3.2. Equilibrium Continuation
 Figure 5 presents water table equilibrium continuation plots for three values of the decay in gross recharge with depth to water table, r2. The figures provide insight into the dependence of the number and location of attractors on the groundwater outflow rate and recharge. Figure 5b shows that two attractors emerge when Caq is between 405 and 1397 m yr–1. Above 1397 and below 405 only the deep attractor and shallow attractor exist, respectively. Figure 5a shows that decreasing r2 from 0.4 to 0.23 m–1 shifts the two-attractor range to higher values of Caq and significantly reduces the extent of the two-attractor range to between 1187 and 1276 m yr–1. Figure 5c shows that increasing r2 to 0.7 m–1 shifts the two-attractor range to lower values of Caq and modestly reduces the extent to between 33 and 618 m yr–1. With regard to the model behavior within the two-attractor range, a common measure of resilience is the distance between the attractor and the repellor. By this measure, a subtler aspect of the results is that the resilience of the deep water table attractor increases as r2 increases, yet there is not a corresponding decrease in the resilience of the shallow attractor. Additionally, by this measure of resilience, Figure 5b shows that, for Caq, the deep attractor has the greater resilience. However, Figure 4 shows the water table moves considerably more rapidly to the shallow than to the deep attractor which, conversely, indicates that the shallow attractor is the most resilient. This contradiction is due to the inability of equilibrium continuation to characterize the shape of each attractor basin.
3.3. Fold-Point Continuation
 Figure 6 presents the fold-point continuation results for Caq against r2 repeated at four values of boundary condition hydraulic gradient. It shows the parameter space over which two attractors emerge under mean climate forcing. Figure 6c shows the fold-point continuation results for the hydraulic gradient adopted in the previous sections, along with the fold points from equilibrium continuation. These points are in good agreement with the numerical approximation when r2 is greater than 1.0, which indicates the numerical approximation to the Lambert W function is acceptable. Figure 6c also shows that two attractors can only exist if r2 is greater than 0.19 m–1. However, the emergence of two attractors when r2 is greater than 0.19 m–1 is very dependent upon the interaction between the groundwater outflow rate and recharge decay rate. Figure 6c also delineates three regions of the Caq and r2 parameter space, where (1) only the shallow attractor emerges; (2) only the deep attractor emerges; and (3) both the shallow and the deep attractors emerge. The other three plots each show similar regions of the parameter space; however, it can be seen that as the hydraulic gradient increases, the area of the two-attractor parameter space increases; two attractors emerge at a lower recharge decay rate; and the area with only the shallow attractor contracts. Clearly, this delineation of the parameter space enhances understanding of model dynamics. For the scenario of a calibrated model with quantified parameter uncertainty, fold point continuation could provide a numerically efficient means of estimating the probability of two attractors existing. However, for all but the simplest of models an analytical method would be inappropriate. Most often a numerical fold-point continuation method, such as that of MatCont [Dhooge et al., 2003], would be required.
Figure 6. Fold-point continuation results for the aquifer conductance parameter, Caq, and the parameter for the recharge decay rate with depth to water table, r2, and repeated at four values of boundary condition head, ho, (expressed as the hydraulic gradient). (a) Hydraulic gradient of 0.005; (b) hydraulic gradient of 0.01; (c) hydraulic gradient of 0.015; and (d) hydraulic gradient of 0.02. The gray region within each plot identifies the parameter space with two attractors. The black dots in Figure 6c identify the parameter space location of the fold points identified from equilibrium continuation. The dotted vertical line in Figure 6c delineates three additional parameter regions such that (1) above the gray region and to the right of the dotted line only the deep attractor exists; (2) below and to the right of the line only the shallow attractor exists; and (3) to the left of the line only one attractor exists for all values of Caq. The black semicircle within Figure 6c is a numerical artifact resulting from the Lambert W function having a complex solution when is less than .
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3.4. Lyapunov Estimation of Attractor Basin Shape
 Figure 7 shows the Lyapunov stability curves for the five values of Caq denoted on the equilibrium continuation contour plot (Figure 3) repeated from the datum centered at the shallow attractor, repellor and deep attractor. If the datum does not exist for a given value of Caq then its curve was not calculated. Key aspects are, first, that for any datum all equilibrium points were identified. For example, when the datum was set to the shallow attractor, the depth to water table of the repellor and deep attractor was estimated. This occurred because the model ODE was used within the Lyapunov stability function, rather than a simpler equation as is often adopted. A second key aspect of the plots is that the relative depth of each attractor basin depends upon the choice of datum. That is, generally the attractor furthest from the datum (as measured by depth to water table) had the deepest attractor basin. Considering that each stability curve satisfies the required Lyapunov conditions, choosing between them depends upon the nature of the problem.
Figure 7. Lyapunov stability curves for depth to water table (DTWT) at five aquifer conductance parameter, Caq, values repeated from three equilibrium datums where: Shall. denotes from the shallow attractor; Rep. denotes from the repellor; and Deep denotes from the deep attractor.
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 In this example, two aspects of the model give some guidance as to the choice of an appropriate stability curve. First, the exponential function for groundwater evaporation produces a strong negative feedback (as shown by the dense contours near to a zero depth within Figure 3). This indicates that the Lyapunov stability curve should be very steep between the shallow attractor and the land surface; a feature not shown in Figure 7 when the datum was set to the shallow attractor. Second, Figure 4 shows convergence of the water table to an attractor is most rapid when within the shallow attractor basin; a feature also evident from the larger number of shallow attractor basin contours within Figure 3 at Caq of 700 m yr–1. This aspect only emerged when the datum was set to the deep attractor. Considering that the deep attractor is also the state to which the system would converge if no positive feedback existed, the deep attractor datum Lyapunov stability curves are the most plausible.
3.5. Assessing Stochastic Model Dynamics
 In this section, the behavior of the model under stochastic forcing is considered. Figure 8a shows the two stochastic time series of precipitation, which differ only in their standard deviation. Figure 8b shows time series of the depth to water table resulting from input of the precipitation series into the model (with Caq set to 700 m yr–1). Each hydrograph shows a switch from the shallow water table attractor basin to the deep attractor within the first 50 years; after which series A switched briefly back to the shallow attractor basin while series B persisted within the deep attractor, thus indicating it to be asymptotically stable (i.e., with sufficient time the model will converge to this attractor basin). The persistence within the deep attractor, and the rapid switching out of the shallow attractor, indicates the deep attractor to be significantly more resilient than the shallow attractor. However, the steady state forcing simulations (Figure 4) and Lyapunov stability plot from the deep attractor datum and Caq at 700 m yr–1 (Figure 7) clearly indicate the shallow attractor to be the most resilient.
Figure 8. Model simulations from two stochastic precipitation series. The simulations show switching between the two attractors. Within Figure 8b, the solid horizontal lines denote attractors, the upper line designating a shallow attractor and the lower line a deep attractor; the dashed horizontal line denotes the repellor. The points on the green line (A-I) refer to data for series A; those on the gray line (X-Z) refer to data from series B.
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 This contradiction can be explained by better understanding what the Lyapunov stability plots are estimating and their use of climate forcing data. Importantly, the Lyapunov curves summarize the model dynamics when the forcing is set to a constant rate, in this case the mean annual precipitation. Therefore, the Lyapunov stability curve must be interpreted as the rate at which the system recovers to an attractor, following a disturbance, when it experiences only the constant rate of forcing. In no way can it characterize the displacement resulting from, or the resilience to, stochastic forcing input to the model. With regard to equilibrium continuation, for applications like those in Figure 5 that investigate a nonforcing parameter, the state-space distance from an attractor to the repellor cannot be interpreted as the resilience to stochastic forcing input to the model. Like Lyapunov stability curves, this state-space distance can only be interpreted as the cumulative impact from a disturbance not input to the model that the system can withstand before switching attractor basins. For this model, such disturbances would include groundwater pumping or flooding recharge. This is an important, but rarely considered, weakness of equilibrium continuation for resilience analysis; and even when equilibrium continuation has been undertaken for the forcing, the mechanism by which a switching of basins occurs has been unclear [Guttal and Jayaprakash, 2007; Peterson, 2009].
 To explore the mechanism by which stochastic forcing does cause a change of attractor basins, Figure 9 shows the 3-D Lyapunov stability surface for a range of annual precipitation rates and depth to water table (with Caq set to 700 m yr–1). Overlain is also the equilibrium continuation curve for the precipitation rate and the stochastic model simulations from Figure 8, with the corresponding points of an attractor basin change or repellor crossing also denoted. With regard to the surface, the gradient relative to the depth to water table axis shows how the depth to water table will change for a given rate of precipitation. The discontinuity at a precipitation of 1.35 m yr–1 is a numerical artifact of the methods. For precipitations less than 1.35 m yr–1, the Lyapunov curve was calculated from a datum at the deep attractor, whereas for precipitations greater than 1.35 m yr–1 it was calculated from the shallow attractor because that is the only existent attractor for those precipitation rate (when Caq equals 700 m yr–1). Importantly, this threshold does not cause hysteresis; that is, at the deep attractor, high precipitation can cause a crossing of this threshold and lower precipitation can cause a reversal.
Figure 9. Attractor basin shape for precipitation and depth to water table (DTWT) derived using Lyapunov stability functions from the deep attractor datum with an aquifer conductance parameter, Caq, of 700 m yr–1. In the top left, Lyapunov energy values above 1200 m yr–1 were capped to 1200 m yr–1. Note, the solid thick black line denotes an attractor; the solid thick gray line denotes a repellor; the thin solid green line denotes simulations from stochastic climate series A; and the thin solid gray line denotes simulations from stochastic climate series B. Series A is shown to have a counterclockwise hysteresis loop whereby the system starts within the shallow attractor (point A); a dry period shifts it to a state having only the deep attractor (point C); and after 141 years a wet period shifts the system back to the shallow attractor (points D to F). Points A-I and X-Z are as discussed in Figure 8 and the accompanying text. The color of the 3-D surface illustrates the quantity of Lyapunov energy marked on the vertical axis.
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 With the stochastic forcing overlain, for series A and B it can be seen that the system is initially within the shallow attractor basin but after a period of low rainfall the lower limit to the shallow attractor is exceeded and, in year 6 (point B) and year 26 (point Y) for series A and B, respectively, each shifts toward the deep attractor. Each series arrives at the deep attractor in year 12 (point C) and year 54 (point Z), respectively, and series A remains within the deep attractor for 141 years while series B persists within it for the remaining duration of simulation. In year 154, a period of high rainfall shifts series A to a state having only the shallow attractor (point D). As this wet period is of sufficient duration, series A converges to the shallow attractor (point F) in year 160. Once within the shallow attractor basin, it only switches back to the deep attractor basin during a period of sufficiently low rainfall to shift the system to a state having only the deep attractor (points G and H). Overall, the transition of series A from point A to point I indicates that the existence of two attractors produces a counterclockwise hysteresis loop. Furthermore, counter to what much of the resilience literature implies, this indicates that the system does not switch between attractor basins by crossing the repellor. This process of attractor switching is likely to apply to other systems experiencing stochastic forcing of an additive or multiplicative form (see Guttal and Jayaprakash  for a description of forcing types).
 Identification of this mechanism of attractor basin switching raises the question of whether two attractors can still emerge despite the system being parameterized to have only one attractor (according to equilibrium continuation). For this model, Figure 3 predicts only one attractor exists when Caq equals 1100 m yr–1. To investigate if multiple attractors can still emerge when Caq equals 1100 m yr–1, Figure 10a shows model simulation results derived using stochastic forcing of sufficiently high standard deviation to produce three periods of crossing from the deep attractor basin to the approximate depth of the shallow attractor. Figure 10b shows the equilibrium continuation plot for the annual precipitation rate on the x axis. The stochastic forcing overlain onto Figure 10b clearly shows the same counterclockwise hysteresis loop as in Figure 9. However, combined with the continuation plot, two differences are apparent. First, Figure 10b shows the lower limit of the two-attractor range to have shifted to a higher rate of precipitation such that at the mean precipitation of 0.5 m yr–1 only the deep attractor exists. This explains why Figure 3 shows only one attractor to exist at Caq of 1100 m yr–1. As shown by the overlain simulation, if the water level is within the shallow attractor basin, once the forcing returns to and persists at the mean rate, the water level will switch to the deep attractor. Therefore any transition into the shallow attractor basin is temporary, and very likely to be of a shorter duration than if two attractors existed at the mean rate of forcing. The second difference apparent in Figure 10b is a shift of the upper precipitation fold point to a significantly higher rate. Under stochastic forcing, a greater rate is therefore required to cause a switch to the shallow attractor; which explains why a larger standard deviation for the forcing series was required to produce a switching of attractors. It is interesting that two of the three switches to the shallow attractor occurred by crossing the repellor. This demonstrates that the forcing need not exceed the upper fold point to switch attractor basins but simply be near to it, which is probably related to the very low sensitivity of the Lyapunov energy surface with respect to depth to water table (as is seen for Caq of 700 m yr–1 in Figure 9). A similar counterclockwise hysteresis loop is still evident in these cases.
Figure 10. Demonstration of temporary attractor basin switching at a value of Caq (1100 m yr–1) which Figure 5 indicates has only one attractor. In Figure 10b, the solid thick black line denotes an attractor the dashed thick black line a repellor, the solid thin gray line the stochastic simulation of depth to water table from Figure 10a, and the vertical dotted line the mean annual precipitation of 0.5 m yr–1.
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 With regard to broader resilience science concepts, Figure 10 shows equilibrium continuation (when the x axis parameter is not the forcing rate) to be an inadequate means for identifying multiple attractors within systems with stochastic forcing. Use of such a method would fail to identify the switch to an alternative attractor which, while temporary, may be of sufficient duration relative to management concerns to be deemed significant. It also means that the controls on behavior of systems with multiple attractors may be misinterpreted.
 By undertaking equilibrium continuation for the forcing rate, two new insights are available. First, the likely attractor dynamics under stochastic forcing of a given mean and variance can be estimated by considering if an extreme forcing event(s) would be sufficient to cause an exceedance of a fold point. If neither fold point is likely to be exceeded then the system is unlikely to enter an alternate attractor basin, making the existence of multiple attractors somewhat inconsequential. If only one of the fold points is likely to be exceeded then the system will likely display asymptotic stability whereby the system will eventually converge to the attractor basin for which forcing is insufficient to cause an exit. If both fold points are likely to be exceeded then both attractors will be nonasymptomatic and the system will most likely repeatedly switch between the two attractor basins. Using this framework the implications from a change in the forcing mean or variance on the emergence of multiple attractors could also be explored without undertaking stochastic simulations.
 The second insight is that this framework provides a new measure of resilience that, unlike the equilibrium continuation of Figure 5, is consistent with the stochastic nature of the system. For considering the resilience to stochastic forcing endogenous to the model, the resilience of an attractor is best quantified as the difference between the mean forcing rate and the forcing at the fold point. Figure 11 demonstrates this measure for five values of aquifer conductance. For Caq of 500 m yr–1, it shows the deep and shallow attractor basins to be of near equal resilience. For Caq of 700 and 900 m yr–1, it shows the deep attractor basin to be significantly more resilient than the shallow attractor basin. However, for values of Caq at which the mean forcing is, say, below the left fold point (e.g., the deep attractor when Caq equals 1100 m yr–1) this measure of resilience is inadequate for it would estimate zero resilience and fail to capture the likely transient switching to the attractor above the fold point (as shown to occur in Figure 10). For such instances, a probabilistic measure of resilience would be required that jointly considers the magnitude and probability of the relevant forcing events.
Figure 11. Resilience of each attractor to annual precipitation forcing (in units of annual precipitation rate) at five values of aquifer conductance parameter, Caq.
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 Often resilience studies are interested in the resilience to a transient disturbance that is not simulated within the model but may cause a switching of attractor basins. For this model, such disturbances would include groundwater pumping or flood recharge. To quantify the resilience to such an exogenous disturbance, the resilience of an attractor can be quantified by the standard measure of the state-variable distance from the attractor to the repellor. However, a more informative measure would be a Lyapunov stability curve for a specified constant forcing rate. In addition to providing the standard measure of resilience, its estimate of the depth of an attractor basin would inform the ease with which an exogenous disturbance could shift the system from an attractor and the rate of recovery after the exogenous disturbance. To jointly estimate the resilience to both endogenous and exogenous disturbances and for a single parameter set, a 3-D Lyapunov surface similar to Figure 9 may be useful.