Crisis dynamics of a class of single-degree-of-freedom piecewise linear oscillators

We investigate boundary crises, interior crises, and merging crises of a class of single-degree-of-freedom piecewise linear oscillators. From the perspective of the tangency of manifolds, the mechanisms of boundary crises are revealed, and the critical exponents are determined to distinguish between homoclinic crises and heteroclinic crises. As the parameter changes continuously, a chaotic orbit suddenly disappear at a certain critical point and reappear suddenly at another critical point. This phenomenon of two sudden changes in the chaotic orbit is related to boundary crises caused by the tangency of the stable and unstable manifolds of the same unstable periodic orbit. We call the regions formed by the intersection of the stable and unstable manifolds of the unstable period orbits associated with boundary crises as the escape regions. The change in the area of the escape regions induces the sudden disappearance and reappearance of a chaotic orbit. Detailed numerical simulations and analyses show that boundary crises may interact with the hysteresis loop, which induces complex dynamical behaviors, including transitions between a stable periodic orbit and a chaotic orbit repeatedly. In the two parameters space, changing a parameter value in the same direction will cause the decreases of the distance between the two boundary-crisis curves. When the distance is zero, there exist a coalescence point, which we call the crisis-disappearance point. Beyond this point, the chaotic orbit will no longer contact unstable periodic orbits, leading to the disappearance of the boundary crisis. Besides, the crisis-disappearance points associated with interior crises and merging crises are also uncovered.