limit cycles in quadratic systems with a first-order weak focus

ISSN 1064�5624, Doklady Mathematics, 2010, Vol. 82, No. 3, pp. 923–926. © Pleiades Publishing, Ltd., 2010.Original Russian Text © G.A. Leonov, 2010, published in Doklady Akademii Nauk, 2010, Vol. 435, No. 4, pp. 447–450.

923

This paper generalizes existence conditions forlimit cycles in quadratic systems with a first�orderweak focus obtained in [1]. The development of meth�ods suggested in [2–5] is continued.

Consider the system

(1)

where c ∈ , c ≠ , β = 0, and α = –ε–1, ε is a

small positive parameter. We assume also that

(2)

Theorem. For sufficiently small ε, system (1) hasthree limit cycles, one cycle on the left of the straight line{x = –1} and two cycle on the right of this line.

We outline the proof this theorem under the

assumption that c ∈ . In the case of c ∈ , ,

this result was obtained in [1].

It is well known [2–5] that the transformation

(3)

reduces system (1) to the Liénard system

(4)

where

x· x2 xy y,+ +=

y· ax2 bxy cy2 αx βy,+ + + +=

13�� 1,⎝ ⎠

⎛ ⎞ 12��

bc 1, b a c, 2c b 1,+<+><

4a c 1–( ) b 1–( )2.>

12�� 1,⎝ ⎠

⎛ ⎞ 13��⎝

⎛ 12��⎠

⎞

u y x2

x 1+��+⎝ ⎠

⎛ ⎞ x 1+ c–=

x· u, u· f x( )u– g x( ),–= =

f x( ) Ψ x( ) x 1+ c– 2–, g x( ) Φ x( ) x 1+ 2c–

x 1+( )3��,= =

Ψ x( ) 2c b– 1–( )x2 2 b β+ +( )x– β,–=

In the case under consideration, all assumptions ofTheorems 2 and 4 from [4] hold (except the localinstability of the point x = u = 0); the equilibrium statex = u = 0 is a weak focus with negative first Lyapunovexponent. According to these theorems, the trajecto�ries of system (4) with large initial data behave asshown in Fig. 1. Such a behavior and the instability ofthe equilibrium state on the left of the line {x = –1}imply the existence of a limit cycle on the left of theline {x = –1}.

Let us analyze the asymptotic behavior of trajecto�ries on the right of the line {x = –1}. System (4) isequivalent to the first�order equation

(5)

Making the change F = G, we obtain

(6)

Consider the solution of Eq. (6) with initial dataG(0) = R = ε–0.01.

Φ x( ) x x 1+( )– 2 ax α+( )=

+ x2 x 1+( ) bx β+( ) cx4.–

FdFdx�� f x( )F g x( )+ + 0.=

α–

GdGdx�� f x( )

α–��G g x( )

α–��+ + 0.=

Limit Cycles in Quadratic Systems with a First�Order Weak Focus

Corresponding Member of the Russian Academy of Sciences G. A. LeonovReceived June 9, 2010

DOI: 10.1134/S1064562410060220

St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034 Russiae�mail: [email protected]

MATHEMATICS

u

−1

x

Fig. 1. The asymptotic behavior of trajectories of theLiénard system for large initial data.

924

DOKLADY MATHEMATICS Vol. 82 No. 3 2010

LEONOV

The change z = (x + 1)–c reduces Eq. (6) to the form

(7)

GdGf z

1c��–

1–⎝ ⎠⎛ ⎞

c α––��z

1– 1c��–

Gdz+

+g z

1c��–

1–⎝ ⎠⎛ ⎞

cα��z

1– 1c��–

dz 0.=

Let ν = ε0.001. For z ∈ [νR, 2R], we have

(8)

In this case, for small ε and z ∈ [νR, 2R], Eq. (7) isasymptotically close to the equation

g z1c��–

1–⎝ ⎠⎛ ⎞

α–��z

2– 1c��–

1– O ε0.009

c��

⎝ ⎠⎛ ⎞ ,+=

f z1c��–

1–⎝ ⎠⎛ ⎞ z

1– 2c��–

1 2c O ε0.009

c��

⎝ ⎠⎛ ⎞ .+ +=

1

−2 0 2 4 6 8 10 12 14 16

0

−1

−2

−3

x × 10−6

y × 10−7

Fig. 2. The large stable limit cycle.

300

−2 0 2 4 6 8 10 12

200

100

0

−100

−200

−300

x

y

Fig. 3. The winding trajectory.


LIMIT CYCLES IN QUADRATIC SYSTEMS 925

(9)

Now, consider the change z = (x + 1)1 – c, where x > 0.This change reduces Eq. (6) to the form

(10)

For z ≥ νR, we have

(11)

It follows that, for z ≥ νR, Eq. (10) is close to the equa�tion

(12)

GdG 1 2c+( )

c α––��z

1c��

Gdz 1c��zdz+ + 0.=

GdGf z

11 c–��

1–⎝ ⎠⎛ ⎞

1 c–( ) α–��z

c1 c–��

Gdz+

+g z

11 c–��

1–⎝ ⎠⎛ ⎞

1 c–( ) α–( )��z

c1 c–��

dz 0.=

g z1

1 c–��

1–⎝ ⎠⎛ ⎞

α–��z

c1 c–��

= zc

1 c–��–

ε b a– c–( )z O ε( )zc

1 c–��–

,+ +

f z1

1 c–��

1–⎝ ⎠⎛ ⎞ z

c1 c–��

2c b– 1–( ) O ε0.091 c–��

⎝ ⎠⎛ ⎞ .+=

GdG 2c b– 1–( )

1 c–( ) α–��Gdz+

+ 11 c–�� z

c1 c–��–

ε b a– c–( )z+⎝ ⎠⎛ ⎞ dz 0.=

It is easy to see that, for the solution of Eq. (12) withG(1) = R, the asymptotic estimates

hold. Therefore, the difference G(1)2 – R2 (G(1) < 0)satisfies the relation

where

It follows that

Similar considerations for the solution of Eq. (9) withG(1) = R imply [1]

Therefore, the trajectory of system (4) with initial data

x(0) = 0, u(0) = R unwinds when making a fullrotation around the equilibrium state, as opposed towinding shown in Fig. 1. The assertion of the theorem

G z( )2 R2 22c 1–�� 1 z

1 2c–1 c–

��

–⎝ ⎠⎛ ⎞ ε b a– c–( )

1 c–��z2––≈

≈ R2 ε b a– c–( )1 c–

��z2–

G 1( )2 R2– 4 2c– b 1+ +( )

1 c–( ) α–�� R2 ε b a– c–( )z2

1 c–��– z,d

1

z0

∫>

z0 R 1 c–ε b a– c–( )��.=

G 1( )2 R2– π 2c– b 1+ +( )R2

1 c–( ) b a– c–( )��, G 1( ) 0.<>

G 1( )2 R2–

2 c1 c–

2c��

⎝ ⎠⎜ ⎟⎛ ⎞

1 2c+( )

α–��R

2 1c��+

τsin( )1c��

τcos( )2 τ,d

0

π

∫≈

G 1( ) 0.<

α–

80

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 1.2

40

20

0

−40

−60

−80

x0.8 1.0

−20

60

y

Fig. 4. The unwinding trajectory.

926


LEONOV

follows at once from this observation and from thenegativity of the first Lyapunov exponent.

As an example, consider system (1) with a = –2,

b = 1, c = , α = –10 000, and β = 0. This system (1)

satisfies all conditions of the theorem and has threecycles. Increasing β, we obtain the fourth cycle aroundthe point x = y = 0. Thus, Fig. 2 shows a large stablelimit cycle arising at β = 0.1, and Fig. 3 shows a wind�ing trajectory which tends to an unstable limit cycle ast → –∞ and to a stable limit cycle as t → +∞. This limitcycle is located between a piece of the winding trajec�

tory shown in Fig. 3 and a piece of the unwinding tra�jectory shown in Fig. 4.

REFERENCES

1. G. A. Leonov, Dokl. Math. 81, 236–237 (2010) [Dokl.Akad. Nauk 431, 447–449 (2010)].

2. G. A. Leonov, Int. J. Bifurcat. Chaos 18, 877–884(2008).

3. G. A. Leonov, Dokl. Akad. Nauk 426 (1), 47–50(2009).

4. G. A. Leonov, Dokl. Math. 81, 31–33 (2010) [Dokl.Akad. Nauk 430, 157–159 (2010)].

5. G. A. Leonov, Prikl. Mat. Mekh. 74, 37–73 (2010).

47��

limit cycles in quadratic systems with a first-order weak focus

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