American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ A Note on Bounded Solutions of a Generalized Lienard Type System Paulo M. Guzmána*, Juan E. Nápoles Valdesb*, Luciano M. Lugo Motta Bittencurtc aUNNE, FaCENA, Av. Libertad 5450, Corrientes (3400), Argentina. bUNNE, FaCENA, Av. Libertad 5450, Corrientes (3400), Argentina. UTN, FRRE, French 414, Resistencia (3500), Chaco, Argentina cUNNE, FaCENA, Av. Libertad 5450, Corrientes (3400), Argentina. aEmail: paulomatiasguzman@hotmail.com bEmail: jnapoles@frre.utn.edu.ar cEmail: lmlmb@yahoo.com.ar Abstract In this paper we study the boundedness of solutions of some generalized Liénard type system under non usual conditions on evolved functions using the Second Method of Lyapunov. Keywords: Boundedness; asymptotic behavior; Liénard equation Type. 1. Introduction The Liapunov´s Second Method (o Direct) has long been recognized as the most general method for the study of the stability of equilibrium points of systems described by differential equations. The method was first presented in his classical memoir, which appeared in Russian in 1892 and was translated into French in 1907 and English in 1949 [1]. Statements and proofs of mathematical results underlying the method and numerous examples and references can be found in the books of Antosiewicz [2], Barbashin and Krasovskii [3], Cesari [4], Demidovich [5], Hahn [6] and Yoshizawa [7] and bibliography listed therein. His method is a powerful tool because his simplicity for the research of the stability. ------------------------------------------------------------------------ * Corresponding author. 89 http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 89-94 For instance, if within a neighborhood of the equilibrium point an appropriate "energy" function is always decreasing we expect the equilibrium to be asymptotically stable. The Second Method generalizes this idea, if such function 𝑽𝑽 (𝒕𝒕;𝒙𝒙), called Liapunov Function, can be constructed for the system (or equation) in a neighborhood of the equilibrium point and if in that neighborhood 𝑽𝑽′(𝒕𝒕,𝒙𝒙) ≤ 𝟎𝟎 for 𝒙𝒙 ≠ 𝟎𝟎 being 𝑽𝑽 (𝒕𝒕; 𝒙𝒙) positive defined, then the equilibrium point is asymptotically stable. If one knows only that 𝑽𝑽′(𝒕𝒕,𝒙𝒙) ≤ 𝟎𝟎 for 𝒙𝒙 ≠ 𝟎𝟎, then, in general, one can conclude only the origin is stable. The above need some clarifications. 𝑽𝑽 (𝒕𝒕; 𝒙𝒙) denote an arbitrary Liapunov`s Function defined on an open set 𝑺𝑺 ⊂ 𝑰𝑰𝒙𝒙𝑹𝑹𝒎𝒎 with continuous partial derivatives with respect to all arguments, corresponding to 𝑽𝑽 (𝒕𝒕; 𝒙𝒙); and we define the function 𝑉𝑉(1) ′ (𝑡𝑡, 𝑥𝑥) ≔ lim ℎ→0+ 𝑠𝑠𝑠𝑠𝑠𝑠 𝑉𝑉�𝑡𝑡 + ℎ, 𝑥𝑥 + ℎ𝑓𝑓(𝑡𝑡, 𝑥𝑥)� − 𝑉𝑉(𝑡𝑡, 𝑥𝑥) ℎ , called the total derivative of 𝑉𝑉 (𝑡𝑡; 𝑥𝑥) for system (1). Under the above conditions, 𝑉𝑉(1) ′ (𝑡𝑡, 𝑥𝑥) = 𝜕𝜕𝑉𝑉 𝜕𝜕𝑡𝑡 + 𝜕𝜕𝑉𝑉 𝜕𝜕𝑥𝑥 𝑓𝑓(𝑡𝑡, 𝑥𝑥). The main difficulty is to construct a suitable 𝑉𝑉(𝑡𝑡, 𝑥𝑥), this requires experience and technique. The classical Liénard equation 𝑥𝑥´´ + 𝑓𝑓(𝑥𝑥)𝑥𝑥´ + 𝑔𝑔(𝑥𝑥) = 0, Appears as simplified model in many domains in science and engineering [8]. It was intensively studied during the first half of 20th century as it can be used to model oscillating circuits or simple pendulums. In the simple pendulum case, 𝑓𝑓 and 𝑔𝑔 represents the friction and acceleration terms. One of the first models where this equation appears was introduced by Balthasar Van del Pol [9,10], considering the equation 𝑥𝑥´´ + 𝜇𝜇(𝑥𝑥2 − 1)𝑥𝑥´ + 𝑥𝑥 = 0, for modeling the oscillations of a triode vacuum tube. The Liénard equation, which is often taken as the typical example of nonlinear self-excited vibration problem, can be used to model resistor-inductor-capacitor circuits with nonlinear circuit elements. It can also be used to model certain mechanical systems which contain the nonlinear damping coefficients and the restoring force or stiffness. Moreover, some nonlinear evolution equations (such as the Burguers - Korteweg - de Vries equation) which arise from various physical phenomena can also be transformed to equation (3). Therefore, the study of equation (3) is of physical significance. We recommend [11] for other references about more applications. The main purpose of this note is to present two results about the existence of bounded solutions of system 90 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 89-94 𝑥𝑥´ = 𝛼𝛼(𝑦𝑦) − 𝛽𝛽(𝑦𝑦)𝐹𝐹(𝑥𝑥), (1) 𝑦𝑦´ = −𝑔𝑔(𝑥𝑥), where 𝛼𝛼(𝑦𝑦),𝛽𝛽(𝑦𝑦), 𝑓𝑓(𝑥𝑥) 𝑦𝑦 𝑔𝑔(𝑥𝑥) are continuous functions that satisfies the following conditions: 𝑥𝑥𝑓𝑓(𝑥𝑥) > 0, 𝑥𝑥𝑔𝑔(𝑥𝑥) > 0,𝑦𝑦𝛼𝛼(𝑦𝑦) > 0,𝑦𝑦𝛽𝛽(𝑦𝑦) > 0 (2) for all 𝑥𝑥,𝑦𝑦 ≠ 0 and 𝐹𝐹(𝑥𝑥) = ∫ 𝑓𝑓(𝑠𝑠)𝑑𝑑𝑠𝑠.𝑥𝑥 0 Clearly classical Liénard equation is a particular case of system (1). 2. Results Theorem 1. Let the condition (2) be satisfied. Then every solution (𝑥𝑥(𝑡𝑡),𝑦𝑦(𝑡𝑡)) of (1), with both components positives in 𝑡𝑡𝑡𝑡[𝑇𝑇 − 𝛿𝛿,𝑇𝑇) is continuable to the right of 𝑇𝑇. Proof. Now we consider the following function 𝑉𝑉(𝑥𝑥,𝑦𝑦) = 𝐴𝐴(𝑦𝑦) + 𝐺𝐺(𝑥𝑥) with 𝐴𝐴(𝑦𝑦) = ∫ 𝛼𝛼(𝑦𝑦)𝑑𝑑𝑠𝑠,𝑦𝑦 0 𝐺𝐺(𝑥𝑥) = ∫ 𝑔𝑔(𝑠𝑠)𝑑𝑑𝑠𝑠.𝑥𝑥 0 By the conditions imposed on 𝛼𝛼(𝑦𝑦) and 𝑓𝑓(𝑥𝑥) we obtain 𝑉𝑉(𝑥𝑥,𝑦𝑦) ≥ 0,∀𝑥𝑥,𝑦𝑦 ≠ 0. To verify that 𝑉𝑉 is the required Lyapunov function we derive throughout the system, and we have: 𝑉𝑉(1) ′ (𝑡𝑡, 𝑥𝑥) = 𝛼𝛼(𝑦𝑦)𝑦𝑦´ + 𝑔𝑔(𝑥𝑥)𝑥𝑥´ − 𝛽𝛽(𝑦𝑦)𝑔𝑔(𝑥𝑥)𝐹𝐹(𝑥𝑥) < 0. For 𝑡𝑡𝑡𝑡[𝑇𝑇 − 𝛿𝛿,𝑇𝑇). By integration we get 𝑉𝑉(𝑡𝑡) = 𝑉𝑉(𝑇𝑇 − 𝛿𝛿) −� 𝛽𝛽�𝑦𝑦(𝑠𝑠)�𝑔𝑔�𝑥𝑥(𝑠𝑠)�𝐹𝐹�𝑥𝑥(𝑠𝑠)�𝑑𝑑𝑠𝑠 𝑡𝑡 𝑇𝑇−𝛿𝛿 ≤ 𝑉𝑉(𝑇𝑇 − 𝛿𝛿). Then 0 < 𝐴𝐴(𝑦𝑦) + 𝐺𝐺(𝑥𝑥) ≤ 𝑉𝑉(𝑡𝑡 − 𝛿𝛿) ⟹ |𝑦𝑦| ≤ 𝑘𝑘, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑑𝑑𝑠𝑠 𝑘𝑘 = 𝑐𝑐𝑡𝑡𝑠𝑠. Integrating the first equation of the system (1.1), we get: 𝑥𝑥(𝑡𝑡) = 𝑥𝑥(𝑇𝑇 − 𝛿𝛿) + � 𝛼𝛼(𝑠𝑠)𝑑𝑑𝑠𝑠 𝑡𝑡 𝑇𝑇−𝛿𝛿 − � 𝛽𝛽�𝑦𝑦(𝑠𝑠)�𝐹𝐹�𝑥𝑥(𝑠𝑠)�𝑑𝑑𝑠𝑠 𝑡𝑡 𝑇𝑇−𝛿𝛿 , with ∫ 𝛼𝛼(𝑠𝑠)𝑑𝑑𝑠𝑠𝑡𝑡 𝑇𝑇−𝛿𝛿 ≤ 𝑐𝑐, (𝑐𝑐 = 𝑐𝑐𝑡𝑡𝑠𝑠) because 𝑦𝑦 is bounded. Therefore, 91 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 89-94 0 < 𝑥𝑥(𝑡𝑡) = ℎ − � 𝛽𝛽�𝑦𝑦(𝑠𝑠)�𝐹𝐹�𝑥𝑥(𝑠𝑠)�𝑑𝑑𝑠𝑠 𝑡𝑡 𝑇𝑇−𝛿𝛿 ≤ ℎ − 𝑏𝑏� 𝐹𝐹�𝑥𝑥(𝑠𝑠)�𝑑𝑑𝑠𝑠, 𝑡𝑡 𝑇𝑇−𝛿𝛿 with ℎ = 𝑘𝑘 + 𝑐𝑐. If ∫ 𝐹𝐹�𝑥𝑥(𝑠𝑠)�𝑑𝑑𝑠𝑠𝑡𝑡 𝑇𝑇−𝛿𝛿 = ∞ we have 𝑥𝑥(𝑡𝑡) < 0 in [𝑇𝑇 − 𝛿𝛿,𝑇𝑇). Which contradicts the assumption that 𝑥𝑥(𝑡𝑡) is positive in [𝑇𝑇 − 𝛿𝛿,𝑇𝑇). Hence 𝑥𝑥 is bounded. The proof of the Theorem is complete.∎ Theorem 2. Let the condition (2) be satisfied and suppose that 𝛼𝛼´(𝑦𝑦) ∈ 𝐶𝐶1(𝑅𝑅) with 𝛼𝛼´(𝑦𝑦) > 0, |𝛽𝛽(𝑦𝑦)| ≤ 𝑀𝑀 for all 𝑦𝑦, 𝑔𝑔(𝑥𝑥) is a bounded function on 𝑅𝑅 and 𝐹𝐹(−∞) < ∞. Then every solution (𝑥𝑥(𝑡𝑡),𝑦𝑦(𝑡𝑡)) of (1) which exists and is negative on [𝑇𝑇 − 𝛿𝛿,𝑇𝑇), 𝑇𝑇 < 0, is continuable to the right of 𝑇𝑇. Proof. Let 𝑥𝑥(𝑡𝑡) be a solution with negative values in [𝑇𝑇 − 𝛿𝛿,𝑇𝑇). Hence �𝑥𝑥´(𝑡𝑡) + 𝛽𝛽(𝑦𝑦)𝐹𝐹(𝑥𝑥)�´ = �𝛼𝛼(𝑦𝑦)�´ = 𝛼𝛼´(𝑦𝑦)𝑦𝑦´ = −𝛼𝛼´(𝑦𝑦)𝑔𝑔(𝑥𝑥) > 0. Then 𝑥𝑥´(𝑡𝑡) + 𝛽𝛽(𝑦𝑦)𝐹𝐹(𝑥𝑥) is increasing in [𝑇𝑇 − 𝛿𝛿,𝑇𝑇). Therefore 𝑥𝑥´(𝑡𝑡0) + 𝛽𝛽�𝑦𝑦(𝑡𝑡0)�𝐹𝐹�𝑥𝑥(𝑡𝑡0)� < 𝑥𝑥´(𝑡𝑡) + 𝛽𝛽�𝑦𝑦(𝑡𝑡)�𝐹𝐹�𝑥𝑥(𝑡𝑡)� ≤ 𝑥𝑥´(𝑡𝑡) + 𝛽𝛽�𝑦𝑦(𝑡𝑡)�𝐾𝐾 < 𝑥𝑥´(𝑡𝑡) In 𝑡𝑡0 ≤ 𝑡𝑡 < 𝑇𝑇 because 𝐹𝐹�𝑥𝑥(𝑡𝑡)� ≤ 𝐾𝐾 < ∞. So, we have (𝑥𝑥´(𝑡𝑡0) + 𝛽𝛽�𝑦𝑦(𝑡𝑡0)�𝐹𝐹�𝑥𝑥(𝑡𝑡0)� < 𝑥𝑥´(𝑡𝑡) which implies that 𝑥𝑥´(𝑡𝑡) is bounded from below on [𝑇𝑇 − 𝛿𝛿,𝑇𝑇). Integrating the last inequality, we have: (𝑥𝑥´(𝑡𝑡0) + 𝛽𝛽�𝑦𝑦(𝑡𝑡0)�𝐹𝐹�𝑥𝑥(𝑡𝑡0)�(𝑡𝑡 − 𝑡𝑡0) ≤ 𝑥𝑥(𝑡𝑡) − 𝑥𝑥(𝑡𝑡0) then, we conclude (𝑥𝑥´(𝑡𝑡0) + 𝛽𝛽�𝑦𝑦(𝑡𝑡0)�𝐹𝐹�𝑥𝑥(𝑡𝑡0)�(𝑡𝑡 − 𝑡𝑡0) + 𝑥𝑥(𝑡𝑡0) ≤ 𝑥𝑥(𝑡𝑡) < 0 hence lim𝑡𝑡→𝑇𝑇− 𝑥𝑥(𝑡𝑡) exists and is finite. In addition lim 𝑡𝑡→𝑇𝑇− 𝑥𝑥´(𝑡𝑡) = lim 𝑡𝑡→𝑇𝑇− �𝑥𝑥´(𝑡𝑡) + 𝛽𝛽�𝑦𝑦(𝑡𝑡)�𝐹𝐹�𝑥𝑥(𝑡𝑡)�� − lim 𝑡𝑡→𝑇𝑇− �𝛽𝛽�𝑦𝑦(𝑡𝑡)�𝐹𝐹�𝑥𝑥(𝑡𝑡)�� < ∞. If we suppose that |𝑔𝑔(𝑥𝑥)| ≤ 𝑀𝑀2, it follows immediately that 𝑦𝑦´(𝑡𝑡) is bounded, that is to say −𝑀𝑀2 ≤ 𝑦𝑦´(𝑡𝑡) ≤ 𝑀𝑀2. Integrating the last inequality, we have 92 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 89-94 −𝑀𝑀2(𝑡𝑡 − 𝑡𝑡0) ≤ 𝑦𝑦(𝑡𝑡) − 𝑦𝑦(𝑡𝑡0) ≤ 𝑀𝑀2(𝑡𝑡 − 𝑡𝑡0) in [𝑡𝑡0,𝑇𝑇). From which is obtained −𝑀𝑀2(𝑡𝑡 − 𝑡𝑡0) + 𝑦𝑦(𝑡𝑡0) ≤ 𝑦𝑦(𝑡𝑡) ≤ 𝑀𝑀2(𝑡𝑡 − 𝑡𝑡0) + 𝑦𝑦(𝑡𝑡0). Then the proof is complete.∎ Corollary. Under assumptions on Theorems 1 and 2 every solution (𝑥𝑥(𝑡𝑡),𝑦𝑦(𝑡𝑡)) of (1) is continuable to the right and can be defined on an infinite interval [𝑡𝑡0,∞). 3. Conclusion In this section we present some observations in conclusion. Remark 1. If in (1) we have 𝑥𝑥´ = 𝑦𝑦 + 2𝑥𝑥, (3) 𝑦𝑦´ = −𝑥𝑥. the Theorem 1 is very clear, because this system has an unbounded solution �𝑥𝑥(𝑡𝑡),𝑦𝑦(𝑡𝑡)� = (𝑠𝑠𝑡𝑡 ,−𝑠𝑠𝑡𝑡). Remark 2. Our results contains as particular case, the Theorem 2.1 and 2.6 of [12]. References [1] Liapunov, A. M. Probléme général de la stabilité du movement, in Annals of Math. Studies No.17, Princeton Univ. Press, Princeton, NJ, 1949. [2] Antosiewicz, H. A. A survey of Lyapunov´s second method contributions to the theory of nonlinear oscillations IV in Annals of Math. Studies No.44, Princeton Univ. Press, Princeton, NJ, 1958. [3] Barbashin, E. A. Lyapunov´s Functions, Science Publishers, Moscow, 1970 (Russian). [4] Cesari, L. Asymptotic behavior and stability problems in ordinary differential equations, Springer- Verlag, Berlin, 1959. [5] Demidovich, B. P. Lectures on the Mathematical Theory of the Stability, Science Publishers, Moscow, 1967 (Russian). [6] Hahn, W. Theory and Application of Liapunov´s Direct Method, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. [7] Yoshizawa, T. Stability theory by Liapunov´s second method, The Mathematical Society of Japan, 93 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 17, No 1, pp 89-94 1966. [8] Liénard, A. Étude des oscillations entreteneus, Revue Génerale de l´Électricité 23: 901-912, 946-954 (1928). [9] Van der Pol, B. On oscillation hysteresis in a triode generator with two degrees of freedom, Phil. Mag (6) 43, 700-719 (1922). [10] Van der Pol, B. On “relaxation-oscillations”, Philosophical Magazine, 2(11): 978-992, 1926. [11] Guckenheimer, J. and P. Holmes. Nonlinear ascillations, dynamical systems, and bifurcations of vector fields, volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 2002. Revised and corrected reprint of the 1983 original. [12] Hricisakova, D. Continuability and (non-) oscillatory properties of solutions of generalized Lienard equation, Hiroshima Math. J. 20 (1990), 11-22. 94