310,356 research outputs found
An application of the ince algebraization to the stability of non-linear normal vibration modes
International audienceA normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by !nee (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained
Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays
This paper addresses the qualitative theory of mixed-order positive linear
coupled systems with bounded or unbounded delays. First, we introduce a general
result on the existence and uniqueness of solutions to mixed-order linear
coupled systems with time-varying delays. Next, we obtain the necessary and
sufficient criteria which characterize the positivity of a mixed-order delay
linear coupled system. Our main contribution is in Section 5. More precisely,
by using a smoothness property of solutions to fractional differential
equations and developing a new appropriated comparison principle for solutions
to mixed-order delayed positive systems, we prove the attractivity of
mixed-order non-homogeneous linear positive coupled systems with bounded or
unbounded delays. We also establish a necessary and sufficient condition to
characterize the stability of homogeneous systems. As a consequence of these
results, we show the smallest asymptotic bound of solutions to mixed-order
delay non-homogeneous linear positive coupled systems where disturbances are
continuous and bounded. Finally, we provide numerical simulations to illustrate
the proposed theoretical results
Observations on the Stability Properties of Cooperative Systems
We extend two fundamental properties of positive linear time-invariant (LTI) systems to homogeneous
cooperative systems. Specifically, we demonstrate that such systems are D-stable, meaning
that global asymptotic stability is preserved under diagonal scaling. We also show that a delayed
homogeneous cooperative system is globally asymptotically stable (GAS) for any non-negative delay if
and only if the system is GAS for zero delay
Instabilities and Patterns in Coupled Reaction-Diffusion Layers
We study instabilities and pattern formation in reaction-diffusion layers
that are diffusively coupled. For two-layer systems of identical two-component
reactions, we analyze the stability of homogeneous steady states by exploiting
the block symmetric structure of the linear problem. There are eight possible
primary bifurcation scenarios, including a Turing-Turing bifurcation that
involves two disparate length scales whose ratio may be tuned via the
inter-layer coupling. For systems of -component layers and non-identical
layers, the linear problem's block form allows approximate decomposition into
lower-dimensional linear problems if the coupling is sufficiently weak. As an
example, we apply these results to a two-layer Brusselator system. The
competing length scales engineered within the linear problem are readily
apparent in numerical simulations of the full system. Selecting a :1
length scale ratio produces an unusual steady square pattern.Comment: 13 pages, 5 figures, accepted for publication in Phys. Rev.
Asymptotic behaviour for a class of non-monotone delay differential systems with applications
The paper concerns a class of -dimensional non-autonomous delay
differential equations obtained by adding a non-monotone delayed perturbation
to a linear homogeneous cooperative system of ordinary differential equations.
This family covers a wide set of models used in structured population dynamics.
By exploiting the stability and the monotone character of the linear ODE, we
establish sufficient conditions for both the extinction of all the populations
and the permanence of the system. In the case of DDEs with autonomous
coefficients (but possible time-varying delays), sharp results are obtained,
even in the case of a reducible community matrix. As a sub-product, our results
improve some criteria for autonomous systems published in recent literature. As
an important illustration, the extinction, persistence and permanence of a
non-autonomous Nicholson system with patch structure and multiple
time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017
Robustness of infinite dimensional systems
The results contained within this thesis concern an abstract
framework for a robustness analysis of exponential stability of infinite
dimensional systems. The abstract analysis relies on the strong
relationship between exponential stability and L2-stability which
exists for many classes of linear systems.
In Chapter 1a "stability radius", for systems governed by semigroups,
is developed, for a class of "structured" perturbations of its
generator. The abstract theory is illustrated by examples of perturbations
of the boundary data for homogeneous boundary value problems and also
perturbations arising due to neglected delay terms in differential delay
equations.
In Chapter 2a related problem of a non standard linear quadratic
problem is studied, which leads to a stability analysis for certain nonlinear
systems.
In Chapter 3 an abstract L2-stability theory is developed and
then applied to integrodifferential equations and time-varying systems,
to investigate the robustness of exponential stability of such systems
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