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common way for electrical circuits, computer programs, brain mechanisms and
other complicated systems, even cultural or spiritual movements, to behave when
one or more important control parameters crosses a threshold. Doyne Farmer of the
Los Alamos’s Prediction Company once said about this vulnerability in complex
system, “Those things can hardly wait to roll up.” The /imit cycle lock-up occurs
most often as a sudden, discontinuous change, called a_ bifurcation, into
autonomous Sself-oscillations from an equilibrium state around which there was
some random variation. A bifurcation, a discontinuous change in outcome from a
smooth changes cause, characteristically occurs when the amount of an important
influence, a metabolic state, a drug, a psychodynamic conflict or level of emotional
stimulation crosses some critical value. The switch from one type of dynamical
behavior to another looks like the system has suddenly changed into something
else with an entirely new kind of life of its own. In the new life of rolled up, locked-up
repetitious motion, almost all new starting conditions follow pathways that lead into
the same limit cycle pattern. Evangelical Christians talk about a// born again life
being in Jesus, fixed in a complete set of moral, social and political beliefs, ideas
and judgments. The limit cycle gets its name because the end state of the orbits of
almost all starting points of the dynamics winds up being drawn into the same fixed,
repetitious pattern of a stable cycle. Visualizing the simulation of one kind of
bifurcation to a limit cycle on a computer screen, we see a slightly jiggling point
explode suddenly into an orbit of ceaseless rotations around a circle.
Ralph Abraham, the University of California at Santa Cruz pioneer in
graphical approaches to nonlinear systems, describes, cinemagraphically, the
emergence of limit cycles from a single point. He starts with a picture of an attractor
of water flow in the shape of a basin. All water that enters the basin, rolls down its
sides to the bottom, to what physicists say represents a potential energy minimum.
A little more technically, this attractor basin is composed of the set of all points such
that the orbits that flow from them tend to end up inside the basin as time goes
toward infinity, no matter where they start. Changing the value of a control
parameter of the system changes the shape of this basin-like landscape, of the
surface of the systems dynamical actions called a manifold, which can intuitively
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