Devised originally by René F. Thom (1923-2002) to model mathematically Waddington’s epigenetic landscape. It is a theory of dynamical systems based on analogy with topographical form and dealing with instabilities in a limited sub-class of dynamical systems that can be described by potential functions. If a system consists of n variables, then one of its states can be represented by a region in this space. Such regions are subjected to topographical classifications (e.g., cusp catastrophe, see figure below) and, under particular circumstances, a sudden or catastrophic jump can take place from one state to a qualitatively different one as a consequence of changes in two or more control parameters. The theory has been used to model a number of psychological phenomena, but one of its drawbacks for modeling developmental processes is that it cannot yet deal with longitudinal data.
A cusp model depicting change in grasping. Smooth continuous change between two types of grasping is shown by the path from a to f, which does not pass through the bifurcation set. If the movement along this path was to the opposite point e, then small differences between the combined values of the two control parameters could be attracted along the downward slope to widely different outcomes as the point is approached – one going along the pincer grasp surface, the other along that for the scissor grasp. A move from a to b, by altering task demands at a particular competence level, leads to crossing the bifurcation set and there discontinuous change (from c to d). Such a change may involve a hysteresis cycle (b->c->d->e->b) in which the two forms of grasping co-exist temporarily, but with no intermediate values.
See Analogy (as trope), Bifurcation, Chaos theory, Control parameter, Dynamical system, Dynamical systems approaches, Epigenetic landscape, Hysteresis, Irreversible thermodynamics, Microgenetic method, Non-linear dynamical systems, Non-linear dynamics, Ontogenetic development, Order parameter, Phase transition (or shift), Quantitative and qualitative change, State (or phase) space, Synergetics