On the dynamics in an auto-balancing system

Preprint, 2017

A simple symmetric physically realisable auto-balancing system with a six dimensional phase space is analysed and its dynamics is simulated. If the mass of the moving correction particle is obtaind from static balancing condition, then the system contains two fixed points where one is the balanced state and the other corresponds to a whirling motion of the rotor. By using numerical continuation techniques a bifurcation analysis of codimension one is performed, but some results for codimension two bifurcations are also shown. When the bifurcation parameter is equal to one the motion switches from subcritical to supercritical motion at a branch point where the two fixed points coincide and an exchange of stability occurs through a saddle-node bifurcation, but only in a very small interval of the parameter. Both fixed point becomes unstable through a subcritical Hopf bifurcation almost immediatley after the branch point, leading to a complicated dynamics for an interval of the bifurcation parameter. The balanced state then becomes stable through a supercritical Hopf bifurcation at the end of the interval where the system is in a supercritical motion. Numerical simulations reveal the complicated dynamics in the interval between the two Hopf points and calculation of the Lyapunov exponents and Poincar\'{e} maps shows that chaotic motions exists mixed with mode locking states with k-periodic orbits of different orders of k. Numerical bifurcation diagrams shows Feigenbaum routs of period doubling cascades to chaotic motion.