Dynamic Behavior Analysis of a Rotating Smooth and Discontinuous Oscillator with Irrational Nonlinearity

This paper focuses on a novel rotating mechanical model which provides a cylindrical example of transition from smooth to discontinuous dynamics. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting both smooth and discontinuous characteristics due to the geometry configuration. By using nonlinear dynamical technique, the unperturbed dynamics of the proposed system are studied including the irrational restoring force, stability of equilibria, Hamiltonian function and phase portraits. Note that a pair of double heteroclinic-like orbits connecting two non-standard saddle points are proposed in discontinuous case. For the perturbed system, we introduce a cylindrical approximate system for which the analytical solutions can be obtained successfully to reflect the nature of the original system without barrier of the irrationalities. Melnikov method is employed to detect the chaotic thresholds for the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing in smooth regime. Finally, numerical simulations show the efficiency of the proposed method and demonstrate the predicated periodic solution and chaotic attractors. It is found that a good degree of correlation is demonstrated in the bifurcation diagram, the phase portraits of periodic solution, the chaotic attractor’ structures and the Lyapunov characteristics between the original system and approximate system.