2. Variables n = number of spokes (here n= 6)

3. What are the possible motions? 1. Rocking to a stop on two spokes 2. Rolling down hill at a constant speed 3. Rolling down hill at ever increasing speeds (for large slopes only) 4. Swinging up and balancing on one spoke (takes infinite time) 1. Rocking to a stop on two spokes 2. Rolling down hill at a constant speed 3. Swinging up and balancing on one spoke (takes infinite time)

4. Why not?

5. Balancing on one spoke If PE gain at vertical = KE after collision Uphill Uphill Downhill Downhill

6. Notation Change

7. Balancing on one spoke If PE gain at vertical = KE after collision Uphill Uphill Downhill Downhill

8. Next two cases 1. Rocking to a stop on two spokes 2. Rolling down hill at a constant speed Theta repeats Theta repeats Omega tells us all we need to know Omega tells us all we need to know

9. Poincaré Map If a system has a repeating variable, keep track of the value of the other variables every time that variable repeats. If a system has a repeating variable, keep track of the value of the other variables every time that variable repeats. Since after each collision theta takes the same value, our mapping will be a mapping of angular velocity from after collision n to after collision n+1 Since after each collision theta takes the same value, our mapping will be a mapping of angular velocity from after collision n to after collision n+1

10. Three sections of the map Rolling downhill Rolling uphill Reversing direction

11. Can’t get over vertical The wheel will hit the ground with exactly the same speed as what it left the ground with, just in the opposite direction. We already have a mapping from before to after collisions Combining them

12. Can get over vertical 1. Use conservation of energy to map from after one collision to just before the next collision 2. Combine this with the mapping we have from before to after collisions to get a mapping from after to after

13. Can get over vertical (part 1) Downhill Downhill Uphill Uphill

14. CombineWithYields Can get over vertical (part 2)

15. Graphical Representation (Fig 7)

16. Fixed Points Should be one for each section Stopped on two spokes Stopped on two spokes Constant rolling speed Constant rolling speed

17. Stability of Fixed Points A fixed point of a map is stable if Rocking to a stop Rocking to a stop Constant rolling speed Constant rolling speed Both Stable!

18. Time for some MatLab

19. Existence of fixed points When they exist they are stable, but do they always exits? When they exist they are stable, but do they always exits? For some n and alpha, could they not exist? For some n and alpha, could they not exist?

20. Stopped on two spokes Intuition tells us that as long as slope is not too large, this fixed point should exist. As long as the CG is located between the two resting spokes

21. Continuous Rolling Intuition tells us that as long as slope is large enough, this fixed point will exits. But how large is large enough?

22. Continuous Rolling (Math) Might be a bit too tough at first. How about as long as the value of the fixed point for rolling is greater than the vertical standing speed. Results in the following expression (Eq. 22) Case when exactly equal defines

23. How about when for Continuous Rolling (Intuition) Use the fact that Case when exactly equal defines

24. Existence of fixed points Stopped on two spokes Stopped on two spokes Constant rolling speed Constant rolling speed

26. Basins of attraction Which IC’s get mapped to which fixed point? Which IC’s get mapped to which fixed point? Some are mapped to the rocking to a stop fixed point while some get mapped to the constant rolling fixed point. Some are mapped to the rocking to a stop fixed point while some get mapped to the constant rolling fixed point. Very few are mapped to vertical balancing Very few are mapped to vertical balancing

27. Regions which are mapped to the stopped fixed point and regions which are mapped to the rolling fixed point are separated by points which get mapped to vertical balancing positions. Basins of attraction

28. The vertical boundary points The balancing speeds after impact Any IC which eventually gets mapped to one of these velocities will be a boundary between the different basins of attraction.

29. Notation Time Initially rolling downhill and ending up balancing. (only for very small slopes) Initially rolling downhill and ending up balancing. (only for very small slopes) Initially rolling uphill and balancing on an uphill rotation. Initially rolling uphill and balancing on an uphill rotation. Initially rolling uphill and balancing on a downhill rotation. Initially rolling uphill and balancing on a downhill rotation.

30. Backwards Mapping

31. All three sequences

32. Graphical Representation

33. Final Results!