Birth of strange nonchaotic attractors in a piecewise linear oscillator

Abstract

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.

Multiple tori intermittency routes to strange nonchaotic attractors in a quasiperiodically-forced piecewise smooth system

“…Shen et al have verified the existence of SNAs in a nonsmooth map (and equation) and analyzed the statistical properties of SNAs [33][34]. Duan et al confirmed that strange non-chaotic attractors do exist in non-smooth systems [35]. Li et al have studied the generative mechanism of SNAs in a quasiperiodic forced piecewise smooth system [36][37].…”

Quantifying strange property of attractors in quasiperiodically forced systems

“…The phase sensitivity can detect the sensitive to the initial phase, and the strange property of SNA can be verified by the stairstep diagram and the phase sensitivity exponent [4,6,8,9]. In addition, recurrence quantification analysis, distribution of finite-time Lyapunov exponents, spectral distribution function, singular continuous power spectrum can also be used to verify the strange property of SNA [10][11][12][13][14][15]. Recurrence quantification analysis refers to the time when the state of dynamical system reappears.…”