This paper by Shaw and Holmes explores a periodically forced piecewise linear oscillator, a system where the restoring force is linear but changes slope at a certain point. The authors analyze harmonic, subharmonic, and chaotic motions, focusing on bifurcations that lead to these behaviors. They consider both the general case and the limiting case where one slope approaches infinity, resulting in an impact oscillator. The paper uses a Poincaré section method to study the system, defining a section at the points of stiffness discontinuity and analyzing the induced mapping. The stability of periodic points and bifurcations are determined by the eigenvalues of the linearized map. The authors derive analytical conditions for bifurcations, particularly the flip bifurcation, and verify these results through numerical simulations. They also investigate the impact limit, where the time of flight during impact is taken to be zero, and find that the system exhibits similar bifurcation patterns. The paper concludes with a detailed analysis of period doubling cascades and the transition to chaotic motions, providing insights into the complex dynamics of piecewise linear oscillators.This paper by Shaw and Holmes explores a periodically forced piecewise linear oscillator, a system where the restoring force is linear but changes slope at a certain point. The authors analyze harmonic, subharmonic, and chaotic motions, focusing on bifurcations that lead to these behaviors. They consider both the general case and the limiting case where one slope approaches infinity, resulting in an impact oscillator. The paper uses a Poincaré section method to study the system, defining a section at the points of stiffness discontinuity and analyzing the induced mapping. The stability of periodic points and bifurcations are determined by the eigenvalues of the linearized map. The authors derive analytical conditions for bifurcations, particularly the flip bifurcation, and verify these results through numerical simulations. They also investigate the impact limit, where the time of flight during impact is taken to be zero, and find that the system exhibits similar bifurcation patterns. The paper concludes with a detailed analysis of period doubling cascades and the transition to chaotic motions, providing insights into the complex dynamics of piecewise linear oscillators.