Search results for "Nonlinear system"
showing 10 items of 1446 documents
Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations
2011
We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and we give conditions for such existence and uniqueness in some cases. Finally we illustrate these methods with an example of a collocation problem, and we give some examples of collocation problems that do not fit in the cases studied previously.
Numerical study of the long wavelength limit of the Toda lattice
2014
We present the first detailed numerical study of the Toda equations in $2+1$ dimensions in the limit of long wavelengths, both for the hyperbolic and elliptic case. We first study the formal dispersionless limit of the Toda equations and solve initial value problems for the resulting system up to the point of gradient catastrophe. It is shown that the break-up of the solution in the hyperbolic case is similar to the shock formation in the Hopf equation, a $1+1$ dimensional singularity. In the elliptic case, it is found that the break-up is given by a cusp as for the semiclassical system of the focusing nonlinear Schr\"odinger equation in $1+1$ dimensions. The full Toda system is then studie…
Spiking patterns emerging from wave instabilities in a one-dimensional neural lattice.
2003
The dynamics of a one-dimensional lattice (chain) of electrically coupled neurons modeled by the FitzHugh-Nagumo excitable system with modified nonlinearity is investigated. We have found that for certain conditions the lattice exhibits a countable set of pulselike wave solutions. The analysis of homoclinic and heteroclinic bifurcations is given. Corresponding bifurcation sets have the shapes of spirals twisting to the same center. The appearance of chaotic spiking patterns emerging from wave instabilities is discussed.
Response Power Spectrum of Multi-Degree-of-Freedom Nonlinear Systems by a Galerkin Technique
2003
This paper deals with the estimation of spectral properties of randomly excited multi-degree-of-freedom (MDOF) nonlinear vibrating systems. Each component of the vector of the stationary system response is expanded into a trigonometric Fourier series over an adequately long interval T. The unknown Fourier coefficients of individual samples of the response process are treated by harmonic balance, which leads to a set of nonlinear equations that are solved by Newton’s method. For polynomial nonlinearities of cubic order, exact solutions are developed to compute the Fourier coefficients of the nonlinear terms, including those involved in the Jacobian matrix associated with the implementation o…
Intensity-invariant nonlinear filtering for detection in camouflage.
2005
We introduce a method based on an orthonormal vector space basis representation to detect camouflaged targets in natural environments. The method is intensity invariant so that camouflaged targets are detected independently of the illumination conditions. The detection technique does not require one to know the exact camouflage pattern, but only the class of patterns (e.g., foliage, netting, woods). We use nonlinear filtering and the calculation of several correlations. The nonlinearity of the filtering process also allows high discrimination against false targets. Several experiments confirm the target detectability where strong camouflage might delude even human viewers.
On the existence of bounded solutions to a class of nonlinear initial value problems with delay
2017
We consider a class of nonlinear initial value problems with delay. Using an abstract fixed point theorem, we prove an existence result producing a unique bounded solution.
Le filtre de Kalman étendu à grand-gain adaptatif et ses applications
2010
The work concerns the “observability problem”—the reconstruction of a dynamic process’s full state from a partially measured state— for nonlinear dynamic systems. The Extended Kalman Filter (EKF) is a widely-used observer for such nonlinear systems. However it suffers from a lack of theoretical justifications and displays poor performance when the estimated state is far from the real state, e.g. due to large perturbations, a poor initial state estimate, etc. . . We propose a solution to these problems, the Adaptive High-Gain (EKF). Observability theory reveals the existence of special representations characterizing nonlinear systems having the observability property. Such representations ar…
Wavelength tuning of femtosecond pulses generated in nonlinear crystals by using diffractive lenses
2010
We demonstrate that diffractive lenses (DLs) can be used as a simple method to tune the central wavelength of femtosecond pulses generated from second-order nonlinear optical processes in birefringent crystals. The wavelength tunability is achieved by changing the relative distance between the nonlinear crystal and the DL, which acts in a focusing configuration. Besides the many practical applications of the so-generated pulses, the proposed method might be extended to other wavelength ranges by demonstrated similar effects on other nonlinear processes, such as high-order harmonic generation.
Nonlinear higher-order polariton topological insulator
2020
We address the resonant response and bistability of the exciton-polariton corner states in a higher-order nonlinear topological insulator realized with kagome arrangement of microcavity pillars. Such states are resonantly excited and exist due to the balance between pump and losses, on the one hand, and between nonlinearity and dispersion in inhomogeneous potential landscape, on the other hand, for pump energy around eigen-energies of corresponding linear localized modes. Localization of the nonlinear corner states in a higher-order topological insulator can be efficiently controlled by tuning pump energy. We link the mechanism of corner state formation with symmetry of the truncated kagome…
QUALITATIVE PROPERTIES OF THE SOLUTIONS OF A NONLINEAR FLUX-LIMITED EQUATION ARISING IN THE TRANSPORT OF MORPHOGENS
2011
In this paper we study some qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens in biological systems. Questions related to the existence of steady states, the finite speed of propagating fronts or the regularization in the interior of the support are studied from analytical and numerical points of view.