From physical linear systems to discrete-time series. A guide for analysis of the sampled experimental data

Invariant set based distributed protocol for synchronization of discrete-time heterogeneous systems with nonlinear dynamics

We in this paper propose an invariant set based distributed control protocol for synchronization of discrete-time heterogeneous multiagent systems. Starting with the assumption that the distributed control input will vanish once a multiagent system achieves synchronization, we attain an easily verifiable method for the nonexistence of synchronous trajectories through characterizing the vector fields of agents. Then, we introduce an invariant set to analyze the limit behaviors of all the synchronous trajectories. Afterwards, based on the assumption that the above invariant set can be characterized by the graph of a function, we design a distributed control protocol to transform the heterogeneous system into an equivalent one, which is composed of two lower dimensional systems. Moreover, for this equivalent system, we provide a synchronization criterion via constructing corresponding Lyapunov-type functions for these two lower dimensional systems, arriving at a synchronization criterion for the original heterogeneous system. Especially, we further improve the applicability of this synchronization criterion by using multiple Lyapunov-type functions. Finally, three examples are presented to demonstrate the validity of the corresponding theoretical results.

Identifying ergodicity breaking for fractional anomalous diffusion: Criteria for minimal trajectory length

In this paper, we study ergodic properties of α-stable autoregressive fractionally integrated moving average (ARFIMA) processes which form a large class of anomalous diffusions. A crucial practical question is how long trajectories one needs to observe in an experiment in order to claim that the analyzed data are ergodic or not. This will be solved by checking the asymptotic convergence to 0 of the empirical estimator F(n) for the dynamical functional D(n) defined as a Fourier transform of the n-lag increments of the ARFIMA process. Moreover, we introduce more flexible concept of the ε-ergodicity.

Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes

In this paper we study asymptotic behavior of a dynamical functional for an α-stable autoregressive fractionally integrated moving average (ARFIMA) process. We find an analytical formula for this important statistics and show its usefulness as a diagnostic tool for ergodic properties. The obtained results point to the very fast convergence of the dynamical functional and show that even for short trajectories one may obtain reliable conclusions on the ergodic properties of the ARFIMA process. Moreover we use the obtained theoretical results to illustrate how the dynamical functional statistics can be used in the verification of the proper model for an analysis of some biophysical experimental data.

Time-series methods in analysis of the optical tweezers recordings

In this paper we treat optical tweezers as discrete-time linear filters and analyze the recorded trajectories of the trapped beads using time-series methods. Using these techniques we obtain a simple analytical formula for the aliased power-spectrum density. Moreover, we separate influences of the noise and blur induced by the video camera from the physical content of the measurements, providing simple tools to detect and account for these distortions. Finally, checking how our tools work on the real data, we identify what parameters of video camera calibration the blur is dominating and what the additive noise is dominating. We also detect a range where these two distortions cancel each other so that the data can be mistakenly classified as undisturbed.