47,097 research outputs found
Transverse exponential stability and applications
We investigate how the following properties are related to each other: i)-A
manifold is "transversally" exponentially stable; ii)-The "transverse"
linearization along any solution in the manifold is exponentially stable;
iii)-There exists a field of positive definite quadratic forms whose
restrictions to the directions transversal to the manifold are decreasing along
the flow. We illustrate their relevance with the study of exponential
incremental stability. Finally, we apply these results to two control design
problems, nonlinear observer design and synchronization. In particular, we
provide necessary and sufficient conditions for the design of nonlinear
observer and of nonlinear synchronizer with exponential convergence property
Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators
We describe a flexible and modular delayed-feedback nonlinear oscillator that
is capable of generating a wide range of dynamical behaviours, from periodic
oscillations to high-dimensional chaos. The oscillator uses electrooptic
modulation and fibre-optic transmission, with feedback and filtering
implemented through real-time digital-signal processing. We consider two such
oscillators that are coupled to one another, and we identify the conditions
under which they will synchronize. By examining the rates of divergence or
convergence between two coupled oscillators, we quantify the maximum Lyapunov
exponents or transverse Lyapunov exponents of the system, and we present an
experimental method to determine these rates that does not require a
mathematical model of the system. Finally, we demonstrate a new adaptive
control method that keeps two oscillators synchronized even when the coupling
between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special
theme issue to accompany 2009 International Workshop on Delayed Complex
Systems
Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation
In this paper, we investigate the spectral instability of periodic traveling
wave solutions of the generalized Korteweg-de Vries equation to long wavelength
transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By
analyzing high and low frequency limits of the appropriate periodic Evans
function, we derive an orientation index which yields sufficient conditions for
such an instability to occur. This index is geometric in nature and applies to
arbitrary periodic traveling waves with minor smoothness and convexity
assumptions on the nonlinearity. Using the integrable structure of the ordinary
differential equation governing the traveling wave profiles, we are then able
to calculate the resulting orientation index for the elliptic function
solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem
corrected. Exposition updated and references added
A matrix stability analysis of the carbuncle phenomenon
The carbuncle phenomenon is a shock instability mechanism which ruins all efforts to compute grid-aligned shock
waves using low-dissipative upwind schemes. The present study develops a stability analysis for two-dimensional steady shocks on structured meshes based on the matrix method. The numerical resolution of the corresponding eigenvalue problem confirms the typical odd–even form of the unstable mode and displays a Mach number threshold effect currently observed in computations. Furthermore, the present method indicates that the instability of steady shocks is not only governed by the upstream Mach number but also by the numerical shock structure. Finally, the source of the instability is localized in the upstream region, providing some clues to better understand and control the onset of the carbuncle
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