Bifurcation study of a chaotic model variable-length pendulum on a vibrating base

It is known that in the dissipative system of an inverted pendulum of constant length on an oscillating base, a cascade of bifurcations arises, leading to chaos. In this paper, the appearance of chaotic behavior of a conservative variable-length pendulum on a vibrating base near the upper equilibrium position at high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and the point of its suspension is discovered and investigated. As a mathematical model, a non-autonomous averaged second-order system with dissipation near resonance 1:2 between the oscillation frequencies of the length and oscillations of the suspension point is used. A numerically-analytical bifurcation study of an autonomous control system and a non-autonomous dissipative system is performed at a decrease in the dissipation coefficient to zero. Cascades of bifurcations of limit cycles in the neighborhood of the upper equilibrium position, leading to the formation of a chaotic attractor, are found. The presence of dynamic chaos is proved by graphs and maps of the largest Lyapunov exponent, by maps of dynamic regimes and bifurcation diagrams.