Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system. © 2012 American Institute of Physics

Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given. © 2012 Elsevier B.V. All rights reserved

Structural Behavior of Inflatable, Reinforced, Braided, Tubular Members

The Hypersonic Inflatable Aerodynamic Decelerator (HIAD) system being developed by the National Aeronautics and Space Administration (NASA) is an inflatable structure composed of multiple, concentric, pressurized tori, load straps, and a thermal protection system. The HIAD overcomes limitations inherent with the use of rigid decelerators since the deployed diameter is much larger than the packed size, which makes it an enabling technology for new opportunities in space exploration. The HIAD is designed to decelerate and protect spacecraft during atmospheric re-entry. The objective of this research was to improve understanding of structural behavior of HIAD components through material testing, structural testing of components, and numerical models. The mechanics of inflatable, reinforced braided tubes have been reviewed from a geometric standpoint. Exploratory experimental efforts were performed to quantify the stiffness of the reinforcing cords, which drive axial and bending stiffness of the inflatable tubes. Benchtop inflation tests were performed to quantify longitudinal stiffness and examine instrumentation methods. The constitutive properties of the braided fabric shell of tori were determined as a function of braid angle and inflation pressure. The shear modulus is highly dependent on braid angle and pressure. Independent testing of extracted fiber tow bundles allowed the effect of de-crimping to be examined with straight tow thickness measured as an upper limit. Beam bending tests of straight beams with highly controlled loading and boundary conditions were performed for tubes with five different braid angles over a range of inflation pressures. These data sets are ideal for finite element validation due to the highly controlled conditions. Structural testing of individual tori was performed via radial compression loading. Many improvements were made to the single torus test setup using fixtures provided by NASA. Methods were developed to quantify the 3D shape of the tori and displacements using non-contact photogrammetry methods. The effect of load-control versus displacement-control experiments was investigated and found to result in different response. Finite-element models using three-dimensional shell-elements were developed and compared to the torus experiments. These modeling efforts proved to be challenging and no firm conclusions could be drawn

Chaotic Advection and the Emergence of Tori in the K\"uppers-Lortz State

Motivated by the roll-switching behavior observed in rotating Rayleigh-B\'enard convection, we define a K\"uppers-Lortz (K-L) state as a volume-preserving flow with periodic roll switching. For an individual roll state, the Lagrangian particle trajectories are periodic. In a system with roll-switching, the particles can exhibit three-dimensional, chaotic motion. We study a simple phenomenological map that models the Lagrangian dynamics in a K-L state. When the roll axes differ by $120^{\circ}$ in the plane of rotation, we show that the phase space is dominated by invariant tori if the ratio of switching time to roll turnover time is small. When this parameter approaches zero these tori limit onto the classical hexagonal convection patterns, and, as it gets large, the dynamics becomes fully chaotic and well-mixed. For intermediate values, there are interlinked toroidal and poloidal structures separated by chaotic regions. We also compute the exit time distributions and show that the unbounded chaotic orbits are normally diffusive. Although the map presumes instantaneous switching between roll states, we show that the qualitative features of the flow persist when the model has smooth, overlapping time-dependence for the roll amplitudes (the Busse-Heikes model).Comment: laTeX, 23 pages, 7 figure

Spectral criterion of stochastic stability for invariant manifolds 1

The mean square stability for invariant manifolds of nonlinear stochastic differential equations is considered. The stochastic stability analysis is reduced to the estimation of the spectral radius of some positive operator. For the important case of manifolds with codimension one, a constructive spectral analysis of this operator is carried out. On the basis of this spectral technique, parametrical criteria of the stochastic stability of limit cycle and 2-torus are developed. © 2013 Springer Science+Business Media New York

Analysis of Noise-Induced Transitions in a Thermo-Kinetic Model of the Autocatalytic Trigger

Motivated by the increasingly important role of mathematical modeling and computer-aided analysis in engineering applications, we consider the problem of the mathematical modeling and computer-aided analysis of complex stochastic processes in thermo-kinetics. We study a mathematical model of the dynamic interaction of reagent concentration and temperature in autocatalysis. For the deterministic variant of this model, mono- and bistability parameter zones as well as local and global bifurcations are revealed, and we show how random multiplicative disturbances can deform coexisting equilibrium regimes. In a study of noise-induced transitions, we apply direct numerical simulation and an analytical approach based on the stochastic sensitivity technique. Two variants of bistability with different scenarios of stochastic transformations are studied and compared. © 2023 by the authors.Ministry of Education and Science of the Russian Federation, MinobrnaukaThis research was funded by the Ministry of Science and Higher Education of the Russian Federation (Ural Federal University Program of Development within the Priority-2030 Program)

Uncertainty quantification applied to the analysis and design of a hypersonic inflatable aerodynamic decelerator for spacecraft reentry

The primary objective of this research is to investigate the uncertainty in the multidisciplinary analysis of a Hypersonic Inflatable Aerodynamic Decelerator configuration for Mars entry, subject to uncertainty sources in the high-fidelity computational models and the operating conditions. Efficient uncertainty quantification methods based on stochastic expansions are applied to the analysis of the hypersonic flowfield, fluid-structure interaction, and flexible thermal protection system response of a deformable inflatable decelerator. Uncertainty analysis is first applied to the hypersonic flowfield simulations to quantify the uncertainty in the surface convective and radiative heat flux, pressure, and shear stress of a fixed inflatable decelerator, subject to uncertainties in the binary collision integrals of the transport properties, chemical kinetics, non-Boltzmann radiation modeling, and the freestream conditions. The uncertainty analysis for fluid-structure interaction modeling is conducted to quantify the uncertainty in the deflection and resulting surface heat flux, shear stress, and pressure of a deformable inflatable decelerator, subject to uncertainties in material structural properties, inflation pressure, and important flowfield uncertain variables identified in the initial study. The deflection uncertainty is shown to be primarily driven by the structural modeling uncertain variables and found to be insignificant in contributing to the resulting surface condition uncertainties. Uncertainty analysis is finally applied to the flexible thermal protection system bondline temperature for a ballistic Mars entry trajectory, subject to uncertainties in the material thermal properties and important flowfield variables from the initial study. The uncertainty in the bondline temperature is compared to its allowable temperature limit and shown to be primarily driven by the material thermal properties of the outer fabric and insulator layers, and the freestream density --Abstract, page iv

Stochastic generation and suppression of early afterdepolarizations in a three-dimensional model of cardiac action potential

The influence of random disturbances on a three-dimensional simplification of Luo–Rudy model of the cardiac action potential is studied. We show that in the parameter region, where the deterministic model is in the equilibrium regime, noise can trigger large-amplitude oscillations that correspond with pathological early afterdepolarizations (EADs). For this stochastic excitement, the phenomenon of coherence resonance was discovered. On the contrary, in another parameter zone of the model, noise can suppress EADs. We analyze these stochastic phenomena using the stochastic sensitivity functions technique, Mahalanobis distance, the methods of principal directions, and confidence domains. © 2021 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.Russian Science Foundation, RSF: 21-11-00062This work was supported by the Russian Science Foundation (No. 21-11-00062)