Explosions in Lorenz maps

We introduce a map to describe the systematics of orbit creation and annihilation in Lorenz-like dynamical systems. This map, y′=b-|y|, has a singular maximum and is useful for describing flows that undergo a tear-and-squeeze route to chaos. We call this map the Lorenz map. We find: much of the dynamics is determined by the bifurcations of the period-one and period-two orbits; orbits are created in explosions (singular saddle-node bifurcations) based on two symbols s0,s1s0,s1, and later removed in inverse processes that are implosions. The order in which direct and inverse explosions occur generally follows the inverse order shown by the logistic map. In the entire parameter range only one regular saddle-node bifurcation and one period-doubling bifurcation occurs.