remain the same after perturbation? Note that the inter-data point metric distances

remain the same after perturbation? Note that the inter-data point metric distances are not considered. If they do, the two dynamical objects being compared are topologically equivalent. The test of this equivalence requires the mapping one set onto the other with, at most, smooth distortions of either or both surfaces. In the context of catastrophe-related bifurcation theory, if a 5 converts a steady valued fixed point to an oscillating cycle on a manifold of potential actions, also called a state space, then the fixed point system was not structurally stable. In phase space, this is seen as a change-in-causal-parameter induced transformation of a dot to a circle. If the one frequency circle is perturbed to a manifold of the system’s actions consisting of two independent frequencies, the circle takes the topological form of the crust of a doughnut, one frequency graphed spiral winding around the doughnut, the other winding along the doughnut around its orifice, the circle is not structurally stable. If 5 distorts the frequency-amplitude relations on a surface such that the manifold of possible actions is distorted from a doughnut to a tea cup, both topological manifolds being one holed surfaces and therefore topologically equivalent, the system is structurally stable. Perturbed systems that maintain the sequence of points in time in sequential order (though the distances between the points may be different), are generally structurally stable. The seductive possibility, one which Thom realized so successfully, was that in the language of distance-independent differential topological forms, there would exist a small, finite set of shapes categorically describing the causes and result parameter spaces from which, even without specific quantities, universal qualitative (including discontinuous) behavior could be described and sometimes predicted. A formal yet general categorical system within which a small set of universal discontinuous changes in global qualities could be rationalized seemed seductively applicable to the enlightenment transitions, spiritual transformations, appearing suddenly after months and years of disciplined spiritual practice. The Platonic view is that the universal forms of discontinuous change existed before they could be about anything specific, before the universe was born. In this era of nonlinear dynamics and dynamical system, common dynamical scenarios give accounts of smooth changes in causes leading to discontinuous 57 HOUSE_OVERSIGHT_013557