Catastrophe Theory... - resilience theory in the 1960's
The origins of resilience theory are many... see this site.
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.