Chaotic dynamics of pendulum ring chain with vibrating suspension

Topic and aim. The aim of the work is to introduce into consideration a mechanical system that is a chain of oscillators capable of demonstrating hyperbolic chaos due to the presence of attractor in the form of the Smale–Williams solenoid. Investigated model. We study the pendulum ring chain with parametric excitation due to the vertical oscillating motion of the suspension alternately at two different frequencies, so that the standing wave patterns appear in the chain with a spatial scale that differs by three times. In this case, the spatial phase on a full modulation period is transformed in accordance with the three-fold expanding circle map, and due to the present dissipation, compression in the remaining directions in the state space of the Poincare map gives rise to the Smale–Williams attractor. Results. A numerical study of the dynamics of the mathematical model was carried out, which confirmed the existence of attractor in the form of a solenoid, if the system parameters are selected properly. The illustrations of the dynamics are presented: diagrams illustrating the topological nature of the mapping for the spatial phase of standing waves, portraits of the attractors showing structure characteristic of the Smale–Williams solenoid, power density spectra, Lyapunov exponents. Discussion. Methodically, the proposed material may be interesting for students and post-graduate students for teaching principles of design and analyzing for systems with chaotic behavior. Since equations with nonlinearity intrinsic to a pendulum in a form of sine function occur in electronics (Josephson junctions, phase-locked loops), it may be possible to build electronic analogs of this system, which will operate as chaos generators insensitive to variation of parameters and fabrication imperfections because of the property of structural stability inherent to the hyperbolic Smale–Williams attractor.