Bikes are unstable when standing still, but stable when moving forward. That’s because steering allows us to move the bike's points of support around under the center of gravity and keep it balanced, often with steering adjustments that are small enough not to even realize we're making them. A stationary bike, however, has zero angular momentum, which makes it harder to balance. And that takes away the feeling of cycling.
However, even without angular momentum, a rider can balance a bike, just like a cyclist stopped at a red light. By steering the front wheel to one side or the other, and applying forward and backward pressure to pedals, a rider can keep the combined center of gravity of bike plus rider above the bike's two contact points with the ground.
Cyclists use two main balancing strategies: steering and body movement relative to the bike. Body movements relative to the bicycle – like leaning left and right – have a smaller effect than steering, but allow a rider to make balance corrections by shifting the center of mass side to side relative to the bicycle and base of support. Steering is absolutely necessary to balance a bicycle, whereas body movements are not.
With the patented Tilt bikes we’ve managed to recreate that balancing feeling on a stationary bike. That's because we combined tilting, steering and 'on-pedal' resistance. So for the first time, you can stay on the bicycle when it's standing still AND get the real feel of balancing a bike.
Here is an interesting post for further reading and research on the physics of staying upright on a bicycle.