perhaps not so different in variety and complexity from the range of representations in art and literature of the /eft hand of evil and the right hand of grace.
Oscillations that appear spontaneously in nonlinear systems without external periodic input were known to Henri Poincaré in 1882, and were systematically studied and made accessible to non-mathematicians by early 20" Century Russian mathematicians and physicists, well represented by a 1949 book, Theory of Oscillations by the Russian engineer-mathematicians, A. A. Andropov and C.E. Chaikin. Another relatively early classic is Nonlinear Oscillations by Nicholas Minorsky. The most common form of transition from a fixed point to a limit cycle was pictured as changes in the surface of the action, the bow/-hillock manifold in the paragraphs above, and is called a Hopf bifurcation. Recall that bifurcation means a discontinuous change in an observable over a continuous change in what is known as a control parameter, such as dose of amphetamine or intensity of an experience. The mathematical mechanism resulting in circular directional motion represented by the (eigen)vectorial states, was named for the German mathematician, Eduard Hopf. His 1942 paper was a mathematical proof of its existence and was discussed in the context of fluid flows that role up such that circling vortices arise from smooth, called jaminar, water flow, at a critical value of the flow rate. Hurricanes are another example of these kinds of dynamics.
The Hopf bifurcation to limit cycles has been found in several, many dimensional, physical, chemical and biological systems. The latter include calcium conductance oscillations in the excitable membranes of muscle, heart and the brain, cardiac arrhythmias such as ventricular flutter as well as oscillations in population numbers in foxes and rabbits, predator-prey systems. California Institute of Technology’s Professor, James Old and Johns Hopkins Professor, Joseph Brady made experimentally obvious the potential for the rigid irrationality implied by the brain’s inclination to be locked up into limit cycle behavior. They demonstrated that animals, from rats to monkeys, could get locked up in apparent self torture, repeatedly and endlessly pushing a bar to deliver current to pain systems in the
their brains. These pushes induced almost unremitting screams in monkeys and
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