# On Sensitivity Of The Stability Of Equilibrium Points With Respect To The Perturbations

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### Analysis of fluid systems: stability, receptivity, sensitivity

by PJ Schmid Cited by 90 equilibrium state of the flow render the base flow unstable; similarly, a flow that fluid dynamics: eigenvalue analysis for stability problems, and the resonance Sensitivity analysis is the starting point for many other fluid problems, among at a relation between a matrix perturbation and the resulting eigenvalue shift of the​

### Stability and performance of a variable gain controller with

by M Heertjesa 2004 Cited by 44 variable gain strategy is studied regarding closed-loop stability and regarding performance. Closed-loop sensitivity characteristic and its corresponding disturbance rejection nential stability of the equilibrium point x = 0 can be shown.

### Chapter 5 Dynamic and Closed-Loop Control - Princeton

by CW Rowley 2008 Cited by 24 control, relative to the more common open-loop flow control strategies. We also an example is keeping a laminar flow stable beyond its usual transition point. We will give Feedback can also reduce sensitivity to external disturbances, nonlinear system (1), as long as we are sufficiently close to the equilibrium§. Since.40 pages

### Definition and Classification of Power System Stability - Iowa

by P Kundur 2004 Cited by 3669 the case of equilibrium points) or over a cycle (as in the case of slow cyclical robust the system with respect to large disturbances. The region voltage sensitive, and the connected demand at normal voltage is not met.

### Tutorial: Notes on Nonlinear Stability

11 Jun 2012 The equilibrium φ is stable (that is, nonlinearly stable) if: ∀ϵ > 0,∃δ > 0 such The definition may be very sensitive to the norms 1 and 2 as well This does not provide enough information: nonlinear perturbations can be Let u be the velocity at a point and p be the pressure. We can regard C = R2.23 pages

### ANALYSIS OF A PREY-PREDATOR

by O El Idrissi not depend on k2 (for all k2, the implicit scheme show that this point is stable coexistence equilibrium, the operator L(u, v), measuring the relative weight of the larger sensitivity of the resources relative growth rate g(r) to changes in the amount perturbation of the death rate of the juveniles by a function ε(a, r) which​

### Stability and Motion around Equilibrium Points in the - arXiv

by Y Jiang 2014 Cited by 3 linear stability of equilibrium points as well as periodic orbits around equilibrium points The linearised equations of motion relative to an equilibrium point are derived and chaotic. If a dynamical system is chaotic, then it is sensitive to initial conditions with gravitational orbit attitude coupling perturbation. Celest. Mech.

### Sensitivity to Cumulative Perturbations for a Class of

by A Sharifnassab 2019 Cited by 3 expansive with respect to the Euclidean norm. Furthermore that was posed in [​15], on delay-stability in the presence of heavy-tailed turbed trajectory comes close to a critical point: we show, in (p is not an equilibrium point, i.e., ξ = 0). The.

### A Tutorial on Sensitivity and Stability in Nonlinear

by G Giorgi 2018 Cited by 2 Sensitivity analysis, stability analysis, nonlinear programming, linear We point out also the book of Bank and others (1982) for another classical overview on The following basic results concerning the differentiability of a perturbed solution of a R. HORST (1984a), Shadow prices and equilibrium prices in linear and

### Alternatives to Resilience for Measuring the - JSTOR

by MG Neubert 1997 Cited by 418 at which perturbations to a stable ecological system decay. Key words: compartment models; eigenvalues; pulse perturbations; reactivity; relative stability; earization of a nonlinear system near an equilibrium point. Eq. 1 has the unique solution Sensitivity of resil- matrix whose 1jth entry gives the change in re- /v wT.

### Bounds on Transient Instability for Complex Ecosystems - PLOS

by F Caravelli 2016 Cited by 5 cies is at a stable equilibrium point then it is able to withstand small point then the effect of a small perturbation, such as the loss of information has been extracted from the community matrix regarding transient dynamics: Verdy A, Caswell H (2008) Sensitivity analysis of reactive ecological dynamics.

### The Equilibrium and Stability of Two Dimensional Pendent - JSTOR

centre plane, which occurs at the point of maximum volume. Likewise, of the equilibrium will be most sensitive to weak perturbations, and the latter will provide the test found by multiplying (3) by x' and integrating with respect to y~​~~~~~~.

### Nonlinear control of HIV-1 infection with a singular - CORE

by M Barao 2007 Cited by 58 A sensitivity analysis of the resulting law is performed with respect to the model parameter linearizing control; Stability robustness; LQ control; Singular perturbations approximate linear model around the equilibrium point.

### A Locust Phase Change Model with Multiple Switching States

by C Xiang Cited by 4 This type of switching is sensitive to the intrinsic growth rate and the initial evolving process of the perturbation system is not related to the initial value after some fixed number of (a) Bifurcation curve of equilibrium points with respect to r. (b) (d) H∗3 is a stable outbreak equilibrium point, and H∗2.

### Perturbation theory of dynamical systems - CEL

by N Berglund 2007 Cited by 32 Eliminating x⋆ from these equations, we get the relation. 4λ3. 1 = 27λ2. 2 1, the system has two stable equilibrium points x± and one unstable Such saddle connections are indeed very sensitive to small perturbations.

### Basics of Stability Theory in Nonlinear PDE - Mathew Johnson

by MA Johnson so that now u(t;u1) is considered as a perturbation of u0. From (1) As we will see, in stability theory the point and essential spectrum give drastically different information concerning the dynamics near a given nonlinear wave. In what fol- when considering the stability of a spatially periodic equilibrium solution of a PDE​.

### Stability Analysis for Incremental Nonlinear Dynamic Inversion Control

Lyapunov methods and nonlinear system perturbation theory. Lie derivatives [2​] of the function hi with respect to the vector fields f and gj are compared to the conventional NDI control law, the INDI control method is less sensitive to model mismatches, because dynamics has an asymptotically stable equilibrium point.

### An Introduction to Feedback Control in Systems Biology

Stability of the equilibrium points set-point or to track dynamic reference signals. input) causes an increase in the sensitivity of the system to disturbances,.

### Qualitative and quantitative responses to press perturbations

by G Giordano 2017 Cited by 9 has a stable equilibrium point, the net effect is given by the negative adjoint of the relative to the Perron-Frobenius eigenvalue) after a press perturbation must known, and the sensitivity of some variables to presses may grow unbounded.

### Global stability properties of equilibria, periodic, and chaotic

by MS Soliman 1996 Cited by 5 disturbance to a given attractor can cause a transition across a basin boundary resulting in a qualitative, and often substantial asymptotically stable, state A in relative terms is strongly forth across one stable equilibrium point and (ii) motions that are gesting that it may be extremely sensitive to small distur- bances.

### Variational analysis and related topics: preface - Taylor

by DC Chang 2011 problems of optimization, equilibria, control of systems governed by ordinary special attention to sensitivity/stability analysis with respect to perturbations. collectively fixed point theorem and maximal element theorem for a family of.

### Sensitivity and stability analysis of nonlinear Kalman filters

by MB Rhudy 2013 Cited by 5 invariant (LTI) systems with respect to perturbations in the initial error When analyzing the stability of an equilibrium point of a nonlinear system using.

### Chapter 3 One-Dimensional Systems

by IL Leak sensitive time constant T(V) is very small, i.e. T(V) < 1, in the entire play in neuron dynamics: The I-V relation in the first group has a region of negative (​rest) points the system could have, whether the equilibria are stable, their small noisy perturbations that are always present in biological systems can make V(t).37 pages

### STABILITY ANALYSIS OF A THREE-TIME - SciTePress

by S Esteban equilibrium points of the model; Section 3 introduces the singular perturbation control law formulation de- rived in (Esteban et al., 2005); the stability analysis.

### Power System Stability

Power system stability is defined as the property of a power system that enables it to Small disturbance (small signal) stability. Ability to δmax is an unstable equilibrium point. Local modes - Swinging of units at a generating system with respect to A system is voltage stable if V −Q sensitivity is positive for every bus​.

### OPTIMIZATION PROBLEMS WITH PERTURBATIONS: A

by JF BONNANS Cited by 345 We illustrate the results by computing the equilibrium position of a chain on stability and sensitivity analysis of optimization problems which are subject to perturbations. ∇g(x): gradient of the function g : Rn → R at the point x ∈ Rn,. X​. ∗ Regularity is also related to stability with respect to perturbations of the feasible.37 pages

### Sensitivity of static traffic user equilibria with perturbations in

by S Lu 2007 Cited by 41 This paper deals with sensitivity analysis of static traf c user equilibrium problems​. We apply In the rst step, we establish a result regarding perturbations of polyhedral convex sets. point (u0,d0,x0), where u0 is a point in U, d0 is a point in D and x0 is a solution of (0.7) The existence, uniqueness and stability of traf c.41 pages

### Sensitivity of Air Sea Fluxes to SST Perturbations - Harvard

by I RIVIN Cited by 12 determining the stability and variability behavior of ocean-only model studies of decadal climate variability. Here, N is the number of horizontal grid points per perturbation, the area is in 1012 m2, and While the changes in atmospheric temperature are small compared apparent atmospheric equilibrium temperature

### Sensitivity Analysis for Uncertainty Quantification in

by L Arriola Cited by 54 precisely quantify the ratio of output perturbations with respect to the input per- determine how the stability of an equilibrium point is affected by changes to the.53 pages

### STABILITY ANALYSIS METHODOLOGIES FOR DC POWER

by SD Sudhoff Cited by 61 of these analyses as well as the respective advantages and disadvantages of each of the methods are compared. constant output voltage regardless of input disturbances. From the output equilibrium point, an operating point, and of stability. Definitions for these chaotic like behavior œ they are extremely sensitive to.

### POWER SYSTEM DYNAMICS AND STABILITY

by PW Sauer 1997 Cited by 3142 controlling unstable equilibrium point approaches. The book does Compared to other disciplines within electrical engineering, the foundations of disturbance​, portions of the power system can be considered in quasi-steady state. a participation factor is a sensitivity measure of an eigenvalue to a diagonal entry of the

### Simple albedo feedback models of the icecaps

by IM Held 1974 Cited by 220 from the properties of small perturbations away from equilibrium. It is observed that The sensitivity and stability of the resulting model are non-linear diffusive transport are compared. there is n critical value of Q (turn around point of.

### Perturbation theory of dynamical systems

by N Berglund 2001 Cited by 32 Eliminating x⋆ from these equations, we get the relation. 4λ3. 1 = 27λ2. 2 1, the system has two stable equilibrium points x± and one unstable Such saddle connections are indeed very sensitive to small perturbations.

### The sensitivity of tokamak magnetohydrodynamics stability on

by LJ Zheng 2017 Cited by 2 Due to the X-point singularity, the safety factor tends to infinity as approaching to the last closed flux surface. The numerical stability is sensitive to the quality of the MHD equilibria, perturbed current density, l0 is the magnetic constant, P rep- resents the Fourier space, prime denotes derivate with respect to the poloidal

### Module 4 STABILITY AND BIFURCATION ANALYSIS Lect1

4 Mar 2014 In the previous lecture, we have learned about different perturbation methods to obtain the To study the stability of the equilibrium point few stability criteria are described below. F with respect to the control parameter µ and then by constructing a matrix. µ. Sensitive to initial condition: Butterfly effect.

### Perturbation analysis of transient population dynamics using

by I Stott 2016 Cited by 16 dicted long-term asymptotic stable, stationary or equilibrium dynamics. I discuss these with reference to the application of transient perturbation analyses biology, population management, population viability analysis, sensitivity, transfer function identified eight key points to consider when conducting a tran-.

### b) equilibrium calculation

axisymmetric perturbation provides a link between equilibrium and stability i theory, suggesting We will see that saturated island widths can be sensitive to the presence of the equation in that region with respect to r and turning it into a jump condition. The point 7 = 0 corresponds to a bifurcation point of the solu- tion.

### Control Theory Tutorial Basic Concepts Illustrated by - OAPEN

by SA Frank Cited by 21 the relation between various adaptive engineering control systems and the ways in setpoint often makes a system more susceptible to perturbations and more likely to become Stabilization considers controller design for robust stability. an equilibrium point, then the system will be approximately linear near that point.

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by H Ríos 2016 disturbances, then it is robustly stable (input-to-state stable. (ISS) or integral ISS (​iISS) some frequencies and be extremely sensitive to a particular frequency of with respect to its particular equilibrium-point is concluded.

### Destabilization by noise of tranverse perturbations to

by JR Gog 1999 Cited by 15 We show that transverse perturbations from structurally stable heteroclinic cycles properties with regard to the transverse modes, with the growth rate of transverse part of the transverse stability problem for the equilibrium point Pj. Then the mean steady velocity fields is very sensitive to the level of truncation (​see,

### Aircraft stability and control - University of Bath

M(x) = M0 + Lx. We have some freedom in where we take our reference and we should make best use of it. The obvious reference point on a wing.

### Increased sensitivity to strong perturbations in a - MPG.PuRe

by BM Jobst 2021 Cited by 1 Increased sensitivity to strong perturbations in a whole-brain model of working point further away from a stable equilibrium as compared to

### Khalil NonlinearSystems 5.pdf - UCLA PDC Lab

Then, the origin is an exponentially stable equilibrium point of the perturbed of the origin is robust with respect to a class of perturbations that satisfy (5.6)-. (5.7). that V(2, a) satisfies the last inequality (5.69), note that the sensitivity function.

### Nonlinear Ordinary Differential Equations - Math User Home

by PJ Olver Cited by 10 equilibrium points, and stability properties, one would not be able to understand when numerical so- and then integrate with respect to t; the result is. ∫. 1 Thus, the sensitive dependence of solutions on initial conditions already equilibrium point, all solutions to the nonlinear system are slight perturbations of solu-.

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by JF Arnoldi 2016 Cited by 19 In deterministic dynamical systems, vanishing points of the generating vector field it must be robust with respect to small perturbations, uncertainties and so forth the spectral sensitivity of the Jacobian matrix at the equilibrium, with respect.

### Feedback Fundamentals - Control and Dynamical Systems

represents the control signal and the input w represents the reference r, the load disturbance reference signal, when the system has settled to equilibrium a step in the load dis- The sensitivity is thus less than one for all points outside a circle with A technical condition, namely that the perturbation ∆P is a stable trans-.

### Bridging theories for ecosystem stability through - bioRxiv

by JJ Kuiper 2019 Cited by 1 sensitive analyses of classical consumer-resource models in perturbation to a system in equilibrium decays (Pimm and Lawton 1977; Neubert and Caswell 1997). 2017) did not discuss their results in relation to ecological resilience Local (fixed-point) stability analysis of the Jacobian matrix provides

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changes in a parameter often lead to hysteresis between two stable equilibria. forcing's drift rate; and (c) the sensitivity of the TP to the period and amplitude of results obtained in Exp1 and Exp2 with respect to perturbations of the forcing.

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by A Diveev 2021 Cited by 4 The method consists in making the control object stable relative to some point Keywords: optimal control; Lyapunov stability; equilibrium point; timal control [​10], modifies the nominal optimal trajectory such that it is less sensitive complex plane, so that uncertainties and perturbations do not make the

### Robust Control Methods for Nonlinear Systems with Uncertain

by CT Ton 2013 Cited by 2 equilibrium point; Lyapunov's Second Stability Theorem, or the direct method sensitivity to variations in the system parameters, improve tracking performance, SMC while retaining robustness with respect to norm-bounded disturbances.