The Fold Catastrophe

The fold is a catastrophe which may occur given one control variable or parameter and a single response variable. It is governed by the general equation V(x) = x ax, and is an unfolding of the singularity at V(x) = x3 (Figure 1). The bifurcation or transition point is of co-dimension one, meaning that a single parameter (a) must be varied for the bifurcation to occur.

Figure 1 A fold catastrophe visualized as the family of polynomials V (x) = x3 - ax. There is only one stable state when a <0, and the system loses stability and bifurcates at a = 0.

Consider the discontinuous process illustrated in Figure 2, where, given an external control parameter a, the response variable y exhibits an abrupt transition, or catastrophe, between states I and II. The interpretation in terms of catastrophe theory is that the process may be described smoothly by the family of polynomial curves y = x - ax. The process has a single stable minimum, or basin of attraction, and an unstable maximum for values a <0 (Figure 1). If a is increased gradually, however, the minimum stable state is slowly eroded as both extrema approach each other. The extrema meet at a = 0, at which point (and for values a >0) the system loses stability and transitions very rapidly to a new state. The expected behavior of the system is stable under small perturbations of a, but a steady increase of the variable may be viewed as an erosion of stability in the system. Nevertheless,

Figure 2 Hypothetical ecological time series, showing a rapid and catastrophic transition between two alternative stable states of the system, I and II. The transition is explainable in terms of the fold catastrophe, as the system moves through the parameter space a = 0.
Figure 3 An elastic beam, forced to buckle upward by the application of a force C to either end. As an increasing weight W is applied to the center of the beam, it will buckle downward catastrophically.

because the response to those increases is negligible until a = 0, the transition is a surprise. The fold catastrophe is the simplest of the so-called elementary catastrophes because it explains the relationship between one control parameter and a single response variable. A simple physical experiment serves as a practical example.

Take an elastic beam to which a constant force C is applied at both ends, causing the beam to buckle upward (Figure 3). A weight W, now applied to the center of the beam, is increased gradually until the beam suddenly buckles downward. If the downward vertical displacement is measured as y, then the alternative stable state is illustrated by the dashed outline in Figure 3. The weight Wis decreased gradually until the beam snaps back to the original configuration (which it might do because of the constantly applied force C). This can be understood intuitively as movement along the surface plotted in Figure 1 , which depicts the family of polynomials that govern the behavior of the beam. The catastrophic changes in y encountered as W is increased or decreased are understood literally as a loss of stability and transition to a new stable state illustrated by the bifurcation on the plane. The region around the critical value of W (a = 0), is a singularity, and a region of transition. Approaching the region is hardly discernible as changes in y, but stability is lost and a catastrophic transition occurs at the bifurcation point. Furthermore, it should be noted that the two states of the beam, buckled upward or downward, are stable states attributable to fixed point attractors.

An Ecological Example

The fold catastrophe is easily conceived in ecology as the existence of alternative states in communities. Examples include the eutrophication of lakes under increasing phosphorus concentrations, and the transition from coral to macroalgae-dominated reefs. Both conditions in these communities may be considered as alternative stable states, since both are robust to minor perturbations in

Ecological Trajectory

Figure 4 The cusp manifold. This surface exhibits both the folds controlled by a, and a cusp, where the folds converge at a = 0 and b = 0. A system moving along one of the blue trajectories will eventually encounter a fold catastrophe (e.g., t1), and transition to an upper or lower sheet. Notice that the point of catastrophe differs depending on the direction of the trajectory. The catastrophe is therefore dependent on the path of the system, or its history. The yellow trajectories demonstrate the divergence of systems that are initially close in value (state), but encounter the cusp.

their environments. Catastrophe theory describes the transition between states because the transition is usually sudden; that is, the time taken for transition is brief relative to the time spent in either stable state, or there is very little indication of the approaching transition. For example, changes in lake turbidity tend to occur on a timescale that is brief relative to time spent in clear or turbid states. Similarly, coral-dominated reef systems that may have withstood battering hurricanes for centuries have, in recent years, transitioned rapidly to macroalgal dominance after single-storm events.

While it is common to depict and seek explanations for these transitions by the examination of ecological time series (e.g., Figure 2), consideration of the transitions in a catastrophic fold framework illuminates the relationship between perturbative events, perturbation magnitudes, and the histories of the systems. Certainly, single hurricanes are unlikely to force a transition to macroalgal dominance, nor does a sudden pulse of phosphorus into a pristine lake result in a stable eutrophic state. Instead, communities are likely to undergo transitions after the cumulative effects of increasing perturbation magnitude, or multiple types of perturbation. This relationship indicates that factors, mostly intrinsic to the system, must be accounted for in addition to the external perturbations, to accurately determine not only the appropriate catastrophic dimension and surface, but also the mapping between the observed response and that surface. These considerations lead directly to a full exposition of the key elements of catastrophe theory, namely catastrophes of higher dimension than the fold, and 'Thom's classification theorem'.

Figure 4 The cusp manifold. This surface exhibits both the folds controlled by a, and a cusp, where the folds converge at a = 0 and b = 0. A system moving along one of the blue trajectories will eventually encounter a fold catastrophe (e.g., t1), and transition to an upper or lower sheet. Notice that the point of catastrophe differs depending on the direction of the trajectory. The catastrophe is therefore dependent on the path of the system, or its history. The yellow trajectories demonstrate the divergence of systems that are initially close in value (state), but encounter the cusp.

the observed discontinuity or catastrophe. The relationship becomes more apparent when mapped onto the control plane ab (Figure 5). Here surfaces of the manifold, which are inaccessible when moving in directions of increasing or decreasing b, are mapped as the shaded region. This region is described by a semicubical parabola, which comes

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