ISSN 1064�5624, Doklady Mathematics, 2010, Vol. 81, No. 1, pp. 31–33. © Pleiades Publishing, Ltd., 2010.Original Russian Text © G.A. Leonov, 2010, published in Doklady Akademii Nauk, 2010, Vol. 430, No. 2, pp. 157–159.

31

For nonlinear dynamical systems, necessary andsufficient conditions for the global asymptotic behav�ior of their solutions can rarely be formulated. In[1, 2], a method for the asymptotic integration of tra�jectories of the Liénard equation was developed. Basedon this method, necessary and sufficient conditionsfor the global boundedness of quadratic two�dimen�sional systems can be derived and, additionally, exist�ence criteria for limit cycles can be formulated.

Consider the quadratic system

(1)

where ai, b1, c1, αi, and βi are real constants.Proposition 1 [2–4]. Without loss of generality, it can

be assumed that c1 = 0.This result is proved by making the linear substitu�

tions x = x1 + νy1 and y = y1, where ν solves a third�degree equation [2–4].

Proposition 2. Let c1 = 0 and β1 ≠ 0. Then, withoutloss of generality, it can be assumed that α1 = 0.

This proposition is proved by making the linear

substitutions x = x1 and y = y1 – .

Proposition 3. Let c1 = 0, α1 = 0, a1 ≠ 0, b1 ≠ 0, β1 ≠ 0.Then, without loss of generality, it can be assumed thatc1 = α1 = 0 and a1 = b1 = β1 = 1.

This proposition is proved by making the substitu�tions

Using these results, we assume in what follows that

x· a1x2 b1xy c1y2 α1x β1y,+ + + +=

y· a2x2 b2xy c2y2 α2x β2y,+ + + +=

α1x1

β1

���������

xβ1

b1

����x1, ya1β1

b12

��������y1, tb1

a1β1

��������t1.= = =

(2)

Proposition 4. The half�plane

is positively invariant.

This result follows from the fact that (t) = x(t)2 = 1if x(t) = –1.

Note that, under assumptions (2), system (1) canbe reduced to the Liénard system [2–4]

(3)

Here, the trajectories of systems (1) and (2) are related by

(4)

and

It is easy to see that transformation (4) reduces sys�tem (3) to the form

Theorem 1. A necessary condition for any solution ofsystem (1) with initial data from Γ to be bounded on(0, +∞) is that c2 ∈ (0, 1).

The proof sketch of this theorem is as follows. Notethat

c1 α1 0, a1 b1 β1 1, c2 0,≠= = = = =

c2 1, c2 b2 a2.–≠–≠

Γ x 1–> y, R1∈{ }=

x·

x· u, u· f x( )u– g x( ).–= =

u y x2

x 1+���������+⎝ ⎠

⎛ ⎞ x 1+ q, q c2,–= =

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.–

x· x 1+ q

x 1+( )������������� x2 xy y+ +( ),=

y· x 1+ q

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

Φ x( )x 1–→

lim c2.–=

Necessary and Sufficient Conditions for the Boundedness of Solutions to Two�Dimensional Quadratic Systems

in a Positively Invariant Half�PlaneCorresponding Member of the RAS G. A. Leonov

Received August 18, 2009

DOI: 10.1134/S1064562410010102

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

MATHEMATICS

32

DOKLADY MATHEMATICS Vol. 81 No. 1 2010

LEONOV

Therefore, for c2 < 0, there exists a sufficiently smallε > 0 such that, for x(0) ∈ (–1, –1 + ε) and u(0) < 0, wehave

Here, T is either +∞ or a constant for which x(T) = –1.Combining this with (4) yields

Now let c2 > 1. If c2 > b2 – a2, then, for sufficientlylarge x, we have g(x) < 0. Therefore, system (3) hassolutions unbounded on (0, +∞) with u(0) > 0 and suf�ficiently large x(0). Moreover, x(t) → +∞ as t → +∞.

If c2 < b2 – a2, then Eq. (3) is written as the first�order equation

. (5)

Changing to the variable z = (x + 1)q + 1 gives

(6)

Obviously,

It follows that the point F = 0, z = 0 is a saddle for Eq. (6).Consequently, for sufficiently large initial data F(1),the solution F(z) is positive on [0, 1]. This solution isassociated with that of system (3) with the initial datax(0) = 0, u(0) = F(1), for which

as t → +∞. The proof of Theorem 1 is complete.Theorem 2. The solutions of system (1) in the half�

plane Γ are bounded on (0, +∞) if and only if c2 ∈ (0, 1),c2 < b2 – a2, and 2c2 > b2 + 1 or 2c2 ≤ b2 + 1 and 4a2(c2 –1) > (b2 – 1)2.

The proof sketch of Theorem 2 is as follows. Onceagain, consider the substitution z = (x + 1)q + 1 andEq. (6) and the substitution z = (x + 1)q – 1 and theequation

u t( ) 0, x t( ) 1– –1 ε+,( ), t∀ 0 T,[ ].∈ ∈<

y t( )t T→

lim ∞.–=

F dFdx����� f x( )F g x( )+ + 0=

FdFΨ z

1q 1+���������

1–⎝ ⎠⎛ ⎞

q 1+( )z2

q 1+���������

������������������������FdzΦ z

1q 1+���������

1–⎝ ⎠⎛ ⎞

q 1+( )z4

q 1+���������

������������������������zdz+ + 0.=

Φ z1

q 1+���������

1–⎝ ⎠⎛ ⎞

q 1+( )z4

q 1+���������

������������������������z 0→

limb2 a2– c2–

q 1+( )��������������������� 0.<=

u t( ) 0, t∀ 0, x t( ) +∞→≥>

(7)

Equations (6) and (7) are well suited for the asymp�totic integration of their solutions for large initial dataand for q ∈ (–1, 0) since

It is easy to see that, for b2 < c2 + a2 and sufficientlylarge z0, the solution F(z) of Eq. (6) with initial dataF(z0) > 0 is positive on [z0, +∞]. Therefore, system (1)in Γ has solutions that are unbounded on (0, +∞).

Let b2 > c2 + a2. Then, taking into account that, forlarge initial conditions F(1) = R, the solutions ofEqs. (6) and (7) are close to those of the equations

where z ≥ 1, we find that, for 2c2 > b2 + 1, the trajecto�ries of system (3) lie as shown in the figure [1]. Thesetrajectories are located in a similar manner if 2c2 ≤ b2 + 1and 4a2(c2 – 1) > (b2 – 1)2. Therefore, the solutions ofsystem (1) are bounded in Γ.

If 2c2 ≤ b2 + 1 and 4a2(c2 – 1) > (b2 – 1)2, then it iseasy to see that there are sufficiently large z0 and ρ0 forwhich the solution of Eq. (6) with the initial dataF(z0) = ρ0 is positive on [z0, +∞]. This means that sys�tem (1) has unbounded solutions in Γ. The proof ofTheorem 2 is complete.

Based on the qualitative behavior of solutions tosystem (3) as shown in the figure, we can obtain anexistence condition for limit cycles of system (1).

Theorem 3. Suppose that c2 ∈ (0, 1), b2 > a2 + c2,2c2 > b2 + 1; and g(x) has the only zero x = 0 on theinterval (–1, +∞), which is associated with the unstableequilibrium x = x1, u = 0 of system (3).

Then system (1) has a limit cycle in the half�plane Γ.

FdFΨ z

1q 1–���������

1–⎝ ⎠⎛ ⎞

q 1–������������������������Fdz

Φ z1

q 1–���������

1–⎝ ⎠⎛ ⎞

q 1–������������������������zdz+ + 0.=

Ψ z1

q 1+���������

1–⎝ ⎠⎛ ⎞

z2

q 1+���������

������������������������z +∞→

lim 2c2 b2– 1,–=

Φ z1

q 1+���������

1–⎝ ⎠⎛ ⎞

z4

q 1+���������

������������������������z +∞→

lim b2 a2– c2,–=

Ψ z1

q 1–���������

1–⎝ ⎠⎛ ⎞

z +∞→

lim 1 2c2,+=

Φ z1

q 1–���������

1–⎝ ⎠⎛ ⎞

z +∞→

lim c2.–=

FdF2c2 b2 1–+( )

q 1+( )��������������������������Fdz

b2 a2– c2–( )q 1+( )

�������������������������zdz+ + 0,=

FdF1 2c2+q 1–( )

�������������Fdzc2

q 1–( )�������������zdz–+ 0,=

–1

u

x

Figure.

DOKLADY MATHEMATICS Vol. 81 No. 1 2010

NECESSARY AND SUFFICIENT CONDITIONS FOR THE BOUNDEDNESS OF SOLUTIONS 33

Theorem 4. Suppose that c2 ∈ (0, 1); b2 > a2 + c2; 2c2 ≤b2 + 1; 4a2(c2 – 1) > (b2 – 1)2; and g(x) has only twozeros x = 0 and x = x1 ∈ (–∞, –1), which are associatedwith the unstable equilibria x = x1, u = 0 and x = 0, u = 0of system (3).

Then system (1) has two limit cycles, of which one liesto the left of the straight line {x = –1, y ∈ R1} and theother lies to the right of this line.

The following results are useful in the verificationof the conditions of Theorems 3 and 4 [2].

Proposition 5. Let β2 = 0. Then g(x) has only twozeros x = 0 and x = x1 ∈ (–∞, –1) if and only if α2 < λ,where λ is the minimal root of the equation

Proposition 6. Let β2 = 0. Then x = x1, u = 0 is anunstable equilibrium if and only if

where

Note that β2 = 0 and α2 < 0 are necessary condi�tions for x = y = 0 to be a weak focus of system (1).

In this case, applying local bifurcation theory [1, 2,5, 6] together with Theorem 4, we can single outclasses of systems of form (1) that have four limitcycles.

REFERENCES

1. G. A. Leonov, Dokl. Phys. 54, 238–146 (2009) [Dokl.Akad. Nauk 426, 47–50 (2009)].

2. G. A. Leonov, N. V. Kuznetsov, and E. V. Kudryashova,Vestn. Sankt�Peterburg. Gos. Univ., Ser. 1, No. 3, 25–61 (2008).

3. G. A. Leonov, Differ. Equations Dyn. Syst. 5 (3/4),289–297 (1998).

4. G. A. Leonov, Vestn. S.�Peterburg. Gos. Univ., Ser. 1,No. 4, 48–78 (2006).

5. J. Li, Int. J. Bifurcation Chaos 13 (1), 47–106 (2003).6. P. Yu and G. Chen, Nonlinear Dyn. 51, 409–427

(2008).

4c2λ3– 27c22 12a2 18b2–( )c2 b2

2–+( )λ2+

+ 2 a2b22

2b23– 9a2b2c2 6a2

2c2–+( )λ a22b2

2– 4a23c2+ = 0.

α21

p 1+( )2��������������� –a2p p 1+( )2 b2p2 p 1+( ) c2p3–+( ),<

p2 b2+

2c2 b2– 1–���������������������� .=