Abstract
Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties. A prominent real-life example of this class of problems is the investigation of climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system is as a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto a suitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markov processes. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinear response of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on the evolution law of densities (the Liouville/Fokker–Planck equation), allowing to effectively calculate the sensitivity and response of two representative dynamical systems.
Highlights
Response theory is the scientific research area at the boundary between mathematics and physics dealing with the understanding of how complex systems react to perturbations affecting their dynamics
The construction of response theory can be approached by taking many different scientific point of views, it has been mostly driven by statistical mechanics [1]
We take the point of view of statistics to understand the effect of perturbations on dynamical systems
Summary
Response theory is the scientific research area at the boundary between mathematics and physics dealing with the understanding of how complex systems react to perturbations affecting their dynamics. Even if it addresses is a very classical problem, the mathematical framework to develop such a theory is still a matter of research. The construction of response theory can be approached by taking many different scientific point of views, it has been mostly driven by statistical mechanics [1] In this context, by considering a complex system in a steady state (equilibrium or nonequilibrium) and applying some sort of forcing to the dynamics as, e.g., a change in the governing parameters, Communicated by Alessandro Giuliani.
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