Non-Standard Reduction of Noisy Structural Systems: Shallow Arches

2000 ◽  
Author(s):  
Lalit Vedula ◽  
N. Sri Namachchivaya
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Exit Time ◽  
Stochastic Averaging ◽  
Dimensional System ◽  
Shallow Arches ◽  
Higher Dimensional ◽  
Mean Exit Time ◽  
Stationary Probability Density Function ◽  
Low Dimensional

Abstract The dynamics of a shallow arch subjected to small random external and parametric excitation is invegistated in this work. We develop rigorous methods to replace, in some limiting regime, the original higher dimensional system of equations by a simpler, constructive and rational approximation – a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results such as mean exit time, stationary probability density function.

Unified Approach for Noisy Nonlinear Mathieu-Type Systems

2001 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
Richard B. Sowers ◽  
H. J. Van Roessel
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Degree Of Freedom ◽  
Unified Approach ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Low Dimensional

Abstract The purpose of this work is to develop a unified approach to study the dynamics of a single degree of freedom system excited by both periodic and random perturbations. We consider the noisy Duffing-van der Pol-Mathieu equation as a prototypical single degree of freedom system and achieve a reduction by developing rigorous methods to replace, in some limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a low-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

UNIFIED APPROACH FOR NOISY NONLINEAR MATHIEU-TYPE SYSTEMS

2001 ◽  
Vol 01 (03) ◽  
pp. 405-450 ◽  
Author(s):  
N. SRI NAMACHCHIVAYA ◽  
RICHARD B. SOWERS
Keyword(s):  
Markov Process ◽  
Random Perturbation ◽  
Mathieu Equation ◽  
Type Systems ◽  
Stochastic Averaging ◽  
Unified Approach ◽  
Van Der Pol ◽  
Periodic Hamiltonian System ◽  
Time Periodic ◽  
Stochastic Bifurcations

The purpose of this work is to develop a unified approach to study the dynamics of a single-degree-of-freedom system excited by both periodic and random perturbations. As a prototype, we consider the noisy Duffing–van der Pol–Mathieu equation and achieve a reduction by developing rigorous methods to replace, in a limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a lower-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

Stochastic Response of SDOF Self-Centering System

2020 ◽  
Vol 20 (05) ◽  
pp. 2050062
Author(s):  
Huiying Hu ◽  
Lincong Chen
Keyword(s):  
Stochastic Averaging ◽  
Nonlinear Stochastic System ◽  
Amplitude Envelope ◽  
Stochastic Response ◽  
Equivalent Damping ◽  
Yield Displacement ◽  
Seismic Loadings ◽  
New Type ◽  
Stationary Probability Density Function ◽  
Energy Dissipation Coefficient

As a new type of seismic resisting device, the self-centering system is attractive due to its excellent re-centering capability, but research on such a system under random seismic loadings is quite limited. In this paper, the stochastic response of a single-degree-of-freedom (SDOF) self-centering system driven by a white noise process is investigated. For this purpose, the original self-centering system is first approximated by an auxiliary nonlinear system, in which the equivalent damping and stiffness coefficients related to the amplitude envelope of the response are determined by a harmonic balance procedure. Subsequently, by the method of stochastic averaging, the amplitude envelope of the response of the equivalent nonlinear stochastic system is approximated by a Markovian process. The associated Fokker–Plank–Kolmogorov (FPK) equation is used to derive the stationary probability density function (PDF) of the amplitude envelope in a closed form. The effects of energy dissipation coefficient and yield displacement on the response of system are examined using the stationary PDF solution. Moreover, Monte Carlo simulations (MCS) are used for ascertaining the accuracy of the analytical solutions.

scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

scholarly journals 1PT004 Mean Exit Time Analysis about Water Molecules around Membrane Surfaces(The 50th Annual Meeting of the Biophysical Society of Japan)

2012 ◽  
Vol 52 (supplement) ◽  
pp. S84
Author(s):  
Eiji Yamamoto ◽  
Takuma Akimoto ◽  
Yoshinori Hirano ◽  
Masato Yasui ◽  
Kenji Yasuoka
Keyword(s):  
Annual Meeting ◽  
Exit Time ◽  
Water Molecules ◽  
Time Analysis ◽  
Membrane Surfaces ◽  
Mean Exit Time ◽  
Biophysical Society

scholarly journals Second-order differential equations with random perturbations and small parameters

2017 ◽  
Vol 147 (4) ◽  
pp. 763-779
Author(s):  
M. Kamenskii ◽  
S. Pergamenchtchikov ◽  
M. Quincampoix
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Random Perturbation ◽  
Stochastic Averaging ◽  
Second Order ◽  
Random Perturbations ◽  
Existence And Uniqueness Theorem ◽  
Second Order Differential Equations ◽  
Averaging Theorem ◽  
Small Parameters

We consider boundary-value problems for differential equations of second order containing a Brownian motion (random perturbation) and a small parameter and prove a special existence and uniqueness theorem for random solutions. We study the asymptotic behaviour of these solutions as the small parameter goes to zero and show the stochastic averaging theorem for such equations. We find the explicit limits for the solutions as the small parameter goes to zero.

scholarly journals Evidence accumulation and change rate inference in dynamic environments

2016 ◽  
Author(s):  
Adrian E Radillo ◽  
Alan Veliz-Cuba ◽  
Kresimir Josic ◽  
Zachary Kilpatrick
Keyword(s):  
Rate Of Change ◽  
Ideal Observer ◽  
Moment Closure ◽  
Dimensional System ◽  
Change Rate ◽  
Observer Model ◽  
Low Dimensional ◽  
State Of The Environment ◽  
Rate Correlation ◽  
Moment Closure Approximation

In a constantly changing world, animals must account for environmental volatility when making decisions. To appropriately discount older, irrelevant information, they need to learn the rate at which the environment changes. We develop an ideal observer model capable of inferring the present state of the environment along with its rate of change. Key to this computation is updating the posterior probability of all possible changepoint counts. This computation can be challenging, as the number of possibilities grows rapidly with time. However, we show how the computations can be simplified in the continuum limit by a moment closure approximation. The resulting low-dimensional system can be used to infer the environmental state and change rate with accuracy comparable to the ideal observer. The approximate computations can be performed by a neural network model via a rate-correlation based plasticity rule. We thus show how optimal observers accumulates evidence in changing environments, and map this computation to reduced models which perform inference using plausible neural mechanisms.

scholarly journals Revealing topology with transformation optics

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Lizhen Lu ◽  
Kun Ding ◽  
Emanuele Galiffi ◽  
Xikui Ma ◽  
Tianyu Dong ◽  
...  
Keyword(s):  
Physical Space ◽  
Real Space ◽  
Virtual Space ◽  
Transformation Optics ◽  
Coordinate Space ◽  
Dimensional System ◽  
Edge States ◽  
And Topology ◽  
Higher Dimensional ◽  
Insight Into

AbstractSymmetry deepens our insight into a physical system and its interplay with topology enables the discovery of topological phases. Symmetry analysis is conventionally performed either in the physical space of interest, or in the corresponding reciprocal space. Here we borrow the concept of virtual space from transformation optics to demonstrate how a certain class of symmetries can be visualised in a transformed, spectrally related coordinate space, illuminating the underlying topological transitions. By projecting a plasmonic system in a higher-dimensional virtual space onto a lower-dimensional system in real space, we show how transformation optics allows us to construct a topologically non-trivial system by inspecting its modes in the virtual space. Interestingly, we find that the topological invariant can be controlled via the singularities in the conformal mapping, enabling the intuitive engineering of edge states. The confluence of transformation optics and topology here can be generalized to other wave realms beyond photonics.

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