TY - JOUR

T1 - Стохастическая устойчивость модели авторезонанса с бифуркацией типа центр–седло

AU - Султанов, Оскар Анварович

PY - 2024

Y1 - 2024

N2 - The purpose of this work is to investigate the effect of stochastic perturbations of the white noise type on the stability of capture into autoresonance in oscillating systems with a variable pumping amplitude and frequency such that a center–saddle bifurcation occurs in the corresponding limiting autonomous system. The another purpose is determine the dependence of the intervals of stochastic stability of the autoresonance on the noise intensity. Methods. The existence of autoresonant regimes with increasing amplitude is proved by constructing and justificating asymptotic solutions in the form of power series with constant coefficients. The stability of solutions in terms of probability with respect to noise is substantiated using stochastic Lyapunov functions. Results. The conditions are described under which the autoresonant regime is preserved and disappears when the parameters pass through bifurcation values. The dependence of the intervals of stochastic stability of autoresonance on the degree of damping of the noise intensity is found. It is shown that more stringent restrictions are required to preserve the stability of solutions for the bifurcation values of the parameters. Conclusion. At the level of differential equations describing capture into autoresonance, the effect of damped stochastic perturbations on the center–saddle bifurcation is studied. The results obtained indicate the possibility of using damped oscillating perturbations for stable control of nonlinear systems.

AB - The purpose of this work is to investigate the effect of stochastic perturbations of the white noise type on the stability of capture into autoresonance in oscillating systems with a variable pumping amplitude and frequency such that a center–saddle bifurcation occurs in the corresponding limiting autonomous system. The another purpose is determine the dependence of the intervals of stochastic stability of the autoresonance on the noise intensity. Methods. The existence of autoresonant regimes with increasing amplitude is proved by constructing and justificating asymptotic solutions in the form of power series with constant coefficients. The stability of solutions in terms of probability with respect to noise is substantiated using stochastic Lyapunov functions. Results. The conditions are described under which the autoresonant regime is preserved and disappears when the parameters pass through bifurcation values. The dependence of the intervals of stochastic stability of autoresonance on the degree of damping of the noise intensity is found. It is shown that more stringent restrictions are required to preserve the stability of solutions for the bifurcation values of the parameters. Conclusion. At the level of differential equations describing capture into autoresonance, the effect of damped stochastic perturbations on the center–saddle bifurcation is studied. The results obtained indicate the possibility of using damped oscillating perturbations for stable control of nonlinear systems.

KW - asymptotics

KW - autoresonance

KW - bifurcation

KW - stability

KW - stochastic perturbation

UR - https://andjournal.sgu.ru/ru/articles/stohasticheskaya-ustoychivost-modeli-avtorezonansa-s-bifurkaciey-tipa-centr-sedlo

UR - https://www.mendeley.com/catalogue/f604480f-cf2b-3c2e-acea-6d92a382d615/

U2 - 10.18500/0869-6632-003090

DO - 10.18500/0869-6632-003090

M3 - статья

VL - 32

SP - 147

EP - 159

JO - ИЗВЕСТИЯ ВЫСШИХ УЧЕБНЫХ ЗАВЕДЕНИЙ. ПРИКЛАДНАЯ НЕЛИНЕЙНАЯ ДИНАМИКА

JF - ИЗВЕСТИЯ ВЫСШИХ УЧЕБНЫХ ЗАВЕДЕНИЙ. ПРИКЛАДНАЯ НЕЛИНЕЙНАЯ ДИНАМИКА

SN - 0869-6632

IS - 2

ER -