Timing of Transients: Quantifying Reaching Times and Transient Behavior in Complex Systems

TY - JOUR

T1 - Timing of Transients

T2 - Quantifying Reaching Times and Transient Behavior in Complex Systems

AU - Kittel, Tim

AU - Heitzig, Jobst

AU - Webster, Kevin

AU - Kurths, Juergen

N1 - The authors thank the anonymous referees for their detailed and constructive feedback. This paper was developed within the scope of the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP. This work was conducted in the framework of PIK’s flagship project on coevolutionary pathways (copan). The authors thank CoNDyNet (FKZ 03SF0472A) for their cooperation. The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and the Land Brandenburg for supporting this project by providing resources on the high performance computer system at the Potsdam Institute for Climate Impact Research. The authors thank the developers of the used software: Python[47], Numerical Python[48] and Scientific Python[49]. The authors thank Sabine Auer, Karsten Bolts, Catrin Ciemer, Jonathan Donges, Reik Donner, Jasper Franke, Frank Hellmann, Jakob Kolb, Chiranjit Mitra, Finn Muller-Hansen, Jan Nitzbon, Anton Plietzsch Stefan Ruschel, Tiago Pereira da Silva, Francisco A. Rodrigues, Paul Schultz, and Lyubov Tupikina for helpful discussions and comments.

PY - 2017/8/7

Y1 - 2017/8/7

N2 - In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, Area under Distance Curve and Regularized Reaching Time, that capture two complementary aspects of transient dynamics. The first, Area under Distance Curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are "reluctant", i.e. stay distant from the attractor for long, or "eager" to approach it right away. Regularized Reaching Time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much "earlier" or "later" than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.

AB - In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, Area under Distance Curve and Regularized Reaching Time, that capture two complementary aspects of transient dynamics. The first, Area under Distance Curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are "reluctant", i.e. stay distant from the attractor for long, or "eager" to approach it right away. Regularized Reaching Time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much "earlier" or "later" than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.

KW - Early-Warning Signals

KW - Complex Systems

KW - Nonlinear Dynamic

KW - Long Transients

KW - Stability against Shocks

KW - Ordinary Differential Equations

U2 - 10.1088/1367-2630/aa7b61

DO - 10.1088/1367-2630/aa7b61

M3 - Article

VL - 19

JO - New Journal of Physics

JF - New Journal of Physics

SN - 1367-2630

M1 - 083005

ER -