Sensitivities and linear stability analysis around a double-zero eigenvalue

Luongo, A.; Paolone, A.; Egidio, A. Di

doi:10.2514/3.14463

Cited by 5 publications

(5 citation statements)

References 3 publications

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“…The tools which seem to be more promising are the Center Manifold Theory and the Normal Form Theory (see, for example, [22,167,61,134,114,140,115], where the two theories are connected to each other and to the Bifurcation Theory and to Chaos); the Tikhonov Theorem (see for example [158,47,67,76]); the Renormalization Group Theory applied to singular perturbations, recently rediscovered (see for example [58,26,45,77,116,133]); the Geometric Singular Perturbation Theory (GSPT) [52,68,78]. All these theories cope on one side with small perturbation parameters, on the other side with multiscale phenomena and timescale separation, which both allow the dimensional system reduction and the simplification of the model, according to the innovative ideas and techniques introduced and applied in several recent works by Luongo and coworkers [55,85,[87][88][89][90][91][92][93][94][95][96][97][98][99][100][101]123].…”

Section: Discussionmentioning

confidence: 99%

New trends and perspectives in nonlinear intracellular dynamics: one century from Michaelis–Menten paper

Bersani

Bersani²,

Dell’Acqua

et al. 2014

Continuum Mech. Thermodyn.

124

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One century after the seminal work by Leonor Michaelis and MaudMenten devoted to the theoretical study of the enzymatic reactions, in this paper we give an overview of the most recent trends concerning the mathematical modeling of several enzymatic mechanisms, focusing on its asymptotic analysis, which needs the use of advanced mathematical tools, such as Center Manifold Theory, Normal Forms, and Bifurcation Theory. Moreover we present some perspectives, linking the models here presented with similar models, arising from different research fields.

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Section: Discussionmentioning

confidence: 99%

New trends and perspectives in nonlinear intracellular dynamics: one century from Michaelis–Menten paper

Bersani

Bersani²,

Dell’Acqua

et al. 2014

Continuum Mech. Thermodyn.

124

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show abstract

“…Systems of the first type see algebraic and geometric multiplicity coincide (each eigenvalue with multiplicity r generates an eigenspace of dimension r ); the remaining systems belong to the second type. In Kato, 49 it is rigorously shown that non‐nilpotent systems admit solution as a perturbation series of integer powers of the ‘small’ parameter (Taylor series); on the contrary, fractional powers (Puiseux series) are needed for nilpotent systems 50 …”

Section: Damage As a Perturbationmentioning

confidence: 99%

Dynamic damage identification using complex mode shapes

Lofrano

Paolone

Ruta

2020

Struct Control Health Monit

Self Cite

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Damage, be it a material or a geometric degradation, modifies some features of the response foreseen by the original structural design. These variations, once the dependence on the damage causing them is established, can be used for identification purposes. In the literature, vibration-based approaches usually compare some responses of linear elastic structures with dissipative properties that are assumed proportional to the mass and stiffness measures. However, such an assumption is reasonable for new, undamaged structures, but can be unreliable in existing, potentially damaged structures, particularly for damages localised in narrow areas. The eigenmodes of a proportionally damped system can be reduced to the real ones of the relevant ideal undamped system. On the other hand, non-proportional damping exhibits complex eigenmodes that cannot be reduced to those of the ideal, or of the proportionally damped, structure. Thus, we may assume the complexity of the eigenmodes as a measure of nonproportional damping, hence of damage. On this basis, some contributions in the literature verified the relationship among presence of damage and amount of complexity. Here, we propose a perturbation approach and an objective function able to identify presence, location and amplitude of localised damages, intended as sources of non-proportionality in viscously damped linear systems. A prototype naturally discrete structure with four degrees-of-freedom is chosen to test and show capability and accuracy of the proposed method.

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“…If the follower force is larger than 1, the system undergoes a Hopf bifurcation, which means it has a pair of complex eigenvalues with positive real parts (Luongo et al, 2000). For example, if the follower force equals to 2, then the system matrix decoupled by the linear transformation is as follows:…”

Section: Doc Of the System Varying With A Parametermentioning

confidence: 99%

Degree of controllability for linear unstable systems

Lee

Park

2014

Journal of Vibration and Control

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The degree of controllability is a measure to check how controllable a given system is, which makes up for the weak points of traditional methods for checking controllability. There are many definitions of the degree of controllability, and the most widely used measure is represented by controllability Gramian. This measure has physical meaning of minimum input energy to change the states. However, it is hard to use this method for unstable systems because the steady-state value of Gramian cannot be defined. Therefore, the degree of controllability for unstable systems is proposed in this paper. The definition of the measure is minimum control input energy to change the state from an arbitrary initial condition to an arbitrary final condition. The derivation result shows that the measure is the sum of two terms, one represented by Gramian of a stable subsystem and the other represented by Gramian of a subsystem that has mirror image poles of an anti-stable subsystem. These Gramians can be obtained by solving two Lyapunov equations. A simple numerical example shows the usefulness of the proposed measure of degree of controllability.

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Sensitivities and linear stability analysis around a double-zero eigenvalue

Cited by 5 publications

References 3 publications

New trends and perspectives in nonlinear intracellular dynamics: one century from Michaelis–Menten paper

New trends and perspectives in nonlinear intracellular dynamics: one century from Michaelis–Menten paper

Dynamic damage identification using complex mode shapes

Degree of controllability for linear unstable systems

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