The origin of this special issue took place at the 9th European Nonlinear Dynamics Conference (ENOC 2017) in Budapest, Hungary. Specifically, the mini-symposium on Random Dynamical Systems – Recent Advances and New Directions brought together novel perspectives on analyzing stochastic dynamics with applications including biology, structural dynamics, control, energy and mechanics. The expanded use of stochasticity in more realistic models exposes questions related to bifurcations, meta-stability, tipping and early warning signals, multiscale dynamics, and connections between chaos and stochastic dynamics. The observed phenomena in applications drive new methodologies and analyses, needed to understand the interplay between different sources of stochastic effects and nonlinearities, network structure, multi-mode and multi-scale behavior, non-smooth dynamics, energy transfer, and spatio-temporal phenomena. Of course, a single issue cannot hope to cover all of the new topics in stochastic analysis for applications. Nevertheless, we hope that the collection of applications and stochastic models presented in this issue illustrates some of the exciting advances and perspectives relevant for broad classes of stochastic models and demonstrates the need in advancing the theory of stochastic processes.

Metastable transitions in Langevin dynamics can exhibit rich behaviours that are markedly different from its overdamped limit. In addition to local alterations of the transition path geometry, more fundamental global changes may exist. For instance, when the dissipation is weak, heteroclinic connections that exist in the overdamped limit do not necessarily have a counterpart in the Langevin system, potentially leading to different transition rates. Furthermore, when the friction coefficient depends on the velocity, the overdamped limit no longer exists, but it is still possible to efficiently find instantons. The approach, we employed for these discoveries, was based on (i) a simple rewriting of the Freidlin–Wentzell action in terms of time-reversed dynamics and (ii) an adaptation of the string method, which was originally designed for gradient systems, to this specific non-gradient system.

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

This study investigates the phenomenon of targeted energy transfer (TET) from a linear oscillator to a nonlinear attachment behaving as a nonlinear energy sink for both transient and stochastic excitations. First, the dynamics of the underlying Hamiltonian system under deterministic transient loading is studied. Assuming that the transient dynamics can be partitioned into slow and fast components, the governing equations of motion corresponding to the slow flow dynamics are derived and the behaviour of the system is analysed. Subsequently, the effect of noise on the slow flow dynamics of the system is investigated. The Itô stochastic differential equations for the noisy system are derived and the corresponding Fokker–Planck equations are numerically solved to gain insights into the behaviour of the system on TET. The effects of the system parameters as well as noise intensity on the optimal regime of TET are studied. The analysis reveals that the interaction of nonlinearities and noise enhances the optimal TET regime as predicted in deterministic analysis.

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

Results on the behaviour of a pendulum which is parametrically excited by large amplitude random loads at its pivot are presented, including a novel experimental case study. Thereby, it is dealt with a random excitation by a non-white Gaussian stochastic process with prescribed spectral density. Special focus is devoted to stochastic processes resulting from random sea wave elevation and the question whether random sea waves can lead to rotational motion of the parametrically excited pendulum. The motivation for such an experimental study is energy harvesting from ocean waves.

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

Vibration energy harvesting aims to harness the energy of ambient random vibrations for power generation, particularly in small-scale devices. Typically, stochastic excitation driving the harvester is modelled as a Brownian process and the dynamics are studied in the equilibrium state. However, non-Brownian excitations are of interest, particularly in the nonequilibrium regime of the dynamics. In this work we study the nonequilibrium dynamics of a generic piezoelectric harvester driven by Brownian as well as (non-Brownian) Lévy flight excitation, both in the linear and the Duffing regimes. Both the monostable and the bistable cases of the Duffing regime are studied. The first set of results demonstrate that Lévy flight excitation results in higher expectation values of harvested power. In particular, it is shown that increasing the noise intensity leads to a significant increase in power output. It is also shown that a linearly coupled array of nonlinear harvesters yields improved power output for tailored values of coupling coefficients. The second set of results show that Lévy flight excitation fundamentally alters the bifurcation characteristics of the dynamics. Together, the results underscore the importance of non-Brownian excitation characterised by Lévy flight in vibration energy harvesting, both from a theoretical viewpoint and from the perspective of practical applications.

This work investigates analytically, the use of piezoelectric tiles placed on stairways for vibrational energy harvesting – harnessing electrical power from natural vibrational phenomena – from pedestrian footfalls. While energy harvesting from pedestrian traffic along flat pathways has been studied in the linear regime and realised in practical applications, the greater amounts of energy naturally expended in traversing stairways suggest better prospects for harvesting. Considering the characteristics of two types of commercially available piezoelectric tiles – Navy Type III and Navy Type V – analytical models for the coupled electromechanical system are formulated. The harvesting potential of the tiles is then studied under conditions of both deterministic and carefully developed random excitation profiles for three distinct cases: linear, monostable nonlinear and an array of monostable nonlinear tiles on adjacent steps with linear coupling between them. The results indicate enhanced power output when the tiles are: (1) placed on stairways, (2) uncoupled and (3) subjected to excitation profiles with stochastic frequency. In addition, the Navy Type V tiles are seen to outperform the Navy Type III tiles. Finally, the strongly nonlinear regime outperforms the linear one suggesting that the realisation of commercially available piezoelectric tiles with appropriately tailored nonlinear characteristics will likely have a significant impact on energy harvesting from pedestrian traffic.

The stochastic resonance phenomenon implies “positive” changing of a system behaviour when noise is added to the system. The phenomenon has found numerous applications in physics, neuroscience, biology, medicine, mechanics and other fields. The present paper concerns this phenomenon for parametrically excited stochastic systems, i.e. systems that feature deterministic input signals that affect their parameters, e.g. stiffness, damping or mass properties. Parametrically excited systems are now widely used for signal sensing, filtering and amplification, particularly in micro- and nanoscale applications. And noise and uncertainty can be essential for systems at this scale. Thus, these systems potentially can exhibit stochastic resonance. In the present paper, we use a “deterministic” approach to describe the stochastic resonance phenomenon that implies replacing noise by deterministic high-frequency excitations. By means of the approach, we show that stochastic resonance can occur for parametrically excited systems and determine the corresponding resonance conditions.

Estimates for sample paths of fast–slow stochastic ordinary differential equations have become a key mathematical tool relevant for theory and applications. In particular, there have been breakthroughs by Berglund and Gentz to prove sharp exponential error estimates. In this paper, we take the first steps in order to generalise this theory to fast–slow stochastic partial differential equations. In a simplified setting with a natural decomposition into low- and high-frequency modes, we demonstrate that for a short-time period the probability for the corresponding sample path to leave a neighbourhood around the stable slow manifold of the system is exponentially small as well.

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.