## Limit Cycles

I’m reading Colin Adams’ Zombie & Calculus. Zombies are not an interest of mine. There are no Zombie Rescue stickers on my car. I’ve never seen a Zombie movie and won’t. All right, I’ve escaped the Zombies. My differential game.

In the book the protagonists, those who have not yet been caught by the Zombies, and the antagonists, the Zombies engage in differential games of capture and pursuit. A differential game is a game played between two or more players where each player has their own differential equation. A differential equation expresses a rate like iterations per year. Some design considerations might involve rates like how fast will we drive away our geeks vs.how fast will we attract consumers. Or, bringing it back to capture and pursuit, what can we do about that fast follower?

In the book, the Zombies, playing these games, eventually caught almost everyone. Well, eventually caught up with almost everyone is more like it. Almost? Well, the author lived to tell the tail. Meanwhile back in the games, some of the protagonists were able to see each individual’s attempt to stay away from the Zombies. They saw every experiment. In one of the experiments, the protagonist took a circular path. The Zombies kept their eyes on the runner. The Zombies ran in a circle, so they could keep their eyes on the runner. The Zombies kept moving, but could never get closer to the runner. The runner was faster; the Zombies, slower. The ratio of their speeds determined how far the runner’s circle was from the Zombies’ circle. The runner circle is the limit cycle. Cycles like in trigonometry class where we spent time going round and round. Not cycles like those in The Lincoln Lawyer. Nobody was spiraling. Nope, just circles. So, lets look at a limit cycle.

The faster person, the red dot, runs on the black circle, the outer circle, the limit cycle. The slower person, the blue dot, runs on the gray circle, the inner circle. The blue arrow illustrates the speed of the slower person. The blue arrow is a vector. Enough of that. The slower person keeps the faster person in sight along the thin blue line, the tangent line. Stop that. Oh, another differential game. Colin Adams, the author didn’t throw down a theorem to define the limit cycle. No, he evaded, the proof-based mathematics, he had his own differential game going. He was pursuing an audience that wouldn’t stand for that.

In the next figure we’ll illustrate what happens when the slower person speeds up or slows down.

The faster person, the red dot is still on the black circle. At the top, the second person has sped up, which made their circle bigger and closer to the limit cycle. In the middle, the second person has slowed down, which made their circle smaller and more distant from the limit cycle. The blue circle is just there to remind us that there are larger scopes even for the larger player. Ultimately, a limit cycle comes with a category. The market leader as determined by market power sells more, has more customers, more money, and more people than anyone else. If you compete with this company you won’t exceed their limit cycle until they bail out of the category. That’s something they wouldn’t do if they saw the point in staying behind.

And, yes, nobody competes for market power these days. Here in the late main street/consumer phase of the technology adoption lifecycle, its all about promo spend and the myth that you can beat the limit cycle. Las Vegas will take that bet.

But, just for grins, lets say your pursuer finally managed to match your speed, what happens next?

Once you’re pursuer is as fast as you are, they won’t be inside the circle looking at you. They will realize that they still can’t catch you. They’ll stumble on the Wayne Gretzky quote about being where the puck is going, instead of where it was just now. Well, your staff saw the day when a chunk of their offer that that this pursuer competes on would commoditize, and they made sure that they had some discontinuous technological innovation that everyone could ride for the next twenty years. They put a real option on the circumference of the current limit cycle, an unweighted control point ready to bare some weight.

So your pursuer was surprised–“Oops,” but they keep trying to catch you. Unfortunately a miracle happens.

They end up inside your new limit cycle. Your pursuer can’t be faster than you, so being slower, the parallelogram flips rotating them inside your new limit cycle. An odd thing happens though. The center of their cycle is not concurrent, is not the same, as the center of your cycle. The more knowledge it took to create your new offering, the more learning they will have to do. They will be elliptical, not shown, until their process maturity has them following you again without additional learning. Mean while, they dream of the day they own the limit cycle, aka they day you cash out of the category. The elliptical brings with it some messier math.

So as product managers, we compete about and in many limit cycles.

Each of our features are either about a limit cycle, like the red dot on black circles, or in a limit cycle, like the red dot on the thin blue circle. Each red line in the factor analysis on the right corresponds to the radius of a limit cycle on the left, except inside the blue circle. The red dot is you. Inside the blue circle the factor line is attached to your red dot. All the limit circles are inside a hyperbolic surface. We could just say F2 is a belief function under distribution. Or, we could learn to draw the hyperbolic surface first and fit the circles inside it later.

Realize that you are a fast follower of your whole product components’ providers. You are using someone’s APIs, you follow them, you chase them. It doesn’t feel like pursuit and capture, because they keep their third-party developers informed. Or, do they? Value chains involve limit cycles as well. And, after doing the math, it might be quicker just to buy their company and integrate their functionality–the pursed captures their pursuer, not a good thing when Twitter or Microsoft did it.

As for the Zombies, the last page still eludes me.