Constrained basin stability for studying transient phenomena in dynamical systems

Adrian van Kan, Jannes Jegminat, Jonathan F. Donges, and Jürgen Kurths

Phys. Rev. E 93, 042205 – Published 13 April 2016

Abstract

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.

Received 13 August 2015
Revised 9 January 2016

DOI:https://doi.org/10.1103/PhysRevE.93.042205

Physics Subject Headings (PhySH)

Dynamical systems

Chaos & nonlinear dynamics

Nonlinear Dynamics

Authors & Affiliations

Adrian van Kan^1,2,*, Jannes Jegminat^1,†, Jonathan F. Donges^3,4, and Jürgen Kurths^3,5,6,7

¹Department of Physics and Astronomy, University of Heidelberg, Im Neuenheimer Feld 226, D-69120 Heidelberg, Germany
²Department of Physics, Imperial College London, Prince Consort Rd, London SW7 2BB, United Kingdom
³Potsdam Institute for Climate Impact Research, P.O. Box 601203, D-14412 Potsdam, Germany
⁴Stockholm Resilience Centre, Stockholm University, Kräftriket 2B, 114 19 Stockholm, Sweden
⁵Department of Physics, Humboldt University Berlin, Newtonstr. 15, D-12489 Berlin, Germany
⁶Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom
⁷Department of Control Theory, Nizhny Novgorod State University, Gagarin Avenue 23, 606950 Nizhny Novgorod, Russia

^*van_kan@stud.uni-heidelberg.de
^†jegminat@iup.uni-heidelberg.de

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 93, Iss. 4 — April 2016

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Subspaces of a schematic two-dimensional phase space containing a fixed point $x_{☆}$ : perturbations are sampled from the domain of the perturbation probability density $ρ$ (area within dashed line). Their transients return to the fixed point when sampled from the basin of attraction $B (x_{☆})$ (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 $C \cap B (x_{☆})$ (striped area). Thus CBS reflects the stability with respect to perturbations whose transients fulfill a given constraint.
Reuse & Permissions
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 $B (x_{*}^{\pm})$ and their intersections $C^{\pm} \cap B$ 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 $C^{\pm} \cap B$ . Top panel: the total basin of a fixed point is composed of successive layers: it is given by $C^{\pm} \cap B$ combined both with the region in the respective other half-space ( $x > 0$ or $x < 0$ ) directly surrounding $C^{\pm} \cap B$ 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 $r_{3}$ .
Reuse & Permissions
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 $(r_{1})$ , the boundary crisis $(r_{2})$ , and the stability loss of the fixed point $(r_{3})$ . 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 $\pm 3 σ$ according to Eq. (10).
Reuse & Permissions
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 $B / C$ (green, lower part of triangle) for $α = 0.15$ (upper panel) and $α = 0.35$ (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 $α$ increases, in agreement with Fig. 5.
Reuse & Permissions
Figure 5
BS (straight lines) and CBS (curved red line) of $x_{☆}^{d}$ and $x_{☆}^{u d}$ vs the human carbon offtake rate $α \in [0, 0.6]$ for $t_{crit} = 90$ . BS of $x_{☆}^{d}$ is represented by the green straight line $(α \leq α_{crit} \approx 0.4)$ and BS of $x_{☆}^{u d}$ by the blue straight line $(α \leq α_{crit})$ . At low values of $α$ , the desirable state is stable against any strength of perturbation while the desert state is unstable. For $α > 0.03$ , an increasing fraction of perturbation-induced trajectories takes longer than $t_{crit}$ to return to the desirable fixed point until, at $α \approx 0.32$ , CBS vanishes. By contrast, BS of both fixed points exhibits a jump at $α_{crit}$ and thus no precursory phenomena can be observed there. The gray envelope represents $\pm 3 σ$ according to Eq. (10).
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics