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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 δ 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 δ 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

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