phase portrait of the motion of this “harmonic oscillator,” is composed of a

phase portrait of the motion of this “harmonic oscillator,” is composed of a (continuous) series of points representing its location, graphed along x, its rate of motion graphed along , y= = x and = co-localize the circular orbit as it speeds up and slows down while it bobs up and down. The transition from a fixed point (the mass at rest) to a circle (the bobbing mass), a bifurcation in phase space, results in the loss of topological equivalence. That is, the phase space geometries before and after the bifurcation cannot be smoothly distorted into each other. Continuity and connectedness of the space is lost. For topological equivalence, stretching, bending and warping are allowed but not tearing apart and/or gluing together. Following the bifurcation of a fixed point into a circle, even limitless shrinking of the ring leaves a hole. The appearance or disappearance of an equilibrium fixed point (called a “saddle-node” bifurcation), splitting into two (“period doubling” bifurcation), its exploding into a circle (“Hopf’ bifurcation to a limit cycle), a circle splitting into two or more incommensurate cycles (“secondary Hopf’ bifurcation) and these multiperiodic (“quasiperiodic”) dynamics breaking down into a recursive spirals (“homoclinic bifurcation to chaos”) are among the common bifurcations in nonlinear dynamical systems, and all of them have been observed in many neurobiological settings. In the forced-dissipative (energetically driven and energy consuming) dynamical systems relevant to the neurosciences---this characteristic contrasts with the dissipation free momentum of the classical mechanics of astrophysical bodies--- there are four “most generic” bifurcation scenarios as a parameter changes that may, but need not, lead to chaos (see below for definition) (for early and physically oriented treatments see Eckmann, 1981; Ott, 1981; Berge’ et al, 1984, Kaneko, 1983). These scenarios are: (1) Fixed point or cycle splittings into twice-as-long period lengths 1>2—>4-—8->16...called the subharmonic or “period doubling route”; (2) The transformation of fixed points to one and then more periodic orbits, multiple independent (nonharmonic, incommensurate) frequency oscillations, their mode lockings and then breakdown called the “quasiperiodic route”; (3) Fixed point or cyclic equilibria metamorphosed into irregular bursting patterns called the “intermittency route”; and (4) In the context of quasiperiodic dynamics, adjacent 195 HOUSE_OVERSIGHT_013695