A beautiful bifurcation

 The world of work is often viewed as a sterile pursuit of checklists, project plans, meetings, coding.. the whole grind, without many emotions or drama. Those last things belong in a movie or TV soaps that we watch after returning home from work. But work does have its moments, of inspiration, serendipity and clarity.. all dramatic enough to be put in a movie, which are the breadcrumbs that show us the way to our life's calling. One such moment happened to me in Bangalore early in my career, and made me decide to pursue control design as a career path. I wanted to pen this down for a long time.. but was not sure if I could share this publicly, despite it being of no commercial value. Since it has been almost 20 years since that incident, writing it down to share an oddly satisfying science discovery and its personal impact on me, without disclosing the business context around it. Hoping it spurs others to remember what drew them to their life's work, and for others who might be like me 20 years back, with a college degree but clueless about what to do next in life, to have faith that the dots will connect.

The location was Peenya industrial estate in outskirts of Bangalore. This was a thriving hub of small and medium scale businesses that sprung up to support Indian space and defense industry, around ISRO and DRDO. The Indian space program, which started from the back of bullock-cart has now reached skies, space, moon & now the mars, in part due to the ingenuity of entrepreneurs from Peenya, the cottage industry that helped India reach the space. It was my first time venturing out of an air-conditioned corporate campus and walk with engineers in Peenya's hot and dusty bi-lanes. We wanted to build a small test setup consisting of two metal wheels, one rubbing against and driving another. Motor connecting to one, and generator to another. A supplier in Peenya had built it per our spec, and we were going to see it in action for the first time today. I was accompanied by a diverse team engineers from our company, some who were veterans from ISRO, BARC, as well as young engineers who had returned from USA to pursue good work in India. Before then, my work was mostly confined to software and simulations.. so I was looking forward to work with the hardware for a change and actually touch the devices I had modelled only in software then. 

The supplier showed us the test setup.. and turned it on. The motor started spinning and driving the two wheels.. We all knew what to expect, the speed will increase and reach a steady state and stay there happily ever after. Just to confirm nothing breaks down, we decide to keep it spinning for 10 more minutes, before approving the rig and shipping it to our lab. The motor was nicely spinning and humming. People started chatting to help pass time, while watching the clock.. That's when something surprising happened. With 1 minutes left to end the test, the pitch of the motor hum changed and motor speed reduced to almost a third of its original. We rushed to see what happened.. of if something broke. Supplier was worried too. He stopped the motor to check its conditions. Everything was nominal, and nothing had broken. He tightened the fixures holding two wheels thinking they were loose. But repeating the experiment longer showed the same pattern, fast speed for ~10 minutes followed by excruciating slow speed for 4-5 minutes, and then to our relief and re-surprise, fast speed again... and repeat. The trasition from slow to fast speed was very quick, within seconds and vice versa. After watching this toggle of slow and fast spinning for a few cycles, we decide to accept the setup and get it shipped to the lab, because after all the supplier had built per our spec. It was not his problem that the physics defied our expectations of how it should work.

But on the way back, I was frankly puzzled and could not understand what happened and why.. I knew bits about control and stability theory from college. But honestly the only impression it left me was how tedious it is to draw root locus diagrams by hand (which our professors made us do, and drained last ounce of interest we had left in the subject) and how complex and scary are the equations it threw at us. I had little idea of how those connect to real world. But today I saw something in Peenya that I could not explain, and knew those textbook may offer help. So I started writing down the equations from first principles and see what is happening. Textbooks of linear control theory cover how things stabilize or runaway to instability  because of either virtuous or vicious feedbacks, respectively. But this phenomena was neither. Things were stable but only for less than 10 minutes, before veering off to either slow/fast speed. Luckily for me, my roommate then, Jimmy was a sponge for knowledge. He would pursue knowledge for its sake from all places. His passion was fluid dynamics, so we would check out books from National Aerospace Labooratory library, squat in few courses in IISC on a back bench, read up photo-copied (this is a copyright violation btw. Please do at your own risk.) spiral-bound books on turbulence on the bus to office, basically nerd out about ingenuity of scientist that lived more than a century ago (Komogorov's 4/5th law of Turbulence anybody?). I frankly used to look up to him, and he was a good influence on me. Well.. he had passed on to me a copy of Steve Strogat'z "Nonlinear dynamic and chaos". This gem of a book mentioned something called a bifurcation, where an otherwise stable system can undergo a bifurcation under some conditions, and exhibit both a stable and unstable equilibrium. It also mentioned that instead of stability or instability, Nonlinear systems can have limit cycles. But these are only possible within non-linear systems.  I came to realize that what happened in Peenya, and later many times in our lab was both a bifurcation and a very slow limit cycle, cycling every 10mins or so. 

So here is what happened. Every motor has a torque-speed curve, in steady-state this is balanced by the torque from rubbing or the friction between the two wheels. This friction has a similar curve of torque vs. slip (relative contact velocity). At small slip, friction increases with slip (also called front-side), but for large enough slip it decreases (called backside), and is thus non-linear. This is why we should not break too hard when going very fast.. act of braking too hard can increase slip and decrease the friction. Anti-lock brakes in cars avoid precisely this. If the motor torque-speed curve intersects friction torque vs. slip curve from top, if speed increases motor torque decreases more than braking torque, reducing the speed. This negative feedback keeps the speed stable. On the other hand, if it cuts from below, small speed increase leads to even more increase and the vicious cycle drives system into run-away speed increase. When we start the motor, it found a stable equilibrium at the backside of the slope curve at high slip. But as the rubbing increased, so did energy dissipation and the two wheels started getting hotter. Heat caused expansion of two wheels, but since the wheel centers could not move, the normal force increases raising the whole friction vs. slip curve. The intersection of the motor and friction curve moves at slower motor speed, until the intersection turns unstable and motor quickly slows down to the front of slip curve a new stable equilibrium is reached at much lower speed and slip velocity. This starts cooling off the two wheels and reduction in normal force/friction curve until motor curve climbs up the front side of slip curve and quickly speeds up to backside as the intersection turns unstable.  Due to thermal time constants, all of this happens over many minutes, and we get this beautiful dance of bifurcations modulated by rubbing, heat and normal force, shift from unstable to stable equilibriums and eventually a limit cycle that repeats.

Unfortunately non-linear systems were left out of my college curriculum due to perceived complexity, but that day I realized how reality is mostly non-linear, complex and yet beautiful.  Now a days, with each passing day, the non-linearity of neural network basis functions is powering the GenAI revolution. Who knows.. it may well be the foundation of our own creativity and complexity. That day I was so stoked to see a phenomena that I just read in a text-book few weeks ago, that I considered it a good omen and decided to learn more about the control theory, and spent two decades working with and optimizing non-linear electro-mechanical systems.