Constrained basin stability for studying transient phenomena in dynamical systems

Transient dynamics are of large interest in many areas of science. Here, a generalization of basin stability (BS) is presented: constrained basin stability (CBS) that is sensitive to various different types of transients arising from finite size perturbations. CBS is applied to the paradigmatic Lorenz system for uncovering nonlinear precursory phenomena of a boundary crisis bifurcation. Further, CBS is used in a model of the Earth's carbon cycle as a return time-dependent stability measure of the system's global attractor. Both case studies illustrate how CBS's sensitivity to transients complements BS in its function as an early warning signal and as a stability measure. CBS is broadly applicable in systems where transients matter, from physics and engineering to sustainability science. Thus CBS complements stability analysis with BS as well as classical linear stability analysis and will be a useful tool for many applications.

Figure 1

Subspaces of a schematic two-dimensional phase space containing a fixed point ${\mathbf{x}}_{☆}$: perturbations are sampled from the domain of the perturbation probability density $\rho$ (area within dashed line). Their transients return to the fixed point when sampled from the basin of attraction $\mathcal{B}\left({\mathbf{x}}_{☆}\right)$ (gray area). The set $C$ (white within solid line) is the set of initial conditions which lead to transients that satisfy a given constraint, e.g., on the $x_1$ component. While BS is computed as the fraction of perturbations within the basin of attraction, CBS is the fraction of perturbations within the intersection $C\cap \mathcal{B}\left({\mathbf{x}}_{☆}\right)$ (striped area). Thus CBS reflects the stability with respect to perturbations whose transients fulfill a given constraint.

Figure 2

Illustrations of the phase space structure of the L63 system defined by Eqs. (11, 12, 13). Both panels show cross sections, obtained by cutting along the plane containing the origin with normal $(1,1,0)$, of the basins of attraction $\mathcal{B}\left({\mathbf{x}}_{*}^{\pm }\right)$ and their intersections $C^{\pm }\cap \mathcal{B}$ with the sets $C^{\pm }$ for $r=15$ (upper panel) and $r=23$ (lower panel). In both panels, the blob-shaped region around each fixed point (black dots) is $C^{\pm }\cap \mathcal{B}$. Top panel: the total basin of a fixed point is composed of successive layers: it is given by $C^{\pm }\cap \mathcal{B}$ combined both with the region in the respective other half-space ($x>0$ or $x<0$) directly surrounding $C^{\pm }\cap \mathcal{B}$ and with the next layer of the same color in the fixed point's half plane. In the lower panel (coloring identical), the fractal structure of the basins, visible as intermingled green and blue sets, is apparent; it is associated with transient chaos. One observes that the fraction of the window covered by the sets $C^{\pm }$ shrinks as $r$ increases. This illustrates the general behavior observed in the L63 system and quantified by CBS, namely that the volume fraction of the three-dimensional sampling region occupied by $C^{\pm }$ decreases continuously as $r$ approaches $r_3$.

Figure 3

BS [green (upper) line] and CBS [red (lower) line] of the positive fixed points in the L63 system as the parameter $r$ is varied. From left to right, the vertical lines indicate the appearance of the chaotic set $\left(r_1\right)$, the boundary crisis $\left(r_2\right)$, and the stability loss of the fixed point $\left(r_3\right)$. The difference between the two curves represents the fraction of the basin from where chaotic transients evolve. Note that BS exhibits a jump at $r_2$ which is blurred by the finite simulation time: the simulation stopped before all (increasingly long) transients had reached the fixed point. Because of the system's x-y symmetry the negative fixed point has the same BS and CBS. The gray envelopes represent $\pm 3\sigma$ according to Eq. (10).

Figure 4

Illustrations of the phase space structure of the Anderies system, Eqs. (15) and (16). Both panels show the (globally attracting) desirable fixed point (black dot) and the set $C$ (red, upper part of triangle) and its complement $\mathcal{B}/C$ (green, lower part of triangle) for $\alpha =0.15$ (upper panel) and $\alpha =0.35$ (lower panel). The white dot at $(0,0.5)$ is the desert state fixed point. One observes that the fraction of the two-dimensional finite phase space covered by the set $C$ shrinks as $\alpha$ increases, in agreement with Fig. 5.

Figure 5

BS (straight lines) and CBS (curved red line) of ${\mathbf{x}}_{☆}^d$ and ${\mathbf{x}}_{☆}^{ud}$ vs the human carbon offtake rate $\alpha \in [0,0.6]$ for $t_{\mathrm{crit}}=90$. BS of ${\mathbf{x}}_{☆}^d$ is represented by the green straight line $\left(\alpha \le {\alpha }_{\mathrm{crit}}\approx 0.4\right)$ and BS of ${\mathbf{x}}_{☆}^{ud}$ by the blue straight line $\left(\alpha \le {\alpha }_{\mathrm{crit}}\right)$. At low values of $\alpha$, the desirable state is stable against any strength of perturbation while the desert state is unstable. For $\alpha >0.03$, an increasing fraction of perturbation-induced trajectories takes longer than $t_{\mathrm{crit}}$ to return to the desirable fixed point until, at $\alpha \approx 0.32$, CBS vanishes. By contrast, BS of both fixed points exhibits a jump at ${\alpha }_{\mathrm{crit}}$ and thus no precursory phenomena can be observed there. The gray envelope represents $\pm 3\sigma$ according to Eq. (10).