 w J 1 + cos9 s 0 \ 0

where F^{w) = (3 + 2cos2w + \ cos4w)2.

System (9) is a cubic system that has a non-algebraic limit cycle whose expression in the polar coordinates (r, 0) is

' fe( 10^(f) rxn Г 10+10 sin2 S d Л J , ^

//j \ , Jo V 1+cos w eXPJo 1+cos s USJUW^I^ \ r(0,r*) = -7j-;-

V ехп Г io+ io s л,

Jo 1+cos2 s

r i P. r

Fig. 3. The phase portrait for system (9).

Fig 4. The phase portrait for system (10).

where 0 G R, and the intersection of the limit cycle with the OX+ axis is the point with r*

~ 0.72379.

The non-algebraic limit cycle of system (9) is shown in Fig. 3. Example 4. If n = b2 = A3 = 3, Ai = 1,c2 = A2 = 2, a1 a2 = c1 = 5, a3 = 4, b3 = -1, and c3 = 3, then (3) takes the form

x' = — (x — 2y)x2 — (2x — y)y2

7, bi

,4x3

+ x(7x4 - 2x2y2 + 5y4)(5x4 + 3x2y2 + 2y4)2(4x4 - x2y2 + 3y4)

2,

(10)

y' = - (2x + y)x2 - (x + 2y)y2

+ y(7x4 - 2x2y2 + 5y4)(5x4 + 3x2y2 + 2y4)2(4x4 - x2y2 + 3y4)3.

The curve U(x, y) = 2x4 +3x2y2 + y4 = 0 is an invariant algebraic curve of (10) with the cofactor

K(x, y) = -4x2 + 2xy - 8y2 + 4(7x4 - 2x2y2 + 5y4)

x (5x4 + 3x2y2 + 2y4)2(4x4 - x2y2 + 3y4)3.

Ssystem (10) has the first integral

H(x,y) = (x2 + y2)-ii exp

22 + 22 sin2 s 1 + cos2 s

22F3(w)

ds

exp

22 + 22 sin2 s

1 + cos2 w J 1 + cos2 s 0

ds I dw,

r * =

w

where

F3(0) = cos29 + \ cos4(9

x + | cos26» + i cos46»^ Q + i cos26> + cos46>

System (10) is a cubic system that has a non-algebraic limit cycle whose expression in the polar coordinates (r, 0) is

, r0( 22 F3(w) rw 22+22 sin2 s i \ > , -22 x /fl X / Jo I 1 + COS2 eXP Jo l + COS2S «Sjaw+r,, r(V,r*) = -5---

V exD f 22+22si"2s ds

\ exPJ0 l+cos2s Ub

where 0 G R, and the intersection of the limit cycle with the OX+ axis is the point with

, r2ic,22F3(w) rw 22+22 sin2 s j \ j x

Jo ll+cos^eXPJo l+cos^ s ^ ^^^^^

1 I rxn f^ 22+22sin2s J J ~ '

I -h exp j0 1+cos2 s as /

The non-algebraic limit cycle of system (10) is shown in Fig. 4.

REFERENCES

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2. Perko L., Differential Equations and Dynamical Systems, Springer, New York (2001). (Texts Appl. Math.; vol. 7).

3. Boukoucha R., "Explicit limit cycles of a family of polynomial differential systems," Electron. J. Differ. Equ., No. 217, 1-7 (2017).

4. Gasull A., Giacomini H., and Torregrosa J., "Explicit non-algebraic limit cycles for polynomial systems," J. Comput. Appl. Math, 200, 448-457 (2007).

5. Odani K., "The limit cycle of the van der Pol equation is not algebraic," J. Differ. Equ., 115, 146-152 (1995).

6. Hilbert D., "Mathematische Probleme," Lecture, Inter. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Gott., Math.-Phys. Kl., 253-297 (1900).

7. Benterki R. and Llibre J., "Polynomial differential systems with explicit non-algebraic limit cycles," Electron. J. Differ. Equ., No. 78, 1-6 (2012).

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Submitted February 05, 2021 Revised February 11, 2022 Accepted May 31, 2022

Rachid Boukoucha

Laboratory of Applied Mathematics,

Department of Mathematics,

Faculty of Exact Sciences,

University of Bejaia,

06000 Bejaia, Algeria

rachid_boukecha@yahoo.fr