TY - JOUR

T1 - Discerning non-autonomous dynamics

AU - Clemson, Philip

AU - Stefanovska, Aneta

N1 - Open Access funded by Engineering and Physical Sciences Research Council. The final, definitive version of this article has been published in the Journal, Physics Reports 542 (4), 2014, © ELSEVIER.

PY - 2014/9/30

Y1 - 2014/9/30

N2 - Structure and function go hand in hand. However, while a complex structure can be relatively safely broken down into the minutest parts, and technology is now delving into nanoscales, the function of complex systems requires a completely different approach. Here the complexity clearly arises from nonlinear interactions, which prevents us from obtaining a realistic description of a system by dissecting it into its structural component parts. At best, the result of such investigations does not substantially add to our understanding or at worst it can even be misleading. Not surprisingly, the dynamics of complex systems, facilitated by increasing computational efficiency, is now readily tackled in the case of measured time series. Moreover, time series can now be collected in practically every branch of science and in any structural scale—from protein dynamics in a living cell to data collected in astrophysics or even via social networks. In searching for deterministic patterns in such data we are limited by the fact that no complex system in the real world is autonomous. Hence, as an alternative to the stochastic approach that is predominantly applied to data from inherently non-autonomous complex systems, theory and methods specifically tailored to non-autonomous systems are needed. Indeed, in the last decade wehave faced a huge advance in mathematical methods, including the introduction of pullback attractors, as well as time series methods that cope with the most important characteristic of non-autonomous systems—their time-dependent behaviour. Here we review current methods for the analysis of non-autonomous dynamics including those for extracting properties of interactions and the direction of couplings. We illustrate each method by applying it to three sets of systems typical for chaotic, stochastic and non-autonomousbehaviour. For the chaotic class we select the Lorenz system, for the stochastic the noiseforced Duffing system and for the non-autonomous the Poincaré oscillator with quasiperiodic forcing. In this way we not only discuss and review each method, but also present properties which help to clearly distinguish the three classes of systems when analysed in an inverse approach—from measured, or numerically generated data. In particular, this review provides a framework to tackle inverse problems in these areas and clearly distinguish non-autonomous dynamics from chaos or stochasticity.

AB - Structure and function go hand in hand. However, while a complex structure can be relatively safely broken down into the minutest parts, and technology is now delving into nanoscales, the function of complex systems requires a completely different approach. Here the complexity clearly arises from nonlinear interactions, which prevents us from obtaining a realistic description of a system by dissecting it into its structural component parts. At best, the result of such investigations does not substantially add to our understanding or at worst it can even be misleading. Not surprisingly, the dynamics of complex systems, facilitated by increasing computational efficiency, is now readily tackled in the case of measured time series. Moreover, time series can now be collected in practically every branch of science and in any structural scale—from protein dynamics in a living cell to data collected in astrophysics or even via social networks. In searching for deterministic patterns in such data we are limited by the fact that no complex system in the real world is autonomous. Hence, as an alternative to the stochastic approach that is predominantly applied to data from inherently non-autonomous complex systems, theory and methods specifically tailored to non-autonomous systems are needed. Indeed, in the last decade wehave faced a huge advance in mathematical methods, including the introduction of pullback attractors, as well as time series methods that cope with the most important characteristic of non-autonomous systems—their time-dependent behaviour. Here we review current methods for the analysis of non-autonomous dynamics including those for extracting properties of interactions and the direction of couplings. We illustrate each method by applying it to three sets of systems typical for chaotic, stochastic and non-autonomousbehaviour. For the chaotic class we select the Lorenz system, for the stochastic the noiseforced Duffing system and for the non-autonomous the Poincaré oscillator with quasiperiodic forcing. In this way we not only discuss and review each method, but also present properties which help to clearly distinguish the three classes of systems when analysed in an inverse approach—from measured, or numerically generated data. In particular, this review provides a framework to tackle inverse problems in these areas and clearly distinguish non-autonomous dynamics from chaos or stochasticity.

KW - Non-autonomous systems

KW - Complex systems

KW - Nonlinear dynamics

KW - Time series analysis

KW - Inverse approach

U2 - 10.1016/j.physrep.2014.04.001

DO - 10.1016/j.physrep.2014.04.001

M3 - Literature review

VL - 542

SP - 297

EP - 368

JO - Physics Reports

JF - Physics Reports

SN - 0370-1573

IS - 4

ER -