Chaotic Surface Waves in Limited Power-Supply Cylindrical Tank Vibrations

Deterministic chaos of a spherical pendulum under limited excitation

“…In the case of idealization of an excitation source, the system of equations (9) decomposes into two subsystems. One subsystem consists of the first, second, fourth, and fifth equations of system (9), and the second subsystem consists of the third equation of system (9). Since the influence of oscillations of the pendulum on the rotation of the shaft of the motor is neglected, we set D = 0 in this equation.…”

“…In what follows, we investigate possible types of attractors of the system of equations (9). Since this is a fairly complicated nonlinear system of equations, a combination of numerical methods and algorithms is used for the construction of its attractors.…”

“…The initial conditions have been varied in the neighborhood of the origin of coordinates of the phase space of the system of equations (9). As a bifurcation parameter, we have used the slope angle of the static characteristic of the electric motor E, which depends on the type of the motor used.…”

“…This approach, together with the wide application of various reduction methods, enables one to reduce the order of dynamical systems used for the investigation of oscillations of pendulums. However, in many cases, this idealization leads to substantial errors in the qualitative and quantitative description of dynamical modes [9,[13][14][15]. The radical change in the world outlook caused by the discovery of deterministic chaos in dynamical systems considerably extended our notion of possible stable oscillation modes in pendulum systems.…”

“…However, an interest in the investigation of various aspects of the dynamical behavior of pendulum systems is mostly caused by the fact that many effects and phenomena first discovered in pendulum systems were later established for much more complicated physical systems such as rings, shells, plates, and various media in cylindrical and spherical cavities. Moreover, various pendulum systems were successfully used for the approximate mathematical simulation of the dynamics of the indicated complicated oscillating systems [6][7][8][9][10][11]. Sometimes, despite the considerable simplification, the obtained differential equations of motion fairly exactly describe the dynamics of these complicated oscillating systems.…”