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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 limit cycle lock-up occurs most often as a sudden, discontinuous change, called a bifurcation, into autonomous self-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 all 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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