The origins of resilience theory are many... see
this site.
Originated by the French mathematician Rene Thom in the 1960s, catastrophe theory is a special branch of dynamical systems theory [of which Henri Poincare played a key role in developing]. It studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances.
Catastrophes are bifurcations between different equilibria, or fixed point attractors. Due to their restricted nature, catastrophes can be classified based on how many control parameters are being simulataneously varied. For example, if there are two controls, then one finds the most common type, called a "cusp" catastrophe. If, however, there are move than five controls, there is no classification.
The image of the folded structure - the 'landscape' - demonstrates neatly that it is not necessary for a 'disturbance' to 'push' a system into a new state (dynamic attractor). Even only slow changes in state variables (slow variables in resilience language) can suddenly bring the system into a region of the folded structure that has inherent non-lineraties. Indeed it was Henrik Poincare's interst in geometric and topological structure of allowed states that could explain sudden shifts in dynamic behaviour of different systems, i.e. the shape of state space surfaces brings us understanding. The hand drawn figure above is really nice in demonstrating this.
From a different 'angle', we can understand system bifurcation points as originating out of this folded structure.
Roughly speaking, a bifurcation is a qualitative change in an attractor's structure as a control parameter is smoothly varied. For example, a simple equilibrium, or fixed point attractor, might give way to a periodic oscillation as the stress on a system increases. Similarly, a periodic attractor might become unstable and be replaced by a chaotic attractor. [...] In a dripping faucet at low pressure, drops come off the faucet with equal timing between them. As the pressure is increased the drops begin to fall with two drops falling close together, then a longer wait, then two drops falling close together again. In this case, a simple periodic process has given way to a periodic process with twice the period, a process described as "period doubling". If the flow rate of water through the faucet is increased further, often an irregular dripping is found and the behavior can become chaotic.
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