Double grazing bifurcation route in a quasiperiodically driven piecewise linear oscillator

Considering a piecewise linear oscillator with quasiperiodic excitation, we uncover the route of double grazing bifurcation of quasiperiodic torus to strange nonchaotic attractors (i.e., SNAs). The maximum displacement for double grazing bifurcation of the quasiperiodic torus can be obtained analytically. After double grazing of quasiperiodic orbits, the smooth quasiperiodic torus wrinkles increasingly with the continuous change of the parameter. Subsequently, the whole quasiperiodic torus loses the smoothness by becoming everywhere non-differentiable, which indicates the birth of SNAs. The Lyapunov exponent is adopted to verify the nonchaotic property of the SNA. The strange property of SNAs can be characterized by the phase sensitivity, the power spectrum, the singular continuous spectrum, and the fractal structure. Our detailed analysis shows that the SNAs induced by double grazing may exist in a short parameter interval between 1 T quasiperiodic orbit and 2 T quasiperiodic orbit or between 1 T quasiperiodic orbit and 4 T quasiperiodic orbit or between 1 T quasiperiodic orbit and chaotic motion. Noteworthy, SNAs may also exist in a large parameter interval after double grazing, which does not lead to any quasiperiodic or chaotic orbits.

Automated summary

Considering a piecewise linear oscillator with quasiperiodic excitation, the route of double grazing bifurcation of quasiperiodic torus to strange nonchaotic attractors (SNAs) is uncovered. The maximum displacement for double grazing bifurcation of the quasiperiodic torus can be obtained analytically. SNAs are born when the smooth quasiperiodic torus loses its smoothness and becomes everywhere non-differentiable after double grazing. The nonchaotic property of SNAs is verified using the Lyapunov exponent, while their strange properties are characterized by phase sensitivity, power spectrum, singular continuous spectrum, and fractal structure.