Sufficient conditions for a period increment big bang bifurcation in one-dimensional maps∗

Typically, big bang bifurcation occurs for one (or higher)-dimensional piecewise-defined systems whenever two border collision bifurcation curves collide transversely in the parameter space. At that point, two (feasible) fixed points collide with the boundary in state space and become virtual. Depending on the properties of the map at the colliding fixed points, there exist different scenarios regarding how the infinite periodic orbits are born, mainly the so-called period adding and period increment. In our work we prove that, in order to undergo a big bang bifurcation of the period increment type, it is sufficient for a one-dimensional map to be contractive near the boundary and to have slopes of opposite sign at each side of the discontinuity. Kewywords: organizing centers, piecewise-smooth maps, border collision bifurcations