THE FAMILY OF FUNCTIONS $S_{\alpha, k}$ AND THE LIENARD EQUATION
Main Article Content
Abstract
In this paper we study qualitatively the Lienard Equation $\ddot x+f(x)\dot x+g(x)=0$ with aid of the non-usual family of funct10ns given by
\[ S_{\alpha, k}(x, y)=\int^{y+F(x)-\alpha G(x)-k}_0 \frac{s}{\alpha s+1} ds +\int_0^x g(u) du\]
where$F(x)=\int_0^x f(u) du$, $G(x)=\int_0^x g(u) du$ and $\alpha, k\in R$.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
H. L. Guidorizzi, "On the existence of periodic solutions for the equation $ddt x +f(x)dot x +g(x) =0," Bol. Soc. Bras. Mat., 22(1) (1991), 81-92.
G. Villari, "On the qualitative behaviour of solutions of Lienard equation," J. Differential Equations, 67(1987), 269-277.
A. V. Dragilev, "Periodic solutions of the differential equations of non-linear oscillations," Prikl. Mat. Mekh., 16(1952), 85-88 (in Russian).
Y. E. Yan Qian et al., "Theory of limit cycles," Translations of Mathematical Monographs, 66(1984), American Mathematical Society.
H. L. Guidorizzi, "Oscillating and periodic solutions of equations of type $ddot x +dot x sum_{i=1}^n f_i(x)|dot x|^{delta_i}+g(x) =0$," Journal of Mathematical Analysis and Applications, 176(2)(1983), 330-345.
Z. Zuo Huan, "On the nonexistence of periodic solutions for Lienard equations," Nonlinear Analysis, Theory, Methods and Applications, 16(2)(1991), 101-110
G. Villari, "On the existence of periodic solutions for Lienard's equation," Nonlinear Analysis, Theory, Methods and Applications, 7(1)(1983). 71-78.
G. Villari, "Cicio limite di Lienard e controllabilita," Bolletino U.M.l. A(S) 17(1980), 406-413.
G. Villari, "A new system for Lienard equation," Bolletino U.M.l. (7)1.A(l987), 375-381
G. Villari, "Periodic solutions of Lienard's equation," J. Math. Analysis and Applications, 86(1982), 406-413.
A. F. Filipov, "Sufficient conditions for the existence of stable limit cycles of second order equations," Mat. Sbornik, 30(1952), 171-180 (in Russian).
Z. Opial, "Sur untheoreme de A. Filipov," Annals Soc, Pol. Math., 5(1958), 67-75.
J. G. Wendel, "On a Van der Pol equation with odd coefficients," J. London Math. Soc., 24 (1949), 65-67.
N. Levinson and 0 . K. Smith, "A general equation for relaxation oscillation," Duke Math. J., 9(1942), 382-403.
P. Ponzo and N. Wax, "On periodic solutions of the system x = y- F (x), ii= - g(x)," J. Differential Equations, 18(1974), 262-269.
T. A. Burton, "On the equation x +f (x)h(x)x +g(x) =e(t)," Ann. Mat. Pura et Appl., 85(1970), 277-286.
J. R. Graef, "On the generalized Lienard equation with negative damping," J. Differential Equations, 12(1972), 34-62.
J. R. Graef, "Asymptotic behavior of solutions of a second order linear differential equations," J. Differential Equations, 17(1975), 461-476.
J. W. Heidel, "A Liapunov function for a generalized Lienard equation," J.Math. Anal. Appl., 39(1972), 192-197.
M. Yamamoto and S. Sakata, "On the boundedness of solutions and the attractivity properties for nonlinear second order differential equations," Math. Japon, 27(1982), 231-257.
T. Hart and T. Yoneyama, "On the global center of generalized Lienard equation and its application to stability problems," Funkcialaj Ekvacioj, 28(1985), 171-192.
B. Van Der Pol, "Forced oscillations in a circuit with resistence(recepatance with reactive triode) " Lon don , Endibrugh and Dublin Phil. Mag., 3(1927), 65-80.
A. Lienard, "Etude des oscillations entretenues," Rev. Generale de l'Electricite, 23(1928), 901-912.