941 research outputs found
An algebraic criterion for the onset of chaos in nonlinear dynamic systems
The correspondence between iterated integrals and a noncommutative algebra is used to recast the given dynamical system from the time domain to the Laplace-Borel transform domain. It is then shown that the following algebraic criterion has to be satisfied for the outset of chaos: the limit (as tau approaches infinity and x sub 0 approaches infinity) of ((sigma(k=0) (tau sup k) / (k* x sub 0 sup k)) G II G = 0, where G is the generating power series of the trajectories, the symbol II is the shuffle product (le melange) of the noncommutative algebra, x sub 0 is a noncommutative variable, and tau is the correlation parameter. In the given equation, symbolic forms for both G and II can be obtained by use of one of the currently available symbolic languages such as PLI, REDUCE, and MACSYMA. Hence, the criterion is a computer-algebraic one
Revisiting the 1D and 2D laplace transforms
Foundation for Science and Technology of Portugal, under the projects UIDB/00066/2020.The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined.publishersversionpublishe
On Algebraic Approach for MSD Parametric Estimation
This article address the identification problem of the natural frequency and the damping ratio of a second order continuous system where the input is a sinusoidal signal. An algebra based approach for identifying parameters of a Mass Spring Damper (MSD) system is proposed and compared to the Kalman-Bucy filter. The proposed estimator uses the algebraic parametric method in the frequency domain yielding exact formula, when placed in the time domain to identify the unknown parameters. We focus on finding the optimal sinusoidal exciting trajectory which allow to minimize the variance of the identification algorithms. We show that the variance of the estimators issued from the algebraic identification method introduced by Fliess and Sira-Ramirez is less sensitive to the input frequency than the ones obtained by the classical recursive Kalman-Bucy filter. Unlike conventional estimation approach, where the knowledge of the statistical properties of the noise is required, algebraic method is deterministic and non-asymptotic. We show that we don't need to know the variance of the noise so as to perform these algebraic estimators. Moreover, as they are non-asymptotic, we give numerical results where we show that they can be used directly for online estimations without any special setting.International audienceThis article address the identification problem of the natural frequency and the damping ratio of a second order continuous system where the input is a sinusoidal signal. An algebra based approach for identifying parameters of a Mass Spring Damper (MSD) system is proposed and compared to the Kalman-Bucy filter. The proposed estimator uses the algebraic parametric method in the frequency domain yielding exact formula, when placed in the time domain to identify the unknown parameters. We focus on finding the optimal sinusoidal exciting trajectory which allow to minimize the variance of the identification algorithms. We show that the variance of the estimators issued from the algebraic identification method introduced by Fliess and Sira-Ramirez is less sensitive to the input frequency than the ones obtained by the classical recursive Kalman-Bucy filter. Unlike conventional estimation approach, where the knowledge of the statistical properties of the noise is required, algebraic method is deterministic and non-asymptotic. We show that we don't need to know the variance of the noise so as to perform these algebraic estimators. Moreover, as they are non-asymptotic, we give numerical results where we show that they can be used directly for online estimations without any special setting
Parameters estimation of systems with delayed and structured entries
International audienceThis paper deals with on-line identification of continuous-time systems with structured entries. Such entries, which may consist in inputs, perturbations or piecewise polynomial (time varying) parameters, can be defined as signals that can be easily annihilated. The proposed cancellation method allows to obtain non asymptotic estimators for the unknown coefficients. Application to delayed and switching hybrid systems are proposed. Numerical simulations with noisy data but also experimental results on a delay process are provided
The Likelihood of Mixed Hitting Times
We present a method for computing the likelihood of a mixed hitting-time
model that specifies durations as the first time a latent L\'evy process
crosses a heterogeneous threshold. This likelihood is not generally known in
closed form, but its Laplace transform is. Our approach to its computation
relies on numerical methods for inverting Laplace transforms that exploit
special properties of the first passage times of L\'evy processes. We use our
method to implement a maximum likelihood estimator of the mixed hitting-time
model in MATLAB. We illustrate the application of this estimator with an
analysis of Kennan's (1985) strike data.Comment: 35 page
Extended Plefka Expansion for Stochastic Dynamics
We propose an extension of the Plefka expansion, which is well known for the
dynamics of discrete spins, to stochastic differential equations with
continuous degrees of freedom and exhibiting generic nonlinearities. The
scenario is sufficiently general to allow application to e.g. biochemical
networks involved in metabolism and regulation. The main feature of our
approach is to constrain in the Plefka expansion not just first moments akin to
magnetizations, but also second moments, specifically two-time correlations and
responses for each degree of freedom. The end result is an effective equation
of motion for each single degree of freedom, where couplings to other variables
appear as a self-coupling to the past (i.e. memory term) and a coloured noise.
This constitutes a new mean field approximation that should become exact in the
thermodynamic limit of a large network, for suitably long-ranged couplings. For
the analytically tractable case of linear dynamics we establish this exactness
explicitly by appeal to spectral methods of Random Matrix Theory, for Gaussian
couplings with arbitrary degree of symmetry
Two-point Functions and Quantum Fields in de Sitter Universe
We present a theory of general two-point functions and of generalized free
fields in d-dimensional de Sitter space-time which closely parallels the
corresponding minkowskian theory. The usual spectral condition is now replaced
by a certain geodesic spectral condition, equivalent to a precise thermal
characterization of the corresponding ``vacuum''states. Our method is based on
the geometry of the complex de Sitter space-time and on the introduction of a
class of holomorphic functions on this manifold, called perikernels, which
reproduce mutatis mutandis the structural properties of the two-point
correlation functions of the minkowskian quantum field theory. The theory
contains as basic elementary case the linear massive field models in their
``preferred'' representation. The latter are described by the introduction of
de Sitter plane waves in their tube domains which lead to a new integral
representation of the two-point functions and to a Fourier-Laplace type
transformation on the hyperboloid. The Hilbert space structure of these
theories is then analysed by using this transformation. In particular we show
the Reeh-Schlieder property. For general two-point functions, a substitute to
the Wick rotation is defined both in complex space-time and in the complex mass
variable, and substantial results concerning the derivation of Kallen-Lehmann
type representation are obtained.Comment: 51 p, uuencoded, LaTex, epsf, 2 figures include
Parameter estimation via differential algebra and operational culculus
Parameter estimation is approached via a new standpoint, based on differential algebra and operational calculus. Some applications such as, the estimation of a noisy damped sinusoid, the analysis of chirp signal, the detection of piecewise polynomial signals and their discontinuities are presented with numerical simulations
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