ON SOME BOUNDED AND STABLESOLUTIONS OF A NON-AUTONOMOUS

LIENARD EQUATION

Juan E. Nápoles Valdes1;2,Luciano M. Lugo Motta Bittencurt1 and Samuel I. Noya1

September 1, 2014

Abstract

In this note using a bounded region we obtain su¢ cient conditionsunder which we can guarantee the boundedness and stability of solutionsof a non-autonomous Liénard equation.

1UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina;2UTN, FRRE, French 414, (3500) Resistencia, Chaco, Argentina;jnapoles@exa.unne.edu.ar; jnapoles@frre.utn.edu.ar

AMS Subject Classi�cation (2010): 34C11

Key words and phrases: Boundedness, stability, Liénard equation.

1 Preliminars

The Liapunov´s Second Method (o Direct) has long been recognized as themost general method for the study of the stability of equilibrium points ofsystems described by di¤erential equations. The method was �rst presented inhis classical memoir, which appeared in Russian in 1892 and was translated intoFrench in 1907 and English in 1949 [9]. Statements and proofs of mathematicalresults underlying the method and numerous examples and references can befound in the books of Antosiewicz [1], Barbashin and Krasovskii [2], Cesari [3],Demidovich [4], Hahn [6] and Yoshizawa [17] and bibliography listed therein.First we consider the system

x0 = f(t; x); (1)

where f 2 C(IxSr) and f(t; 0) � 0. With I we denote the interval 0 � t <+1, Rm will stand for Euclidean m-space, k:k will be an arbitrary norm in Rmand Sr = fx 2 Rm : kxk < rg.This method is a powerful tool because his simplicity for the research of the

stability. For instance, if within a neighborhood of the equilibrium point anappropriate "energy" function is always decreasing we expect the equilibriumto be asymptotically stable. The Second Method generalizes this idea, if suchfunction V (t; x), called Liapunov Function, can be constructed for the system(1) in a neighborhood of the equilibrium point and if in that neighborhoodV 0(t; x) � 0 for x 6= 0 being V (t; x) positive de�ned, then the equilibrium pointis asymptotically stable. If one knows only that V 0(t; x) � 0 for x 6= 0, then, ingeneral, one can conclude only the origin is stable.The main di¢ culty is to construct a suitable V (t; x), this requires experience

and technique. So, many times it may be di¢ cult to construct a LiapunovFunction which satis�es all above conditions, in such cases it is often useful toprove the boundedness and stability properties by using qualitative methodsbased in a bounded closed set. We apply this to the well-know non-autonomousLiénard equation

x00 + f(x)x0 + a(t)g(x) = 0; (2)

under suitable assumptions. The classical Liénard equation

x00 + f(x)x0 + g(x) = 0; (3)

appears as simpli�ed model in many domains in science and engineering [10].It was intensively studied during the �rst half of 20th century as it can be usedto model oscillating circuits or simple pendulums. In the simple pendulum case,f and g represents the friction and acceleration terms. One of the �rst modelswhere this equation appears was introduced by Balthasar van der Pol [15,16],considering the equation

x00 + �(x2 � 1)x0 + x = 0; (4)

for modeling the oscillations of a triode vacuum tube. The Liénard equa-tion, which is often taken as the typical example of nonlinear self-excited vi-bration problem, can be used to model resistor-inductor-capacitor circuits withnonlinear circuit elements. It can also be used to model certain mechanical sys-tems which contain the nonlinear damping coe¢ cients and the restoring forceor sti¤ness. Moreover, some nonlinear evolution equations (such as the Burgers-Korteweg-de Vries equation) which arise from various physical phenomena canalso be transformed to equation (3). Therefore, the study of equation (3) isof physical signi�cance. We recommend [5] for other references about moreapplications.The main purpose of this note is to present qualitative methods rather than

to obtain general results of boundedness and stability properties of solutionsof (2), without making uses of common conditions. By this, we need some

clari�cations. V (t; x) denote an arbitrary Liapunov�s Function de�ned on anopen set S � IxRm with continuous partial derivatives with respect to allarguments, corresponding to V (t; x); we de�ne the function

V 0(1)(t; x) := Lim suph!0+

V (t+ h; x+ hf(t; x))� V (t; x)h

;

called the total derivative of V (t; x) for system (1). Under the above condi-tions,

V 0(1)(t; x) =@V

@t+@V

@xf(t; x): (5)

We now mention two theorems which will play an important role in theproofs of our main results.Theorem A. Suppose that there exists a Liapunov�s function V (t; x) de�ned

on IxSr satisfying the following conditions:1) V (t; 0) � 0,2) a(x) � V (t; x); where a(x) is a positive de�nite function and continuous

increasing function on R.3) V 0(1)(t; x) � 0:Then the trivial solution x(t) � 0 of (1) is stable.For the proof, see [17, Theorem 8.1].Theorem B (Barbashin and Krasovskii). If there exists a function

V(t,x) which is everywhere positive de�nite, radially unbounded, decreasing,and whose total derivative (4) for system (1) is negative de�nite, then the solu-tion x(t) � 0 of (1) is asymptotically stable in the whole.For the proof, see [4, p. 248].

2 Results

In (2) consider that f : R ! R is a continuous function, g : R ! R is a locally

lipschitzian continuous function, with G(x) =

xZ0

g(s)ds, and a : [0;1)! R is a

function of class C1 and such that 0 < a � a(t) � A < +1.For a continuous functionM(t) de�neM(t)+ = maxfM(t); 0g andM(t)� =

maxf�M(t); 0g. It is clear that M(t) =M(t)+ �M(t)�.

Theorem 1. Suppose thata) g(x)F (x) > 0, for 0 < jxj < k;b) G(x) < l, implies jxj < k.Then, every solution x(t) of (2) are bounded and the trivial solution is stable.

Proof. Letting F (x) =R x0f(s)ds and y = x0 we obtain an equivalent system

to equation (2):

x0 = y � F (x);

(6)

y0 = �a(t)g(x):

Here a suitable Liapunov Function is

V (t; x; y) =y2

2a(t)+G(x): (7)

Then along the solutions of (5) we have

V 0(3)(t; x; y) � �a0(t)+

2a2(t)y2 � g(x)F (x): (8)

Let l and m arbitrary positive numbers and consider the region D de�nedby the inequalities

�l < W (x; y) < l and (y + F (x))2 < m2

where W (x; y) = y2

2a + G(x). The conditions a) and b) assure us that theregion D is bounded (see also [8] or [11]) and that within D, V 0(3)(t; x; y) � 0.We select l and m so large that (x0; y0) will be in the interior of D. We shall

show than any solution (x(t); y(t)) with initial conditions (x0; y0) cannot leavethe bounded region D, so we can obtain that all solutions of (2) are bounded,since (x(t); y(t)) is arbitrary.Is clear that in order to leave D, the solution (x(t); y(t)) must either cross

the locus of points determined by W (x; y) = l or one of loci determined byy+F (x) = �m. We can selectm su¢ ciently large that the part of y+F (x) = mwhich is the boundary of D corresponds to x > 0 and the part of y+F (x) = �mcorresponds to x < 0. Since V 0(3)(t; x; y) � 0 and V (t; x; y) �W (x; y) a solutionstarting inside cannot cross W (x; y) = l.Now

d

dt(y + F (x))

2= �2(y + F (x))a(t)g(x):

Along the part of y + F (x) = m or y + F (x) = �m which makes up theboundary of D, we have

d

dt(y + F (x))

2= �2am(y + F (x)) jg(x)j < 0:

Hence, the solution (x(t); y(t)) cannot cross outside of D through the partof the boundary determined by y + F (x) = �m. Therefore, every solution of(2) is bounded for t � 0. From this and taking a(x; y) = y2

2A +G(x) we obtainthe desired result under Theorem A.

Remark 1. A di¢ culty in using the Theorem A often is that one can constructa Liapunov Function satisfying all assumptions but not the existence of functiona(x,y). We illustrate in the next result that it may then be easier to establish theboundedness of the solutions as a separate problem. So, we continue our workwith a weaker assumption on g(x) and a stronger assumption on the damping,but taking 0 < a � a(t) < +1.

Theorem 2. Under hypothesesa) xg(x) > 0, for x 6= 0.b) f(x) > 0, for x 6= 0.c) jG(x)j ! 1 as jxj ! 1,then all solutions satis�es jx(t)j < M , jx0(t)j < M , where M may depend on

the solution.

Proof. We use the same Liapunov Function as before. From (4) and con-ditions a), b) and c) we obtain again that V 0(3)(t; x; y) � 0. So, as above everysolution of (2) is bounded for t � 0.

De�nition 1. By F(I) we denote the class of non decreasing continuous func-tions ' on I such '(u) > 0 for all u 2 I and

R10

du'(u) =1.

Theorem 3. Under conditionsa) 0 < a � a(t) � A < +1,b) xg(x) > 0 for x 6= 0;c) f(x) > 0 for all x,d) G(x)!1, when x!1:Then the trivial solution x � 0 of the system (5) is asymptotically stable in

the whole.

Proof. Let V (t; x; y) as in (5) and let

U(t;x) :=

Z V (t;x;y)

0

dr

(r);

where 2 F(R+); then U(t;x) satis�es the conditions of the Barbashin-Krasovski Theorem (Theorem B above), and the proof of the theorem is com-plete.

Remark 2. The simple case

x00 � 2x0 + x = 0; (9)

with unbounded solution x(t) = et, shows that positivity of f is probably necessaryin some sense. This is an open problem.

Remark 3. If we take f and g strictly increasing, our results are consistent withthose of [12], mainly Theorem 2 a).

Remark 4. Finally we give examples of functions f(x) which show that ourresults contains those in [13] and [14] refer to the boundedness and stability ofsolutions of equation (2).

Example 1. f(x) =�

x; if jxj � 1;x�1; if jxj > 1:

Example 2. f(x) =

8<: 1; if x � 1;x; if jxj < 1;�1; if x � �1:

These examples do not satisfy the conditions of Repilado and Ruiz, but theyare covered by results of this paper.

Remark 5. When a(t) � 1 our results improve the obtained in [7] and those of[8] refer to classical Liénard Equation (3).

References

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[2] Barbashin, E. A.,Liapunov�s Functions, Science Publishers, Moscow, 1970(Russian).

[3] Cesari, L., Asymptotic behavior and stability problems in ordinary di¤er-ential equations, Springer-Verlag, Berlin, 1959.

[4] Demidovich, B. P.,Lectures on the Mathematical Theory of the Stability,Science Publishers, Moscow, 1967 (Russian).

[5] Guckenheimer, J. and P. Holmes, Nonlinear oscillations, dynamical sys-tems, and bifurcations of vector �elds, volume 42 of Applied MathematicalSciences. Springer-Verlag, New York, 2002. Revised and corrected reprintof the 1983 original.

[6] Hahn, W., Theory and Application of Liapunov�s Direct Method, Prentice-Hall, Englewood Cli¤s, New Jersey, 1963.

[7] Hasan, Y. Q. and L. M. Zhu, The bounded solutions of Liénard equation,J. of Applied Sciences 7(8): 1239-1240, 2007.

[8] LaSalle, J. P., Some extensions of Liapunov�s second method, IRE Trans-action on Circuit Theory, Dec. 1960, 520-527.

[9] Liapounov, A. M., Problème général de la stabilité du mouvement, in Annalsof Math. Studies No.17, Princeton Univ. Press, Princeton, NJ, 1949.

[10] Liénard, A., Étude des oscillations entretenues, Revue Génerale del�Électricité 23: 901�912, 946�954 (1928).

[11] Miller, R. K. and A. N. Michel, Ordinary Di¤erential Equations, New York:Lawa State University, 1982.

[12] Nápoles Valdes, Juan E., On the boundedness and asymptotic stability inthe whole of Liénard equation with restoring term, Revista de la UniónMatemática Argentina, 41(4) 2000, 47-59 (Spanish).

[13] Repilado, J. A. and A. I. Ruiz, On the behavior of solutions of equationx�+f(x)x�+a(t)g(x)=0 (I), Revista Ciencias Matemáticas, University of Ha-vana, VI (1), 1985, 65-71 (Spanish).

[14] Repilado, J. A. and A. I. Ruiz, On the behavior of solutions of equationx�+f(x)x�+a(t)g(x)=0 (II), Revista Ciencias Matemáticas, University ofHavana, VII (3), 1986, 35-39 (Spanish).

[15] Van der Pol, B., On oscillation hysteresis in a triode generator with twodegrees of freedom, Phil. Mag (6) 43, 700�719 (1922).

[16] Van der Pol, B., On �relaxation-oscillations�, Philosophical Magazine,2(11):978�992, 1926.

[17] Yoshizawa, T., Stability theory by Liapunov�s second method, The Mathe-matical Society of Japan, 1966.