MS14.3: Random Dynamical Systems - Recent Advances and New Directions

Targeted energy transfer (TET) is one of the passive approaches for the attenuation of vibration by means of irreversible energy transfer from the primary linear oscillator (LO) to the auxiliary system. Recent studies on TET through vibro-impact (VI) based NES have shown improved performance over a broad spectrum. In VINES, where a ball oscillates inside the LO, energy transfers through the impacts and mitigates the vibration of the LO. Previous studies of VINES consider limited parameter ranges with the smaller mass ratio and external excitation predominantly near the resonant frequency. In this study, we are considering fully non-smooth system, applying novel analytical and numerical analyses of externally excited VINES over a broad range of parameters for different periodic dynamics. Additionally, the external excitation can have random fluctuations, called noise, which have the potential to affect the energy transfer mechanism. The previous studies are restricted to the stochastic analysis for conventional TET mechanisms. Preliminary results for VINES reveal scenarios where certain types of noise improve the performance of the system. This study investigates the stochastic bifurcation structure of the TET phenomenon combining the non-smooth analysis with a probabilistic framework. The results are directly related to several performance measures of VI-NES within the noisy environment.

Vibro-impact (VI) systems are nonlinear systems that can be employed in many engineering applications, such as energy harvesting (EH). We present a VI system that consists of an externally forced inclined cylindrical capsule and a bullet that is allowed to freely move inside the capsule. Dielectric elastomer membranes cover the capsule ends, and impacts between the bullet and membranes generate an excess in electrical energy that can be harvested. Parametric studies reveal smooth and non-smooth bifurcations of (ir)regular impact sequences, including grazing bifurcations, leading to additional low-velocity impacts. Here, we focus on the interplay between dry friction, forcing frequency, and noise in the restitution coefficient, investigating its influence on EH.

A novel method is proposed to obtain global solutions of fuzzy optimal control with fixed state terminal conditions and control bounds. The global solution implies that the optimal control solutions are valid for all the initial conditions in a region of the state space. The method makes use of Bellman’s principle of optimality and fuzzy generalized cell mapping method (FGCM). A discrete form of fuzzy master equation with a control dependent transition membership matrix is generated by using the FGCM. This allows to evaluate both the transient and the steady-state responses of the controlled system. The method, simply called FGCM with BP, is applied to a nonlinear system leading to excellent control performances.

Within the nucleus, structural maintenance of chromosome protein complexes, namely condensin and cohesin, create an architecture to facilitate the organization and proper function of the genome. Condensin creates localized clusters of chromatin in the nucleolus through transient crosslinks. Large-scale simulations revealed three different dynamic behaviors as a function of timescale: slow crosslinking leads to no clusters, fast crosslinking produces rigid clusters, while intermediate timescales are optimal for producing flexible clusters that mediate gene interaction. By mathematically analyzing different relative scalings of the two sources of stochasticity, thermal fluctuations and the force induced by the transient crosslinks, we predict these three distinct regimes of cluster behavior. Standard time-averaging that takes the fluctuations of the transient crosslink force to zero can predict the existence of clusters, but not their timescale-dependent lifetimes. Accounting for the interaction of both fluctuations from the crosslinks and thermal noise with an effective energy landscape does capture the timescale-dependent flexible cluster lifetimes. No clusters are predicted when the fluctuations of the transient crosslink force are taken to be large relative to thermal fluctuations. This perturbation analysis illuminates the importance of accounting for stochasticity in local incoherent transient forces to predict emergent complex biological behavior.

The paper delves into the stochastic dynamics of a nonlinear two-degree-of-freedom system coupled with a nonlinear energy sink (NES). Under random excitation on the primary mass, the primary objective is to maximise targeted energy transfer efficiency, investigating the impact of various nonlinear damping models. A surrogate optimisation algorithm, applied within the stochastic framework, is introduced. The optimisation involves both nonlinear stiffness and nonlinear damping terms simultaneously. Three distinct cost functions, rooted in mean energy or mean dissipated energy of system components, are considered. Results highlight the nonlinear damping's influence on the sink's parameters and illustrate how different cost functions affect optimal values of the nonlinear system's coefficients.

Particle filters provide a general and flexible approach to numerically approximate the nonlinear filtering solution, a conditional probability distribution, of continuous-time partially observable stochastic signal processes. In this paper we consider the case of a continuous-time signal and discrete-time observation process. Standard implementations of particle methods suffer from particle degeneracy (also known as impoverishment or collapse), where after repeated application of Bayes' formula for updating the weights of the particles, few particles have any significant weight. The most common remedy in practice is to apply heuristic resampling methods. Control and flow-based particle methods have been introduced in the past decade as alternatives for handling the issue of degeneracy. The former method (control-based) modifies the dynamics between observations, whereas the latter (flow-based) introduces a pseudo-time for particle flow at the instance of observation. Both have advantages and disadvantages. In this paper, we develop theory and algorithms for a method that attempts to blend the advantages of both control and flow-based approaches. Motivated by chaotic systems from the geophysical sciences, we benchmark the hybrid method against standard, control, and flow-based particle methods on the atmospheric models of Lorenz.