lyapunov quantities and limit cycles of two-dimensional dynamical systems. analytical methods and...

ISSN 1560-3547, Regular and Chaotic Dynamics, 2010, Vol. 15, Nos. 2–3, pp. 354–377. c© Pleiades Publishing, Ltd., 2010.

L.P. SHILNIKOV–75Special Issue

Lyapunov Quantities and Limit Cyclesof Two-dimensional Dynamical Systems.

Analytical Methods and Symbolic Computation

G. A. Leonov* and O. A. Kuznetsova**

Faculty of Mathematics and Mechanics,St. Petersburg State University

Universitetsky pr. 28, St.Petersburg 198504, RussiaReceived November 16, 2009; accepted March 1, 2010

Abstract—In the present work the methods of computation of Lyapunov quantities andlocalization of limit cycles are demonstrated. These methods are applied to investigation ofquadratic systems with small and large limit cycles. The expressions for the first five Lyapunovquantities for general Lienard system are obtained. By the transformation of quadratic systemto Lienard system and the method of asymptotical integration, quadratic systems with largelimit cycles are investigated. The domain of parameters of quadratic systems, for which fourlimit cycles can be obtained, is determined.

MSC2000 numbers: 34C07DOI: 10.1134/S1560354710020218

Key words: Hilbert’s 16-th problem, small and large limit cycles, Lyapunov quantities, symboliccomputation, localization of limit cycles, quadratic system, Lienard system.

The study of limit cycles of two-dimensional dynamical systems was stimulated by as purelymathematical problems (Hilbert’s sixteenth problem, the center-and-focus problem) as manyapplied problems such as the investigation of oscillations of electronic generators and electricalmachines, the dynamics of populations, and the safe and dangerous boundaries of stability (see,e.g., [1–4] and many others).

In 1901 Hilbert, in his famous 16-th problem, posed a problem of the analysis of relativedisposition and the number of limit cycles for two-dimensional polynomial systems. By now, inmore than a century, within the framework of investigation of this problem numerous theoreticaland numerical results were obtained (see bibliography in [5]). But the problem is still far from beingresolved even for the class of quadratic systems.

In the works of Bautin [6, 7] by the method of Lyapunov quantities and small perturbations isshown that in quadratic system three small limit cycles could be bifurcated from one critical point.Then in the works [8, 9] by the global analysis of a class of quadratic systems with three small limitcycles in the neighborhood of one equilibrium the conditions of existence of large limit cycle roundanother equilibrium were obtained.

At present, numerous publications are devoted to the further development of the methods ofglobal analysis, the computation of Lyapunov quantities and the search of the largest number oflimit cycles for different classes of two-dimensional polynomial systems (see, e.g., [10–17] and manyothers).

In the present work various methods of computation of Lyapunov quantities are consideredand symbolic expressions of the first five Lyapunov quantities for general Lienard system, whichare used for investigation of small limit cycles, are obtained. Following the work [18], for Lienardsystem with discontinuous right-hand side, the method of asymptotical integration, permitting one

*E-mail: [email protected]**E-mail: o a [email protected]

354

LYAPUNOV QUANTITIES AND LIMIT CYCLES 355

to localize large limit cycles, is described. By the procedure of reduction to Lienard system [19, 20]and the method of asymptotical integration simple conditions of existence of one and two largelimit cycles in quadratic systems are obtained.

The method of disturbance of Lyapunov quantities together with the method of asymptoticalintegration allow one to obtain the conditions of existence of four limit cycles in quadratic systems:two large limit cycles in the case of a weak focus of second order and one large limit cycle in the caseof a weak focus of third order. The conditions obtained here have very simple form and generalizewidely known theorem of Shi [9].

1. COMPUTATION OF LYAPUNOV QUANTITIES AND SMALL LIMIT CYCLES

The computation of Lyapunov quantities is one of the central problems in considering small limitcycles in the neighborhood of equilibrium of two-dimensional dynamical systems (see, e.g, [6, 21–27] and others). Lyapunov quantities are also substantial in considering the important problem ofengineering mechanics on the behavior of dynamical system in the case when its parameters areclose to the boundary of stability domain. Followed by the work of Bautin, one differs the safeand dangerous boundaries, a slight shift of which yields small (invertible) or non-invertible changesof system behavior, respectively. Such changes correspond, for example, to scenario of “soft” and“hard” excitations of oscillations, which was considered by Andronov [1].

Consider a system of two autonomous differential equations

dx

dt= f10x + f01y + f(x, y),

dy

dt= g10x + g01y + g(x, y).

(1.1)

Suppose x, y ∈ R, the conditions

f(0, 0) = 0, g(0, 0) = 0

are valid, and the matrix ⎛⎝ f10 f01

g10 g01

⎞⎠has two purely imaginary eigenvalues. The functions f(·, ·) and g(·, ·) are sufficiently smooth andtheir expansions begin with the terms of at least the second order, namely

f(x, y) =n∑

i+j=2

fijxiyj + o

((|x| + |y|)n

)= fn(x, y) + o

((|x| + |y|)n

),

g(x, y) =n∑

i+j=2

gijxiyj + o

((|x| + |y|)n

)= gn(x, y) + o

((|x| + |y|)n

).

(1.2)

Without loss of generality, one can assume that

f10 = 0, f01 = −1, g10 = 1, g01 = 0.

Then system (1.1) takes the form

dx

dt= −y + f(x, y),

dy

dt= x + g(x, y).

(1.3)

At present, there exist various methods for computation of Lyapunov quantities and computerrealizations of these methods. This makes it possible to find Lyapunov quantities in the formof symbolic expressions, depending on expansion coefficients of the right-hand sides of equationsof system (1.3) (see, e.g., [12, 13, 17, 19, 23, 24, 26–32]). These methods differ in complexity of

REGULAR AND CHAOTIC DYNAMICS Vol. 15 Nos. 2–3 2010

356 LEONOV, KUZNETSOVA

algorithms, compactness of obtained symbolic expressions, and a space in which computations areperformed.

While the general expressions for the first and second Lyapunov quantities (in terms of expansioncoefficients fij, gij of f, g in the original space) were obtained in the 40-50s of last century [6, 33],the third Lyapunov quantity was computed only in 2008 [32, 34]. The expression for this quantityoccupies more than four pages. While the expression for the fourth Lyapunov quantity, obtainedfirst by the authors, occupies more than 45 pages.

1.1. Classical Poincare method

The first method for the computation of Lyapunov quantities was suggested by Poincare [28]and Lyapunov [29]. This method consists in sequential obtaining time-independent integrals for theapproximations of system. Since the expression

V2(x, y) =(x2 + y2)

2

is an integral of the first approximation of system (1.3) and the system is sufficiently smooth, in acertain small neighborhood of zero point we can sequentially construct Lyapunov function of theform

V (x, y) =x2 + y2

2+ V3(x, y) + . . . + Vk(x, y). (1.4)

Here Vk(x, y) =∑

i+j=k

Vi,jxiyj are homogeneous polynomials with the unknown coefficients

{Vi,j}i+j=k, i,j�0 and k � n + 1. These coefficients can be found sequentially (via the coefficients ofexpansions of functions f and g and the coefficients {Vi,j}i+j<k, obtained at the previous steps ofiteration) in such a way that the derivative of V (x, y) by virtue of system (1.3)

V (x, y) =∂V (x, y)

∂x(−y + f(x, y)) +

∂V (x, y)∂y

(x + g(x, y)) (1.5)

takes the form

V (x, y) = w1(x2 + y2)2 + w2(x2 + y2)3 + . . . + o((|x| + |y|)k+1

). (1.6)

Here the coefficients wi depend only on the coefficients of expansions of f and g.

Sequentially determining the coefficients of the form Vk for k = 3, . . . (for which purpose at eachstep it is necessary to solve a system of (k + 1) linear equations), from (1.5) and (1.6) we can obtainthe coefficient wm being the first not equal to zero coefficient:

V (x, y) = wm(x2 + y2)m+1 + o((|x| + |y|)2m+2

).

The coefficient wm is usually called [13] a Poincare–Lyapunov constant (2πwm is m-th Lyapunovquantity). Under the additional conditions [31]

V2m,2m+2 + V2m+2,2m = 0, V2m,2m = 0

at the k-th step of iteration the coefficients {Vi,j}i+j=k are uniquely determined.

Note that one of the well-known variants of classical Poincare method (see, e.g, [35]) consists inthe computation of Lyapunov quantities in complex variables, for example, u = x + iy, v = x − iy.



1.2. Method of Computation in Polar Coordinates

Another approach to the computation of Lyapunov quantities involves the finding the approxi-mations of a solution of system in polar coordinates [29].

Having performed the changes x = r cos φ, y = r sin φ, we obtain system (1.3) in the polarcoordinates r and φ. It can be represented as

dr

dφ=

f(r cos φ, r sinφ) cos φ + g(r cos φ, r sin φ) sin φ

1 − f(r cos φ, r sin φ) sin φ

r+

g(r cos φ, r sin φ) cos φ

r

= R(φ, r).

Here R(0, 0) = 0 and for sufficiently small r the function R(φ, r) is smooth (due to the smoothnessof f and g). Consider the representation

dr

dφ= rR1(φ) + r2R2(φ) + . . . + rnRn(φ) + o(rn).

Substitute into this equation the following representation of solution with the initial data (0, r0):

r(φ, 0, r0) = u1(φ)r0 + u2(φ)r20 + . . . + un(φ)rn

0 + o(rn0 ) , u1(0) = 1, ui>1(0) = 0. (1.7)

Using this substitution, we obtain the equations for sequential calculation of ui(φ)

u1(φ) = R1(φ)u1(φ),

u2(φ) = R1(φ)u2(φ) + R2(φ)u21(φ),

u3(φ) = R1(φ)u3(φ) + R2(φ)u1(φ)u2(φ) + R3(φ)u31(φ),

· · ·un(φ) = R1(φ)un(φ) + . . . + Rn(φ)un

1 (φ).

(1.8)

By the solution for φ = 2π we have

r = r(2π, 0, r0) = α1r0 + α2r20 + α3r

30 + . . . + αnrn

0 + o(rn0 ) , αi = ui(2π),

where αi are called the focus values. Here α1 = 1 and if α2 = . . . = α2m = 0, then α2m+1 is calledm-th Lyapunov quantity.

1.3. Direct Method for Computation of Lyapunov Quantities (in Euclidean Coordinates and inthe Time Domain)

In the works [34, 36] a new method for computation of Lyapunov quantities is considered. It isbased on the construction of approximations of solution (as a finite sum in powers of initial datum)in the original Euclidean coordinates and in the time domain. The advantages of this method arein its ideological simplicity and obviousness. This approach can also be applied to the problem ofdistinguishing of isochronous center [13, 17, 30, 37] since it permits one to find an approximationof “turn” time of trajectory as a function of initial data.

Let x(t, x(0), y(0)), y(t, x(0), y(0)) be a solution of system (1.3) with the initial data

x(0) = 0, y(0) = h. (1.9)

Denotex(t, h) = x(t, 0, h), y(t, h) = y(t, 0, h).

If smoothness condition (1.2) is satisfied, then one can consider x(t, h) and y(t, h) in the followingform

x(t, h) = xhn(t, h) + o(hn) =n∑

k=1

xhk(t)hk + o(hn),

y(t, h) = yhn(t, h) + o(hn) =n∑

k=1

yhk(t)hk + o(hn).

(1.10)



Here xh1(t, h) = xh1(t)h = −h sin(t), yh1(t, h) = yh1(t)h = h cos(t) and xhk(t), yhk(t) can be foundsequentially by the following lemma.Lemma. Consider the system

dxhk(t)dt

= −yhk(t) + ufhk(t),

dyhk(t)dt

= xhk(t) + ughk(t).

(1.11)

For solutions of system (1.11) with the initial data

xhk(0) = 0, yhk(0) = 0 (1.12)

the equations

xhk(t) = ughk(0) cos(t) + cos(t)

t∫0

cos(τ)((ug

hk(τ))′ + ufhk(τ)

)dτ

+ sin(t)

t∫0

sin(τ)((ug


)dτ − ug

hk(t),

yhk(t) = ughk(0) sin(t) + sin(t)

t∫0

cos(τ)((ug


)dτ

− cos(t)

t∫0

sin(τ)((ug


)dτ

(1.13)

are valid. Here ufhk(t), ug

hk(t) can be found by substitution x(t, h) = xhk−1(t, h) + o(hk−1), y(t, h) =yhk−1(t, h) + o(hk−1) into f and g

f(xhk−1(t, h) + o(hk−1), yhk−1(t, h) + o(hk−1)) = ufhk(t) + o(hk),

g(xhk−1(t, h) + o(hk−1), yhk−1(t, h) + o(hk−1)) = ughk(t) + o(hk).

Consider the “turn” time T (h)(the time of first crossing the halfline {x = 0, y > 0} by the

solution (x(t, h), y(t, h)))

for the initial datum h ∈ (0,H] and let T (0) = 2π. One can prove thatT (h) is n times differentiable function. So we have

T (h) = 2π + ΔT = 2π +n∑

k=1

Tkhk + o(hn), (1.14)

where Tk =1k!

dkT (h)dhk

(so-called period constants [17]).

Substituting relation (1.14) for t = T (h) in the right-hand side of the first equation of (1.10) anddenoting the coefficients of hk by xk, we obtain the series x(T (h), h) in terms of powers of h:

x(T (h), h) =n∑

k=1

xkhk + o(hn). (1.15)

In order to express the coefficients xk by the coefficients Tk we assume that in the first equationof (1.10) t = 2π + τ . Then

x(2π + τ, h) =n∑

k=1

xhk(2π + τ)hk + o(hn). (1.16)



Here by smoothness condition (1.2) one can consider

xhk(2π + τ) = xhk(2π) +n∑

m=1

x(m)

hk (2π)τm

m!+ o(τn), k = 1, . . . , n.

Substituting this representation in (1.16) for τ = ΔT (h) and grouping together the coefficients ofthe same exponents h, we obtain

h : 0 = x1 = xh1(2π),

h2 : 0 = x2 = xh2(2π) + x′h1(2π)T1,

· · ·hn : 0 = xn = xhn(2π) + . . .

Hence, we sequentially find Tj .A similar procedure is applied for the determination of coefficients yk of the expansion

y(T (h), h) =n∑

k=1

ykhk + o(hn). (1.17)

Substitute the following representation

yhk(2π + ΔT (h)) = yhk(2π) +n∑

m=1

y(m)

hk (2π)ΔT (h)m

m!+ o((ΔT (h))n), k = 1, . . . , n

in the expression

y(2π + ΔT (h), h) =n∑

k=1

yhk(2π + ΔT (h))hk + o(hn). (1.18)

Note that here the coefficients Tj (from expansion of T (h) (1.14)) are known. Then equating thecoefficients of the same exponents h, we sequentially determine yi=1,...,n.

Here y1 = 1 (corresponds to the exponential growth [38]) and for n = 2m + 1, if yk = 0 fork = 2, . . . , 2m, then y2m+1 coincides with m-th Lyapunov quantity:

Lm = y2m+1.

Realization of this method in MatLab symbolic computation package can be found in the work[39].

In addition, for the computation of Lyapunov quantities, many other methods, involving, forexample, the reduction of system to normal form (see, e.g., [14, 24, 27]), and special analytical-numerical methods [40] were developed.

1.4. Lyapunov Quantities for Lienard System

Consider system (1.1), where g10 is arbitrary number. Suppose in (1.1) the following

f10 = 0, f01 = −1, f(x, y) ≡ 0, g01 = 0, g(x, y) = gx1(x)y + gx0(x)

andgx1(x) = g11x + g21x

2 + . . . , gx0(x) = g20x2 + g30x

3 + . . .

Then we obtain Lienard system in general form

x = −y, y = g10x + gx1(x)y + gx0(x). (1.19)

Note, in order that the matrix of linear approximation of the system has 2 purely imaginaryeigenvalues, the following condition

g10 > 0 (1.20)



must be satisfied. Since the methods for computation of Lyapunov quantities were described forthe systems with simple linear part (g10 = 1), we need to transform system (1.19) to required form,for which purpose we perform the following change of variables

t →√

1g10

t, x →√

1g10

x.

This change of variables does not change y (see (1.19)), namely

y = −

√1

g10dx√

1g10

dt= −dx

dt

and, therefore, the expansion (1.17) in new variables is just as before. It means that expressionsfor Lyapunov quantities for the systems before and after the change of variables coincide.

Notice that the expressions for the first five Lyapunov quantities for Lienard system with simplelinear part were computed first in the work [41]. For Lienard system of general form with provisionfor (1.20), the expressions for Lyapunov quantities Li=1,...,5 are given below.

For the first Lyapunov quantity we have

L1 =π

4 (g10)5/2

(g21 g10 − g11 g20) .

If g21 = g11 g20

g10, then we obtain L1 = 0 and

L2 =−π

24 (g10)9/2

(3 g11 g10 g40 − 3 g41 g10

2 + 5 g20 g10 g31 − 5 g30 g11 g20

).

If

g41 =3 g11 g10 g40 + 5 g20 g10 g31 − 5 g30 g11 g20

3g102

,

then we obtain L2 = 0 and

L3 = −π

576(g10)15/2 (63 g40 g103g31 − 70 g20

3g10 g31 − 105 g50 g102g11 g20 + 105 g20 g10

3g51 − 45 g61 g104

−105 g30 g102g20 g31 − 63 g30 g10

2g11 g40 + 105 g302g10 g11 g20 + 70 g20

3g30 g11 + 45 g11 g103g60).

If g61 is determined from equation L3 = 0, then we obtain

L4 = −π

17280(g10)21/2 (945 g11 g105g80 + 2835 g20 g71 g10

5 − 4620 g203g51 g10

3 + 3080 g205g31 g10

+1701 g302g11 g40 g10

3 + 8820 g30 g203g31 g10

2 − 1215 g30 g11 g60 g104 − 2835 g70 g10

4g11 g20

−2835 g30 g20 g51 g104 − 1701 g30 g40 g31 g10

4 − 8820 g302g10 g20

3g11 + 4620 g203g50 g10

2g11

+1701 g40 g105g51 + 5670 g30 g50 g10

3g11 g20 + 4158 g30 g102g20

2g40 g11 − 945 g81 g106

−3080 g205g30 g11 + 2835 g30

2g20 g31 g103 − 2835 g30

3g102g11 g20 − 2835 g50 g20 g31 g10

4

−1701 g50 g11 g40 g104 − 4158 g20

2g40 g31 g103 + 1215 g60 g10

5g31).

If g81 is determined from equation L4 = 0, then we obtain

L5 = π3110400(g10)27/2 (−4158000 g30

3g203g10

2g11 − 5613300 g205g30

2g10 g11 + 200475 g30 g60 g31 g106

−935550 g30 g50 g20 g31 g105 − 561330 g30 g50 g11 g40 g10

5 − 486486 g402g30 g10

4g11 g20

+280665 g50 g40 g31 g106 − 2402400 g20

5g51 g103 + 467775 g50 g20 g51 g10

6 − 1601600 g207g30 g11

+579150 g202g60 g31 g10

5 + 155925 g30 g11 g80 g106 − 200475 g30

2g11 g60 g105

−2522520 g204g40 g31 g10

3 + 5613300 g205g30 g31 g10

2 + 467775 g303g20 g31 g10

4

−1351350 g203g10

4g70 g11 + 486486 g20 g402g31 g10

5 + 1621620 g202g40 g51 g10

5

+280665 g30 g40 g51 g106 − 1621620 g20

2g40 g50 g104g11 − 3118500 g20

2g30 g40 g31 g104

+1403325 g302g50 g10

4g11 g20 + 2522520 g204g30 g10

2g40 g11 + 3118500 g302g10

3g202g40 g11



−3014550 g203g30 g51 g10

4 − 467775 g302g20 g51 g10

5 + 280665 g303g11 g40 g10

4 + 467775 g70 g20 g31 g106

+2402400 g205g10

2g50 g11 − 467775 g304g10

3g11 g20 + 127575 g10 1 g108 − 935550 g30 g10

5g70 g11 g20

+467775 g30 g20 g71 g106 − 280665 g30

2g40 g31 g105 − 2113650 g20

3g50 g31 g104 + 280665 g70 g11 g40 g10

6

+4158000 g203g30

2g31 g103 + 5128200 g20

3g30 g50 g103g11 − 579150 g30 g10

4g202g60 g11

+200475 g50 g11 g60 g106 − 467775 g50

2g105g11 g20 + 467775 g90 g10

6g11 g20 − 200475 g60 g107g51

−280665 g40 g71 g107 + 1351350 g20

3g71 g105 − 127575 g11 g10

7g10 0 − 467775 g20 g91 g107

−155925 g80 g107g31 + 1601600 g20

7g31 g10).

If L1,...,n−1 = 0 and Ln �= 0, then using the well-known technique of Bautin, we can construct nsmall limit cycles by small disturbances of coefficients of the system [7, 31, 42].

For example, for quadratic system this technique makes it possible to construct 3 small limitcycles if the coefficients of the system are chosen so that L1,2 = 0 and L3 �= 0. If the coefficientsare chosen so that L1,2,3 = 0, then L4,5,... = 0. This implies that for quadratic system, more than 3small cycles cannot be obtained by this technique.

2. LIMIT CYCLES OF QUADRATIC SYSTEMS

Two-dimensional quadratic systems describe many real systems and play an important role inpure mathematics. Therefore many publications are devoted to these systems. Investigation of limitcycles in quadratic systems is one of the important mathematical problems. The surveys [5, 14, 16]contain numerous references devoted to this problem.

In the last decades together with analytical results for these systems, the computer-assistedproofs and computer calculations play an important role. However the obtaining of simple analyticestimates for two-dimensional quadratic dynamical systems with limit cycles is of great importance.

Here we use the described above method of computation of Lyapunov quantities for investigationof small limit cycles together with the method of asymptotical integration [18] for investigation oflarge limit cycles.

A two-dimensional quadratic system may be written as

x = a1x2 + b1xy + c1y

2 + α1x + β1y,

y = a2x2 + b2xy + c2y

2 + α2x + β2y,(2.1)

where aj , bj , cj , αj , βj are real numbers.

System (2.1) can be reduced to more convenient form. For this purpose we can use the followingsimple assertions.

Proposition 1. Without loss of generality, one can assume that c1 = 0.

For the proof of this Proposition the linear change x → x + νy, y → y is performed. Here ν is areal solution of the following equation

−a2ν3 + (a1 − b2)ν2 + (b1 − c2)ν + c1 = 0. (2.2)

This equation always has a real solution if a2 �= 0.If a2 = 0, then after the change of variables x → y, y → x we obtain c1 = 0 and the statement

of proposition is proved.

Proposition 2. Let be c1 = 0, β1 �= 0. Then, without loss of generality, one can assume thatα1 = 0.

The proof of the assertion is based on the use of the following linear change x → x, y →y − α1x/β1.

Proposition 3. Let be c1 = 0, α1 = 0, a1 �= 0, b1 �= 0, β1 �= 0. Then, without loss of generality,one can assume that c1 = α1 = 0, a1 = b1 = β1 = 1.



The proof of this assertion is by the following linear change

x → β1

b1x, y → a1β1

b21

y, t → b1

a1β1t.

We assume further that

c1 = α1 = 0, a1 = b1 = β1 = 1, c2 �= 0, c2 �= −1, c2 �= b2 − a2.

Then in place of system (2.1) we further consider the system

x = x2 + xy + y,

y = a2x2 + b2xy + c2y

2 + α2x + β2y.(2.3)

2.1. Transformation of Two-Dimensional Quadratic System to Lienard System

First, we prove the following

Proposition 4. The half-plane

Γ = {x > −1, y ∈ R1}

is positively invariant with regard to system (2.3).

This assertion results from the fact that for x(t) = −1 the relation x(t) = x(t)2 = 1 is valid.

System (2.3) can be reduced to Lienard system [19, 20, 43]

x = y, y = −f(x)y − g(x) (2.4)

with the help of the following change of variables

(y +x2

x + 1)|x + 1|q → y, x → x, q = −c2, (2.5)

and

f(x) = (A1x2 + A2x + A3)|x + 1|q−2,

g(x) = (B1x4 + B2x

3 + B3x2 + B4x)

|x + 1|2q

(x + 1)3.

Here the new coefficients Ai, Bi have the form

A1 = −b2 + 2c2 − 1, A2 = −2 − b2 − β2, A3 = −β2,

B1 = −a2 + b2 − c2, B2 = −2a2 + b2 − α2 + β2, B3 = −a2 − 2α2 + β2, B4 = −α2.

It is easily seen that by transformation reverse to (2.5) system (2.4) can be represented as

x = (x2 + xy + y)|x + 1|q(x + 1)

,

y = (a2x2 + b2xy + c2y

2 + α2x + β2y)|x + 1|q(x + 1)

.

(2.6)

By Proposition 4 this system is equivalent to system (2.3) on the left and right of line {x = −1}.Investigation of qualitative behavior of solutions of system (2.4) makes it possible to obtain the

existence conditions for limit cycles of system (2.3). (Note that to stable (unstable) limit cycleof system (2.4) on the right of the line {x = −1} corresponds stable (unstable) cycle of system(2.3) there, while to stable (unstable) limit cycle of system (2.4) on the left of the line {x = −1}corresponds unstable (stable) cycle of system (2.3) there.)



2.2. Transformation of Lienard System to Special Form

We assume further that in system (2.4) q ∈ (−1, 0). Represent the functions f(x) and g(x) fromsystem (2.4) as

f(x) = (C1(x + 1)2 + C2(x + 1) + C3)|x + 1|q−2,

g(x) =(D1(x + 1)4 + D2(x + 1)3 + D3(x + 1)2 + D4(x + 1) + D5

) |x + 1|2q

(x + 1)3,

whereC1 = −b2 + 2c2 − 1, C2 = b2 − 4c2 − β2,

C3 = 2c2 + 1, D1 = −a2 + b2 − c2,

D2 = 2a2 − 3b2 + 4c2 − α2 + β2,

D3 = −a2 + 3b2 − 6c2 + α2 − 2β2,

D4 = −b2 + 4c2 + β2, D5 = −c2, q = −c2.

(2.7)

System (2.4) is equivalent to the following first order equation

FdF

dx+ f(x)F + g(x) = 0,

which can be represented as1) for x � 0

FdF +1

(q + 1)

(C1 +

C2

(x + 1)+

C3

(x + 1)2

)Fd(x + 1)q+1

+1

(q + 1)

(D1 +

D2

(x + 1)+

D3

(x + 1)2+

D4

(x + 1)3+

D5

(x + 1)4

)× (x + 1)q+1d(x + 1)q+1 = 0;

(2.8)

2) for x ∈ (−1, 0)

FdF +1

(q − 1)(C1(x + 1)2 + C2(x + 1) + C3

)× Fd(x + 1)q−1 +

1(q − 1)

(D1(x + 1)4 + D2(x + 1)3

+D3(x + 1)2 + D4(x + 1) + D5

)(x + 1)q−1d(x + 1)q−1 = 0.

(2.9)

Further we use the change z = (x + 1)q+1 for equation (2.8) and the change z = (x + 1)q−1 for(2.9). In this case equations (2.8) and (2.9) take the form

FdF +1

(q + 1)

(C1 + C2z

− 1q+1 + C3z

− 2q+1

)Fdz

+1

(q + 1)

(D1 + D2z

− 1q+1 + D3z

− 2q+1 + D4z

− 3q+1 +D5z

− 4q+1

)zdz, z � 1;

(2.10)

FdF +1

(q − 1)

(C1z

2q−1 + C2z

1q−1 + C3

)Fdz

+1

(q − 1)

(D1z

4q−1 + D2z

3q−1 + D3z

2q−1 + D4z

1q−1 + D5) zdz, z � 1.

(2.11)

The form of equations (2.10) and (2.11) is “well adapted” to the asymptotical analysis [44–46]of trajectories with sufficiently great initial data. Since the terms

z−k

q+1 and zk

q−1 , k = 1, . . . , 4are infinitely small “at infinity”, neglecting them we can proceed to the analysis of second orderequations with constant coefficients. This scheme is realized below.



2.3. Method of Asymptotical Integration of Trajectories of Lienard System

Fix a certain number δ > 0, consider sufficiently great number R > 0, and introduce the followingdenotations

λ = − C1

2(q + 1), ω =

√4D1(q + 1) − C2

1

2(q + 1).

Then we arrive at the following

Lemma 1. Suppose that conditions

C1 � 0, 4D1(q + 1) > C21 (2.12)

are satisfied. Then for the solution of system (2.4) with the initial data x(0) = 0, y(0) = R thereexists a number T > 0 such that

x(T ) = 0, y(T ) < 0, x(t) > 0, ∀ t ∈ (0, T ),

R exp(

λπ

ω− δ

)< |y(T )| < R exp

(λπ

ω+ δ

).

Proof. System (2.4) is equivalent to equation (2.10) with F (1) = R.

Note that for any (arbitrary small) number ε > 0 there exists a number Z such that

C1 − ε < C1 + C2z− 1

q+1 + C3z− 2

q+1 < C1 + ε, ∀ z � Z,

D1 − ε < D1 + D2z− 1

q+1 + D3z− 2

q+1 + D4z− 3

q+1 + D5z− 4

q+1 < D1 + ε, ∀ z � Z.(2.13)

In addition, for great R on [1, Z] the solution F (z) considered is close to the solution F (z) ofequation

dF

dz+

1(q + 1)

(C1 + C2z

− 1q+1 + C3z

− 2q+1

)= 0, F (1) = R.

Thus, on [1, Z] we have

F (z) = R −z∫

1

(1

(q + 1)(C1 + C2s

− 1q+1 + C3s

− 2q+1 )

)ds + κ(R),

where κ(R) → 0 as R → +∞.It follows that

(1 − κ1(R))R � F (Z) � (1 + κ1(R))R, (2.14)

where κ1(R) → 0 as R → +∞.Obviously, for z � Z we can obtain the estimate F (z) � F1(z), where F1(z) is a solution of the

equation

F1dF1

dz+

(C1 + ε)(q + 1)

F1 +(D1 − ε)(q + 1)

z = 0, F1(Z) = F (Z).

For this purpose it is necessary to consider the curve F1(z) and the following relation on this curve

dF1

dz− dF

dz> 0, ∀ z � Z.

We obtain similarly the estimate F (z) � F2(z), where F2(z) is a solution of the following equation

F2dF2

dz+

(C1 − ε)(q + 1)

F2 +(D1 + ε)(q + 1)

z = 0, F2(Z) = F (Z).

A relative disposition of solutions is shown in Fig. 1. The arrows show the vectors, correspondingto the values dF/dz and to a vector field of system (2.4).



Fig. 1.

Since ε is arbitrary small number, the curves F1(z) and F2(z) are close to the derivative z(t) ofsolution z(t) of the equation

z +C1

(q + 1)z +

D1

(q + 1)z = 0, z(0) = Z, z(0) = F (Z), (2.15)

which can be represented as

z(t) = eλt(c1 sin ωt + c2 cos ωt),

c1 =1ω

(F (Z) − λZ), c2 = Z.

Similar reasoning can be used for the lower half-plane {F � 0, z � 1} (Fig. 2).Here F4(z) � F (z) � F3(z), where F4(z) and F3(z) are negative solutions of the equations

F4dF4

dz+

(C1 − ε)(q + 1)

F4 +(D1 − ε)(q + 1)

z = 0,

F3dF3

dz+

(C1 + ε)(q + 1)

F3 +(D1 + ε)(q + 1)

z = 0

with the initial data F4(z1) = 0, F3(z2) = 0, respectively. Here z1 and z2 are zeros of the functionsF1(z) and F2(z), respectively.

Note that for the small ε the curves F3(z) and F4(z) are also close to the derivative z(t) ofsolution z(t) of equation (2.15).

Recall that in the above equations for the parameter ε the relation ε(Z) → 0 is valid as Z → +∞.For the time of crossing the solution z(t), z(t) with the straight line {z = Z} for t > 0:

z(T ) = Z, z(T ) < 0

we obtain the estimateT =

π

ω+ κ2(R),

where κ2(R) → 0 as R → +∞.In this case we have

−F (Z) exp(

λπ

ω+ κ3(R)

)� z(T ) � −F (Z) exp

(λπ

ω− κ3(R)

), (2.16)

where κ3(R) → 0 as R → +∞.Here F (Z) > 0.Note that for the negative F (Z) (Fig. 2) the following estimate

F (Z) − z(T ) = O(ε)

is valid.



Fig. 2.

Then by relation ε(Z) → 0 as Z → +∞ and estimates (2.14) and (2.16) we obtain for Z � Rthe assertion of Lemma 1.

The estimation of negative F (Z) on [1, Z] is similar to the above estimation of positive F (Z)on this interval.

Take a certain number c < −1.Lemma 2. Let conditions (2.12) be valid. Then for the solution of system (2.4) with the initialdata x(0) = c, y(0) = −R there exists a number T > 0 such that

x(T ) = c, y(T ) > 0, x(t) < c, ∀ t ∈ (0, T ),

R exp(

λπ

ω− δ

)< y(T ) < R exp

(λπ

ω+ δ

).

The proof of Lemma 2 is similar to the proof of Lemma 1.In addition, one can formulate the analogues of Lemmas 1 and 2.

Lemma 3. Suppose that the conditions

D1 > 0, C1 > 0 (2.17)

are satisfied. Then for the solution of system (2.4) with the initial data x(0) = 0, y(0) = R thereexists a number T > 0 such that

x(T ) = 0, y(T ) < 0, x(t) > 0, ∀ t ∈ (0, T ),− δR < y(T ) < 0.

Lemma 4. Let conditions (2.17) be satisfied. For the solution of system (2.4) with the initial datax(0) = c, y(0) = −R there exists a number T > 0 such that

x(T ) = c, y(T ) > 0, x(t) < c, ∀ t ∈ (0, T ),

0 < y(T ) < δR.

The proof of Lemmas 3 and 4 is similar to the proof of Lemma 1.

Lemma 5. Let conditionsC3 > 0, C2

3 > 4D5(q − 1) > 0 (2.18)

be satisfied. Then for the solution of system (2.4) with the initial data x(0) = 0, y(0) = −R thereexists a number T > 0 such that

x(T ) = 0, 0 < y(T ) < δR,

x(t) ∈ (−1, 0), ∀ t ∈ (0, T ).



Proof. Note that system (2.4) is equivalent to equation (2.11) with z � 1 and F (1) = −R.Consider this equivalence in more detail.The transformation U : z = (x + 1)q−1, x ∈ (−1, 0] maps the trajectory x(t), y(t) of system (2.4)

(or the solution F (x)) into the solution F (z) in such a way as this is shown in Fig. 3.

Fig. 3.

With provision for this transformation U a scheme of proof repeats, in substance, the techniquedeveloped for the proof of Lemma 1.

For the further proof of lemma we note the following. For any ε > 0 there exists a number Zsuch that

C3

(q − 1)− ε <

1(q − 1)

(C1z

2q−1 + C2z

1q−1 + C3

)<

C3

(q − 1)+ ε, ∀ z � Z, (2.19)

D5

(q − 1)− ε <

1(q − 1)

(D1z

4q−1 + D2z

3q−1 + D3z

2q−1 + D4z

1q−1 + D5

)<

D5

(q − 1)+ ε, ∀ z � Z.

(2.20)For the great values R the solution F (z) (F (1) = −R) is close to a solution of the following

equation

dF

dz+

1(q − 1)

(C1z

2q−1 + C2z

1q−1 + C3

)= 0, F (1) = −R.

Thus, on [1, Z] we have

F (z) = −R −z∫

1

(1

(q − 1)(C1s

2q−1 + C2s

1q−1 + C3

)ds + κ(R),

where κ(R) → 0 as R → +∞.This implies that

(−1 − κ1(R))R � F (Z) � (−1 + κ1(R))R,

where κ1(R) → 0 as R → +∞.Similarly to the Proof of Lemma 1, from (2.19) and (2.20) we obtain the estimates (Fig. 4)

F1(z) � F (z) � F2(z), F2(Z) = F (Z) = F1(Z),

where F1(z) is a solution of the equation

F1dF1

dz+

(C3

(q − 1)− ε

)F1 +

(D5

(q − 1)+ ε

)z = 0



and F2(z) is a solution of the equation

F2dF2

dz+

(C3

(q − 1)+ ε

)F2 +

(D5

(q − 1)− ε

)z = 0.

Fig. 4.

Since ε is an arbitrary small number, the functions F1 and F2 are close to the derivative z(t) ofsolution of the equation

z +C3

(q − 1)z +

D5

(q − 1)z = 0, z(0) = Z, z(0) = F (Z). (2.21)

Taking into account (2.18), the solution of the equation can be presented as

z(t) = eλ1tc1 + eλ2tc2.

Here t < 0 and ci, λi are the following:

λ1 = − C3

2(q − 1)+

√C2

3

4(q − 1)2− D5

(q − 1),

λ2 = − C3

2(q − 1)−

√C2

3

4(q − 1)2− D5

(q − 1),

c1 =F (Z) − λ2Z

λ1 − λ2, c2 =

F (Z) − λ1Z

λ2 − λ1.

Similar reasoning can be used for the upper half-plane {F > 0, z � 0} (Fig. 5).Thus, the derivative z(t) of the solution z(t) of equation (2.21) is close to F (z) also in upper

half-plane, in which case for the time of crossing the straight line {z = Z} : z(T ) = Z (T < 0) bythe solution z(t) we have the estimate

z(T ) < Rκ2(R),

where κ2(R) → 0 as R → +∞.The latter follows from positiveness of λ1 and λ2. This implies the assertion of Lemma 5.Similar assertion occurs also for the case x < −1.

Lemma 6. Let condition (2.18) be satisfied. Then for the solution of system (2.4) with the initialdata x(0) = c, y(0) = R there exists a number T > 0 such that

x(T ) = c, −δR < y(T ) < 0,

x(t) ∈ (c,−1), ∀ t ∈ (0, T ).

The proof of Lemma 6 is similar to the proof of Lemma 5.



Fig. 5.

2.4. Criterion of Existence of Limit Cycles

Note that by (2.7) condition (2.12) can be rewritten as a pair of inequalities

2c2 � b2 + 1 (2.22)

and

4a2(c2 − 1) > (b2 − 1)2, (2.23)

and condition (2.17) as

b2 > a2 + c2, 2c2 > b2 + 1. (2.24)

The condition q ∈ (−1, 0) is replaced by the condition

c2 ∈ (0, 1), (2.25)

and (2.18) takes the form

2c2 > −1,

(2c2 + 1)2 > 4c2(c2 + 1) > 0

and is always valid under the condition (2.25).From the above we can formulate the following

Theorem 1. Let conditions (2.25) and either (2.22) and (2.23), either (2.24) be satisfied. Thenthe behavior of trajectories of system (2.4) with sufficiently great initial data

|x(0)| + |y(0)| 1, x(0) �= −1

is as shown in Fig. 6.

Fig. 6.

Theorem 1 implies the following criteria of existence of limit cycles in system (2.4):



Theorem 2. Suppose that function g(x) has a single zero on the interval (−1,+∞), to whichcorresponds the unstable equilibrium x = y = 0 of system (2.4). Let conditions (2.25) and either(2.22) and (2.23), either (2.24) be satisfied. Then system (2.4) has a limit cycle, situated in thehalf-plane {x > −1, y ∈ R

1}.Theorem 3. Suppose that function g(x) has only two zeros x = 0 and x = x1 ∈ (−∞,−1), towhich correspond the unstable equilibria x = y = 0 and x = x1, y = 0 of system (2.4), respectively.Let conditions (2.22), (2.23), and (2.25) be satisfied. Then system (2.4) has 2 limit cycles. One ofthem lies to the left of straight line {x = −1, y ∈ R

1} and another one to the right of this straightline.

For testing the conditions of Theorems 2 and 3, the following results can be useful.

Proposition 5. Let the conditions of Theorem 2 or Theorem 3 be valid. In order that g(x) hasonly two zeros x = 0 and x1 ∈ (−∞,−1) it is necessary and sufficient that the inequality

α2 < λ, (2.26)

where λ is a minimal root of the equation

0 = −4c2λ3 + (−β2

2 + (2b2 + 6c2)β2 + 27c22 + (12a2 − 18b2)c2 − b2

2)λ2−

− 2(−β32 + (−3c2 − a2 + 4b2)β2

2 + (−5b22 + 9b2c2 + 2b2a2 − 3a2c2)β2 − a2b

22+

+ 2b32 − 9a2b2c2 + 6a2

2c2)λ + (−β42 + (2b2 − 4c2 + 2a2)β3

2 + (12a2c2 − b22 − a2

2−− 4a2b2)β2

2 + (2a22b2 − 12a2

2c2 + 2a2b22)β2 − a2

2b22 + 4a3

2c2),

is satisfied.

To prove Proposition 5, it is sufficient to note that the conditions of Theorem 2 or Theorem 3imply the fact that the polynomial

−(x + 1)2(a2x + α2) + x(x + 1)(b2x + β2) − c2x3

necessarily has a root on the interval (−∞,−1). Therefore in order this polynomial has no otherreal roots it is necessary and sufficient that

Δ = (4c3a − c2b2 − 18abcd + 27a2d2 + 4b3d) > 0.

Here a = a2 − b2 + c2, b = α2 + 2a2 − b2 − β2, c = 2α2 + a2 − β2, d = α2. It follows the assertionof Proposition 5.Proposition 6. If β2 > 0, then the equilibrium x = y = 0 is Lyapunov unstable.

Proposition 7. Let the conditions of Proposition 5 be valid. For the equilibrium x = x1, y = 0 tobe Lyapunov unstable, it is necessary and sufficient that p < −1, where p is a minimal root of theequation

(2c2 − b2 − 1)p2 − (2 + b2 + β2)p − β2 = 0,

and

α2 <1

(p + 1)2(−a2(p + 1)2p + (b2p + β2)(p + 1)p − c2p

3). (2.27)

Note also that from the conditions of Proposition 7 it follows that α2 < 0.Theorems 2 and 3 together with Propositions 5–7 select in the space of parameters the sets Ω1

and Ω2 with one and two limit cycles, respectively. Obviously, these sets have infinite Lebesguemeasure and the effectively described here sets Ω1 and Ω2 are not small.

Consider now systems with small and large limit cycles, applying the method of computationof Lyapunov quantities from the first Chapter and the above-described method for localization oflarge limit cycles. Two different configurations of four limit cycles will be examined: 2 small and 2large cycles (two small and one large at zero point together with one large at the point x1 < −1)and 3 small and 1 large cycles (three small cycles at zero point together with one large at the pointx1 < −1).



2.5. Criteria of Existence of Four Limit Cycles

Suppose that in system (2.3) conditions (2.22), (2.23), and (2.25) are valid and the followingrelations

β2 = 0, α2 < 0 (2.28)

are true. In this case matrix of linear approximation of the system (2.3) in the point of equilibriumx = y = 0 has two purely imaginary eigenvalues.

The first Lyapunov quantity is as follows

L1(0) =−π

4(−α2)5/2(α2(b2c2 − 1) − a2(b2 + 2)).

For the equilibrium x = 0, y = 0 of system (2.3) (and, therefore, x = y = 0 of system (2.4)) tobe a weak focus of at least second order, it is necessary and sufficient that the relations

β2 = 0, α2 =a2(2 + b2)b2c2 − 1

(2.29)

are satisfied.Taking into account (2.28), we obtain

a2(2 + b2)b2c2 − 1

< 0. (2.30)

Recall that relations (2.22), (2.23), and (2.25) yield the inequalities b2 > −1, a2 < 0. Then, takinginto account the inequality (2.30), we have

b2c2 > 1 (2.31)

and therefore by (2.25)

b2 > 1. (2.32)

The second Lyapunov quantity is as follows

L2(0) =π(b2 − 3)(b2 c2 − 1)5/2

24(−a2)7/2(2 + b2)7/2×

× ((c2b2 + b2 − 2c2)(c2b2 − 1) − a2(c2 − 1)(1 + 2c2)2).

In the case of weak focus the second Lyapunov quantity L2(0) is positive if b2 < 3 or taking intoaccount (2.32)

b2 ∈ (1, 3). (2.33)

Hence by (2.31) c2 satisfies the inclusion

c2 ∈ (1/3, 1) (2.34)

and the condition (2.26) is valid. Note that, taking into account (2.33) and (2.34), condition (2.22)is valid. The condition (2.27) with provision for relation (2.30) takes the form

(2 + b2)b2c2 − 1

2

[−a2(1 − c2)(1 + 2c2)2 + (b2c2 − 1)(−b2 + 2c2 − c2b2)] > 0

and it is satisfied.Thus, the conditions of Propositions 5 and 7 are satisfied and at a point x = y = 0 there is

weak focus, then g(x) has only two zeros x = 0 and x1 < −1 and both equilibria of system (2.4)are unstable. Therefore by Theorem 3 if the relations (2.23), (2.29), (2.31), (2.33), and (2.34) aresatisfied, then systems (2.3) and (2.4) have two limit cycles.



It is well known that for small disturbance of the parameters

β2 ∈ (0, ε),

α2 ∈(

a2(2 + b2)b2c2 − 1

,a2(2 + b2)b2c2 − 1

+ δ

),

(2.35)

where 0 < ε � δ � 1, both large limit cycles persist and in the neighborhood of zero two smalllimit cycles occur.

Thus, system (2.3) has four limit cycles (2 small and 2 large) if conditions (2.23), (2.31), (2.33),(2.34), and (2.35) are satisfied.

The domain Ω3, distinguished by means of these conditions, has infinite Lebesgue measure.However this domain is small with respect to parameters β2 and α2.

Note that the domain of unperturbed parameters is three-dimensional. It has the form

{b2 ∈ (1, 3), c2 ∈ (1/3, 1), b2c2 > 1, a2(c2 − 1) >(b2 − 1)2

4}. (2.36)

In Fig. 7 are shown 2 large limit cycles (note that 2 small cycles at zero point can be obtainedby small disturbances of parameters of system (2.3)).

Here are the following coefficients:

a2 = −35, b2 = 1.6, c2 = 0.7, α2 = −1050, β2 = 0.

In Fig. 7 in the domain of closeness of the trajectories are presented one stable (on the right)and one unstable (on the left) limit cycles.

Fig. 7. Two large limit cycles.

In the limit case when b2 = 3 we obtain that L2 = 0 and

L3 =π√

5(3c2 − 1)9/2

500000(−a2)9/2(c2 − 2)(4c3

2a2 − 3c22 − 3a2c2 − 8c2 − a2 + 3)

is negative for all a2 and c2, satisfying (2.23) and (2.34). It means that by small positive disturbanceμ, such that

b2 ∈ (3 − μ, 3), (2.37)

the condition L2 > 0 can be satisfied and the third small limit cycle at zero point can be obtained.



Notice that in the case when b2 = 3, taking into account (2.34), condition (2.31) is satisfied.That means that system (2.3) has four limit cycles (3 small at zero point and one large at the

point x1 < −1) if (2.23), (2.37), (2.34), and (2.35) are satisfied. Here 1 μ δ ε � 0.Note that the domain of unperturbed parameters, corresponding to the conditions described

above, is two-dimensional. It has the form

{c2 ∈ (1/3, 1), a2(c2 − 1) > 1}. (2.38)

Below it will be shown that this domain involves entirely the domain, defined by the famous theoremof Shi [9].

Thus, the criterion of the existence of 4 limit cycles can be stated.

Theorem 4. System (2.3) has four limit cycles if the conditions

c2 ∈ (1/3, 1), b2 ∈ (1, 3), 4a2(c2 − 1) > (b2 − 1)2, b2c2 > 1,

β2 ∈ (0, ε), α2 ∈ (a2(2 + b2)b2c2 − 1

,a2(2 + b2)b2c2 − 1

+ δ)

and 1 δ ε � 0 are satisfied.

2.6. Visualization of Domain of Parameters Corresponding to the Existence of Limit Cycles

Consider the system of Shi and show in the figure the domain of Shi and the domain describedabove. The system of Shi is as follows

x = −y + lx2 + 5axy + ny2,

y = x + ax2 + (3l + 5n)xy.(2.39)

Reduce it to the form of system (2.3).Suppose, a �= 0. Having performed the change of variables x → x − ky, y → y and put the

coefficient of y2 in the first equation to zero, we get the system

x = −xk − (1 + k2)y + (l − ka)x2 + (−lk + 5a − 2k2a − 5kn)xy,

y = x + yk + ax2 + (2ka + 3l + 5n)xy + y2k(ka + 3l + 5n).(2.40)

Here k is a real root of equation (2.2), written for system (2.39), namely

n − 2 lk2 + 5 ka − ak3 − 5 k2n = 0.

By the change of variables: x → x, y → kx

(1 + k2)+ y, the coefficient of x in the first equation

can be put to zero.By the following change of variables: x → P 2

R x, y → SPR2 y, t → −R

S t, where

P = 1 + k2, R = −5 k3n − 3 k3l + 7 ak2 + 7 kn + kl − 5 a, S = 6 ka − l − 6 k2n − 3 lk2

we arrive at the system

x = x2 + xy + y, y = a2x2 + b2xy + c2y

2 + α2x.

Here

a2 =R(2 kl − k3n + 5 ak2 + 5 kn − a)

S2, b2 =

5 ak3 + lk2 + 7 k2n − 7 ka − 5n − 3 l

S,

c2 = −(lk2 + 6 ka + 3 l + 6n)kR

, α2 = −R2

S2.

Note that it is possible only in the non-critical case when R,S �= 0.



Below we give the conditions of existence of large limit cycle stated in the work [9]:

3a2 − l(l + 2n) < 0, 25a2 + 12n(l + 2n) < 0,

l(3l + 5n)2 − 5a2(3l + 5n) + na2 < 0,

a2(5l + 8n) − ((2l + 5n)2 + 15a2)(25a2 + 3n(2l + 5n)) > 0.

Shi proved that if these conditions are satisfied, then by small disturbances of coefficients of system(2.39) one can also obtain 3 small limit cycles.

Since a2 and c2 are functions of a, l, n, these conditions yield some conditions on a2 and c2.In Fig. 8 is shown the domain described by Shi (shaded), and the domain (grey) obtained abovewith the help of the method of asymptotical integration, note that, according to (2.38), it can be

described with the following simple conditions {c2 ∈ (1/3, 1), a2 <1

(c2 − 1)}.

Fig. 8. Domain of Shi and its extension.

Finally note that the domain obtained involves entirely the domain of Shi and corresponds tothe domain obtained analytically for the same case in the work [47]. This confirms the effectivenessof the new method of asymptotical integration, described above.

Note, in general case the domain of four limit cycles existence is three-dimensional and, according

to (2.36), has the form {b2 ∈ (1, 3), c2 ∈ (1b2

, 1), a2 <(b2 − 1)2

4(c2 − 1)}.

In Figures 9 are shown the sections of this domain, corresponding to the existence of 2 smalland 2 large limit cycles for b2 = {2.6, 2.2, 1.8, 1.4}, respectively.

Note that, according to (2.36), projection of the whole three-dimensional domain of four limit

cycles existence (for b2 ∈ (1, 3)) can be described with the following inequalities {a2 <(c2 − 1)

4c22

, c2 ∈

(1/3, 1)} and is shown in Fig. 10. It means that for every pair of coefficients a2, c2 from the domainfour limit cycles can be obtained for some b2, namely for b2, satisfying the following condition

1c2

< b2 < 2√

m + 1, m = min(1, a2(c2 − 1)).



Fig. 9. Sections for b2 = 2.6, 2.2, 1.8, 1.4.

Fig. 10. Projection of three-dimensional domain.

ACKNOWLEDGMENTS

This work was partly supported by Finnish Graduate School in Computational Sciences (FICS),projects of Ministry of education and science of RF (2.1.1/3889) and Federal Program “Scientificand scientific-pedagogical cadres Innovative Russia” in 2009–2013 years.

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