Birth of strange nonchaotic attractors in a piecewise linear oscillator

Nonsmooth systems are widely encountered in engineering fields. They have abundant dynamical phenomena, including some results on the complex dynamics in such systems under quasiperiodically forced excitations. In this work, we consider a quasiperiodically forced piecewise linear oscillator and show that strange nonchaotic attractors (SNAs) do exist in such nonsmooth systems. The generation and evolution mechanisms of SNAs are discussed. The torus-doubling, fractal, bubbling, and intermittency routes to SNAs are identified. The strange properties of SNAs are characterized with the aid of the phase sensitivity function, singular continuous spectrum, rational frequency approximation, and the path of the partial Fourier sum of state variables in a complex plane. The nonchaotic properties of SNAs are verified by the methods of maximum Lyapunov exponent and power spectrum. Strange nonchaotic attractors (SNAs) can be regarded as a special class of attractors between quasiperiodic attractors and chaotic attractors. The word strange means that the dependence of the dynamical variables to the phase is not given by the smooth relations but constitutes some fractal ones. The word nonchaotic means that the maximum Lyapunov exponent is nonpositive. However, there are few studies on SNAs in nonsmooth dynamical systems. In this work, we study a quasiperiodically forced piecewise linear oscillator. The complicated and interesting strange nonchaotic dynamics phenomena are revealed via numerical methods.