question (its differential equation) in the form of what is called its Jacobian matrix of partial derivatives (a matrix representation of the differential equation indicating orthogonal directional velocities of change of locations of the components of the motion with respect to changing values of the control parameter), we know that
when the p of the matrix’s set of two complex conjugate eigenvalues is less than zero, p <0, the orbit representing the system, spirals into a stable fixed point. This
is analogous to going to the bottom of the parabolic attractor basin as described above. Values of the invisible eigenvalues and their changes constitute the abstract mathematical mechanisms underlying the observable dynamics of the system observable physically.
The mathematical mechanism underlying the Hopf bifurcation of fixed points into limit cycles (associated with bi-directional splitting that accompanies the amphetamine transformation into limit cycle stereotypy of rigid ideas and equally likely mirror image motions in the directions of good versus evil) is the crossing of the systems real valued parts, p’s, of its complex conjugate eigenvalues into positive territory, o > 0. The mirror image of clock arrows is transformed from 8:00 and 10:00 o'clock to the clock locations of 4:00 and 2:00. At a Hopf bifurcation, a pair of complex conjugate eigenvalues crosses the imaginary (vertical) axis such that is real parts have positive value. In the orbit representing the motions of the system itself, the fixed point disappears to be replaced by the action spiraling out to an invariant circle. This is analogous to our manifold image of the disappearance of the central attractive point and the sudden appearance of a small hill at the bottom of a parabolic basic of attraction.. The new attractor is an invariant circular path around the hill, with the spiraling out to the invariant circle being a two dimensional picture of the disappearance of the bowl-bottom and appearance of a missing point, hill top fixed point and a spiral flow to the path circling the hill. Underlying the transition from a fixed point to a limit cycling, invariant circle, are a pair of mirror image complex conjugate eigenvalues that turn in mirror image, we could say, good versus evil, opposite directions. The Hopf bifurcating system inevitably has both.
The implications of this very abstract metaphor for the emergent limit cycle-
splitting style of spiritual transformation can be made deeper by considering the
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