  This paper considers the problem of switching between two periodic motions,
also known as limit cycles, to create agile running motions. For each limit
cycle, we use a control Lyapunov function to estimate the region of attraction
at the apex of the flight phase. We switch controllers at the apex, only if the
current state of the robot is within the region of attraction of the subsequent
limit cycle. If the intersection between two limit cycles is the null set, then
we construct additional limit cycles till we are able to achieve sufficient
overlap of the region of attraction between sequential limit cycles.
Additionally, we impose an exponential convergence condition on the control
Lyapunov function that allows us to rapidly transition between limit cycles.
Using the approach we demonstrate switching between 5 limit cycles in about 5
steps with the speed changing from 2 m/s to 5 m/s.
