TY - JOUR

T1 - On stability of zero solution of an essentially nonlinear second-order differential equation

AU - Bibikov, Yu. N.

AU - Pliss, V.A.

AU - Trushina, N. V.

N1 - Bibikov, Y.N., Pliss, V.A. & Trushina, N.V. On stability of zero solution of an essentially nonlinear second-order differential equation. Vestnik St.Petersb. Univ.Math. 50, 235–241 (2017). https://doi.org/10.3103/S1063454117030062

PY - 2017/7/31

Y1 - 2017/7/31

N2 - Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.

AB - Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.

KW - STABILITY

KW - periodic perturbation

KW - oscillator

KW - essentially nonlinear differential equations

UR - https://link.springer.com/article/10.3103/S1063454117030062

M3 - Article

SP - 235

EP - 241

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

ER -