Figure 1. The Vector Field of a Dynamical System.

(A) The differential equations, dx/dt = f(x,y) and dy/dt = g(x,y), define a direction and rate of change at each point in the ‘state space’ (x,y). A solution of the differential equations follows the arrows from some starting point until the trajectory (any one of the green dashed lines) reaches a stable attractor (one of the two stable steady-states represented by the black circles). The white circles represent repellers, and the x marks a ‘saddle point’. (B) The vector field in panel A can be associated with this ‘topographic map’, where the contours are plotted at 20 m intervals above the ‘lake’ in the upper left corner. The stable attractors are lakes (blue zones) in two depressions (at elevations of 0 and 20 m), and the repellers are mountain peaks (the white circles, at elevation 260 m). In the middle of the landscape is a saddle point (at an elevation of 130 m), which lies on the boundary (the grey dashed line) between the watersheds of the two lakes.