scholarly journals Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial Jordan block case

2019 ◽  
Vol 19 (03) ◽  
pp. 1950022
Author(s):  
Yuri Bakhtin ◽  
Zsolt Pajor-Gyulai
Keyword(s):  
Correction Term ◽  
Exit Time ◽  
Scaling Limit ◽  
Nonlinear Dynamical System ◽  
Jordan Block ◽  
Exit Point ◽  
Limiting Behavior ◽  
Nonlinear Dynamical ◽  
Noise Limit ◽  
Full Dimension

We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random remainder satisfies a scaling limit.

Asymptotic Analysis of Nonlinear Stochastic Equations With Rapidly Oscillating and Decaying Components

2003 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Dynamical System ◽  
Markov Process ◽  
Martingale Problem ◽  
Nonlinear Dynamical System ◽  
Stochastic Equations ◽  
Limiting Behavior ◽  
Nonlinear Dynamical ◽  
Nonlinear Stochastic Equations ◽  
Lower Dimensional ◽  
Quadratic And Cubic Nonlinearities

We consider a noisy n-dimensional nonlinear dynamical system containing rapidly oscillating and decaying components. We extend the results of Papanicolaou and Kohler and Namachchivaya and Lin; these results state that as the noise becomes smaller, a lower dimensional Markov process characterizes the limiting behavior. Our approach springs from a direct consideration of the martingale problem and considers both quadratic and cubic nonlinearities.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

scholarly journals Robust Tracker of Hybrid Microgrids by the Invariant-Ellipsoid Set

Electronics ◽  
2021 ◽  
Vol 10 (15) ◽  
pp. 1794
Author(s):  
Hilmy Awad ◽  
Ehab H. E. Bayoumi ◽  
Hisham M. Soliman ◽  
Michele De Santis
Keyword(s):  
Sliding Mode ◽  
Tracking Error ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Nonlinear Dynamical ◽  
Load Variations ◽  
Invariant Ellipsoid ◽  
Steady State Responses ◽  
Linearized Model

This paper introduces a new ellipsoidal-based tracker design to control a grid-connected hybrid direct current/alternating current (DC/AC) microgrid (MG). The proposed controller is robust against both parameters and load variations. The studied hybrid MG is modelled as a nonlinear dynamical system. A linearized model around an operating point is developed. The parameter changes are modelled as norm-bounded uncertainties. We apply the new extended version of the attractive (or invariant) ellipsoid for this tracking problem. Convex optimization is used to obtain the region’s minimal size where the tracking error between the state trajectories and the reference states converges. The sufficient conditions for stability are derived and solved based on linear matrix inequalities (LMIs). The proposed controller’s validity is shown via simulating the hybrid MG with various operational scenarios. In each scenario, the performance of the controller is compared with a recently proposed sliding mode controller. The comparison clearly illustrates the superiority of the developed controller in terms of transient and steady-state responses.

DYNAMICAL ESTIMATION OF CALENDAR EFFECTS

2006 ◽  
Vol 06 (01) ◽  
pp. L7-L15
Author(s):  
ALEXANDROS LEONTITSIS
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Noise Level ◽  
Nonlinear Dynamical System ◽  
Main Tool ◽  
Experimental Results ◽  
Accurate Estimation ◽  
Nonlinear Dynamical ◽  
Correlation Integral ◽  
Calendar Effects

The paper introduces a method for estimation and reduction of calendar effects from time series, which their fluctuations are governed by a nonlinear dynamical system and additive normal noise. Calendar effects can be considered deviations of the distribution(s) of particular group(s) of observations that have a common characteristic related to the calendar. The concept of this method is the following: since the calendar effects are not related to the dynamics of the time series, the accurate estimation and reduction will result a time series with a smaller amount of noise level (i.e. more accurate dynamics). The main tool of this method is the correlation integral, due to its inherit capability of modeling both the dynamics and the additive normal noise. Experimental results are presented on the Nasdaq Cmp. index.

A Topological Study of the Genesis of a Horseshoe Vortex in the Symmetry Plane Due to an Adverse Pressure Gradient

2001 ◽  
Author(s):  
Martijn A. van den Berg ◽  
Michael M. J. Proot ◽  
Peter G. Bakker
Keyword(s):  
Pressure Gradient ◽  
Bifurcation Theory ◽  
Nonlinear Differential Equations ◽  
Nonlinear Dynamical System ◽  
Symmetry Plane ◽  
Adverse Pressure Gradient ◽  
Horseshoe Vortex ◽  
Nonlinear Dynamical ◽  
Separating Flow ◽  
Flow Through

Abstract The present paper describes the genesis of a horseshoe vortex in the symmetry plane in front of a juncture. In contrast to a previous topological investigation, the presence of the obstacle is no longer physically modelled. Instead, the pressure gradient, induced by the obstacle, has been used to represent its influence. Consequently, the results of this investigation can be applied to any symmetrical flow above a flat plate. The genesis of the vortical structure is analysed by using the theory of nonlinear differential equations and the bifurcation theory. In particular, the genesis of a horseshoe vortex can be described by the unfolding of the degenerate singularity resulting from a Jordan Normal Form with three vanishing eigenvalues and one linear term which is related to the adverse pressure gradient. The examination of this nonlinear dynamical system reveals that a horseshoe vortex emanates from a non-separating flow through two subsequent saddle-node bifurcations in different directions and the transition of a node into a focus located in the flow field.

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