Global dynamics of a harmonically excited oscillator with symmetric constraints in two-parameter plane.

A harmonically excited oscillator with symmetric constraints is considered and the constraints are assumed to be rigid. To calculate the coexisting periodic motions of the impact oscillator and carry out their stability and bifurcation analysis, the smooth flow maps and impact maps are constructed and the calculation methods of their Jacobi matrices are presented. The Jacobi matrix of global Poincaré map for various types of periodic motions can be obtained according to the chain rule of compound map. The two-parameter transition characteristics between 1–p–p_S and 1–(p + 1)–(p + 1)_S orbits and the global dynamics in beat motion and hysteresis regions are investigated by combining shooting, continuation and cell mapping approaches as well as numerical simulation. The periodic saddles are computed and traced to help explain the evolutions of periodic and chaotic motions. The grazing bifurcation of 1–p–p orbit not only creates U1–(p + 1)–(p + 1)_S saddle but also can give birth to Un–(np + 1)–(np + 1)_S saddles as well as a pair of anti-symmetric n–(np + 1)–np and n–np–(np + 1) saddles, and thereby forming the beat motion regions in the transition from 1–p–p_S orbit to 1–(p + 1)–(p + 1)_S orbit. The collision between the stable attractor and periodic saddle can give rise to rich dynamical behaviors, such as grazing and saddle-node bifurcations as well as interior, boundary and attractor merging crises. The unique characteristics of chaotic crises in impact oscillator with symmetric rigid constraints are revealed. A comparative analysis of two-parameter dynamical characteristics in the impact oscillators with symmetric rigid and elastic constraints is carried out. The pitchfork bifurcation can exhibit subcritical characteristics since it is followed very closely by a saddle-node bifurcation. This type of pitchfork bifurcation is defined as SN-type pitchfork bifurcation, which will further enrich the dynamics of non-smooth systems. [ABSTRACT FROM AUTHOR]