Catastrophes {catastrophe theory}| are space discontinuities [Thom, 1968] [Thom, 1972] [Waddington, 1968] [Woodcock and Davis, 1978]. Discontinuity type depends on dimension number. Number of parameters determines how many system states are possible. Actual behavior depends on present state and past history.
transformations
Spaces with four or less dimensions allow seven discontinuous transformations: fold, cusp, swallowtail, butterfly, parabolic umbilic, elliptic umbilic, and hyperbolic umbilic. No other catastrophe types are possible. Discontinuities can appear in continuous-equation systems.
Folds make a discontinuity line between two planes {fold catastrophe}. It involves one dimension and only one state. From fold point, either stable or unstable behavior can happen.
Folds along two dimensions make two discontinuity lines, which meet at a point between three planes {cusp catastrophe}. From meeting-point state, states can diverge on both folds, with no middle behavior between the states. Different positions and directions make different state changes {hysteresis, catastrophe}.
Folds along four dimensions make four discontinuity lines, which meet at a point between five planes {butterfly catastrophe}.
Folds along three dimensions make three discontinuity lines, which meet at a point between four planes {swallowtail catastrophe}.
Folds along three dimensions make three discontinuity lines, which meet at a line {elliptic umbilic catastrophe}. It involves three dimensions and only two states.
Folds along three dimensions make three discontinuity lines, which meet at a line {hyperbolic umbilic catastrophe}. It involves three dimensions and only two states.
Folds along four dimensions make four discontinuity lines, which meet at a surface {parabolic umbilic catastrophe}. It involves four dimensions and only two states.
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Date Modified: 2022.0225