A cusp catastrophe occurs when there are two control parameters or variables and a single response variable. It is an algebraic unfolding of the singularity at V(x) = x4, and has the general equation V(x) = x4 + ax2 + bx, where a and b are the control parameters. The bifurcation is now of co-dimension two, and may be visualized as a manifold or smooth surface (Figure 4). Notice that this surface exhibits both the fold catastrophe, as well as a cusp at the origin (a = 0 and b = 0). This cusp can be understood as the bifurcation essentially looping back on itself, allowing repeated transitions between alternative stable states. The surface of the manifold represents the set of all possible states of the system as determined by a and b, and because the surface in the neighborhood of any particular point possesses a similar tangent, the states are essentially indistinguishable. If b is increased or decreased, then since the slope of the tangent to the manifold is undefined (or infinite) at point t1, the system will transition at mathematically infinite speed between the lower and upper sheets, yielding

Cusp

Cusp

Figure 6 Light caustics reflecting off the bottom of the swimming pool at Filoli Gardens, California.

Table 1 Thom's classification of catastrophe manifolds for up to five input parameters, and one or two response variables

Catastrophe manifold

Figure 6 Light caustics reflecting off the bottom of the swimming pool at Filoli Gardens, California.

together at the cusp. Any system or process entering the region of transition will move rapidly from one state to the other upon exiting the region.

The projection or mapping of smooth surfaces onto spaces of lower dimension is a practical feature of singularity theory, and many examples may be found in nature. For example, apparent singularities are encountered when folded three-dimensional surfaces are projected onto the human retina. Perhaps the most robust applications of 'catastrophe theory' have been in the field of optics, where caustics, or the apparent concentration of light into bundles, are explained as the mapping or projection of families of light rays, defined by catastrophe manifolds, onto sensitive or reflective surfaces. One classic example is the appearance of light caustics in water, seen for example as reflections off the surface of the ocean, or off the bottom of a swimming pool (Figure 6). Apparent singularities in the reflections, such as the crossing of lines, or three-way branching points, are seen upon closer examination to be the familiar fold and cusp catastrophes. Also apparent are catastrophes of co-dimension greater than two, since multiple parameters, reflective and refractive events, may intervene before final reflection of the original incident rays onto a sensitive receptor. This suggests the existence of manifolds more complex than the fold and cusp catastrophe manifolds.

Thom described seven catastrophe manifolds for up to four input or control parameters and one or two output or response variables, defining the 'classification theorem' (Table 1). All these catastrophe manifolds exhibit three fundamental properties. First, there is the catastrophic transition between states as the system crosses folds or cusps on the manifold. Second, if the system reverses its trajectory, the transition back to a previous state occurs at parameter values that are not equal to the parameter values at which the initial transition was recognized. This is the hysteresis phenomenon, and is visualized by

Table 1 Thom's classification of catastrophe manifolds for up to five input parameters, and one or two response variables

Catastrophe manifold

Number of input |
One response |
Two response |

parameters |
variable |
variables |

1 |
Fold | |

2 |
Cusp | |

3 |
Swallowtail |
Hyperbolic umbilic |

3 |
Elliptic umbilic | |

4 |
Butterfly |
Parabolic umbilic |

5 |
Wigwam |
Second elliptic |

umbilic | ||

5 |
Second hyperbolic | |

umbilic | ||

5 |
Symbolic umbilic |

reversible trajectories on the fold morphology (Figure 4) or its mapping onto the control plane (Figure 5). Third, neighboring system states that are initially indistinguishable (i.e., the manifold is relatively flat on local scales) may diverge as they follow trajectories into the neighborhoods of cusps and folds. Thom's list has since been extended for up to 5 input parameters, yielding a total of 11 catastrophe manifolds. The central point of the theorem, though, is that the manifolds are not arbitrary descriptors of the behavior of systems with r inputs and n outputs; they are in fact the canonical descriptors of such systems, and are expected to explain the presence of discontinuities in those systems. It is therefore reasonable to expect that these manifolds may serve as explanations for the existence of, and transition between, alternative stable states in multiparameter ecological systems.

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