ISSN 1064�5624, Doklady Mathematics, 2010, Vol. 81, No. 2, pp. 236–237. © Pleiades Publishing, Ltd., 2010.Original Russian Text © G.A. Leonov, 2010, published in Doklady Akademii Nauk, 2010, Vol. 431, No. 3, pp. 316–318.

236

In [1] Arnold wrote: “To estimate the number oflimit cycles of quadratic vector fields in the plane,A.N. Kolmogorov distributed, as an assignment, sev�eral hundreds of such fields (with second�degree poly�nomials having randomly chosen coefficients) amongseveral hundreds of students at the Faculty of Mechan�ics and Mathematics of Moscow State University.

Each student had to determine the number of limitcycles of his or her field.

The result of this experiment was completely unex�pected: none of the fields had a limit cycle!

When the coefficients of a field undergo small vari�ations, a limit cycle of the field is preserved. Therefore,the systems with one, two, three (and, as was provedlater, even four) limit cycles form open sets in the coef�ficient space and, if the coefficients of the polynomialsare drawn at random, the probability of hitting thesesets is positive.

The fact that this event did not occur suggests thatthe probability is apparently small.”

In [2–4], effective criteria for the existence of limitcycles of quadratic systems were obtained and it wasshown that domains of existence of limit cycles in thecoefficient space are not small. Moreover, a two�dimensional quadratic system can be synthesized thathas a limit cycle with given initial data. This problem issolved below.

Consider the system

(1)

where aj, bj, cj, αj, and βj are real numbers.Proposition 1 [2, 3]. Without loss of generality, we

can assume that c1 = 0.Proposition 2. Let c1 = 0 and β1 ≠ 0. Without loss of

generality, we can assume that α1 = 0.

x· a1x2 b1xy c1y2 α1x β1y,+ + + +=

y· a2x2 b2xy c2y2 α2x β2y,+ + + +=

Proposition 3. c1 = α1 = 0, a1 ≠ 0, b1 ≠ 0, and β1 ≠ 0.Without loss of generality, we can assume that a1 = b1 =β1 = 1.

Let c1 = α1 = 0, a1 = b1 = β1 = 1, c2 ∈ (0, 1). In thiscase, system (1) can be transformed into the Liénardsystem

(2)

where

It is easy to see that the transformation

brings system (2) to the form

In what follows, we assume that a2 = b2 – 1, b2 = β2,and α2 < –1. Then

x· u, u· f x( )u– g x( ),–= =

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

x 1+( )3���������������,= =

Ψ x( ) 2c2 b2– 1–( )x2 2 b2 β2+ +( )x– β2,–=

Φ x( ) x x 1+( )2 a2x α2+( )–=

+ x2 x 1+( ) b2x β2+( ) c2x4.–

u y x2

x 1+���������+⎝ ⎠

⎛ ⎞ x 1+ q, q c2–= =

x· x 1+ q

x 1+( )������������� x2 xy y+ +( ),=

y· x 1+ q

x 1+( )������������� a2x2 b2xy c2y2 α2x β2y+ + + +( ).=

Ψ x( ) β2 x 1+( )2– 2c2 1–( )x2 2x,–+=

Synthesis of Two�Dimensional Quadratic Systems with a Limit Cycle Satisfying Prescribed Initial Conditions

Corresponding Member of the RAS G. A. LeonovReceived October 20, 2009

DOI: 10.1134/S1064562410020195

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiae�mail: leonov@math.spbu.ru

MATHEMATICS

u

–1

–1 + ε 0 x

Fig. 1. Qualitative behavior of the trajectory for Ψ(x) > 0.

DOKLADY MATHEMATICS Vol. 81 No. 2 2010

SYNTHESIS OF TWO�DIMENSIONAL QUADRATIC SYSTEMS 237

It is easy to see that Φ(x)x > 0, ∀x > –1, x ≠ 0; ifβ2 = 2c2 – 3, then Ψ(x) > 0, ∀x > –1; and, if β2 =

, ε ∈ (0, 1), then Ψ(x) < 0, ∀x > –1 + ε.

Consider the trajectory of system (2) with the ini�tial data u(0) = 0 and x(0) = –1 + ε. Obviously, it fol�lows from Ψ(x) > 0, ∀x > –1 that the trajectory islocated in the phase plane as shown in Fig. 1, while thecondition Ψ(x) < 0, ∀x > –1 + ε implies that this tra�jectory behaves as shown in Fig. 2 or 3.

These findings and the fact that the solutions of sys�tem (2) depend continuously on the parameters implythe following result.

Theorem. For fixed c2 ∈ (0, 1), α2 < –1, and ε ∈ (0, 1),there exists a constant

such that system (2) with β2 = , a2 = – 1, and

b2 = has a periodic solution with the initial datau(0) = 0 and x(0) = –1 + ε.

Obviously, this solution corresponds to a closedtrajectory of system (1).

Note that the equilibrium x = y = 0 is a weak focusonly if β2 = 0. In this case, the first Lyapunov numberof system (1) has the form [5]

Thus, the synthesized system is always nonconser�vative since α2 < 0.

Example. Consider system (1) with a1 = b1 = β1 = 1,

c1 = α1 = 0, a2 = β2 – 1, b2 = , c2 = , α2 = –2, and

the initial data x(0) = y(0) = – . When passing to sys�

tem (2), we have ε = and u(0) = 0. For β2 = 0.22, sys�

tem (1) has a limit cycle with these initial data (Fig. 4).

Note that the approach proposed can be extendedto various spaces of initial data and parameter spaces.

REFERENCES

1. V. I. Arnold, Experimental Mathematics (Fazis, Mos�cow, 2005) [in Russian].

2. G. A. Leonov, Vestn. Sankt�Peterburg. Gos. Univ.,Ser. 1: Mat. Mekh. Astron., No. 4, 48–78 (2006).

3. G. A. Leonov, Int. J. Bifurcation Chaos 18, 877–884(2008).

4. G. A. Leonov, Dokl. Phys. 54, 238–241 (2009) [Dokl.Akad. Nauk 426, 47–50 (2009)].

5. G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova,Vestn. St. Petersburg. Gos. Univ., Ser. 1: Mat. Mekh.Astron., No. 3, 25–61 (2008).

Φ x( ) x 1+( )2x2 α2x x 1+( )2– c2x4.–=

2 c2 1+( )

ε2������������������

β2 2c2 3–2 c2 1+( )

ε2������������������,⎝ ⎠

⎛ ⎞ ,∈

β2 β2

β2

L1 0( ) π

4 α2–( )3/2������������������ 2 α2–( ).–=

β212��

12��

12��

u

x0

–1

–1 + ε

–1

–0.6–2

–0.4 0–0.2 0.2 0.4 0.6 0.8 1.0 1.2

0

1

2

Fig. 4. Limit cycle of the synthesized system.

u

–1

–1 + ε 0 x

Fig. 2. Qualitative behavior of the trajectory for Ψ(x) < 0(case 1). Fig. 3. Qualitative behavior of the trajectory for Ψ(x) < 0

(case 2).