scholarly journals New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2872
Author(s):  
Sergey Kashchenko ◽  
Anna Tolbey
Keyword(s):  
Differential Operator ◽  
Nonlinear Systems ◽  
Original Problem ◽  
Spatial Variable ◽  
Local Behavior ◽  
Spatial Variables ◽  
Irregular Solution ◽  
Spatially Distributed ◽  
A Chain ◽  
Special Differential Operator

For the spatially-distributed Fermi–Pasta–Ulam (FPU) equation, irregular solutions are studied that contain components rapidly oscillating in the spatial variable, with different asymptotically large modes. The main result of this paper is the construction of families of special nonlinear systems of the Schrödinger type—quasinormal forms—whose nonlocal dynamics determines the local behavior of solutions to the original problem, as t→∞. On their basis, results are obtained on the asymptotics in the main solution of the FPU equation and on the interaction of waves moving in opposite directions. The problem of “perturbing” the number of N elements of a chain is considered. In this case, instead of the differential operator, with respect to one spatial variable, a special differential operator, with respect to two spatial variables appears. This leads to a complication of the structure of an irregular solution.

A chain observer for a class of nonlinear systems with long multiple delays in output measurements

2018 ◽  
Author(s):  
Boubekeur Targui ◽  
O. Hernandez-Gonzalez ◽  
C.M. Astorga-Zaragoza ◽  
M. Pouliquen ◽  
O. Gehan
Keyword(s):  
Nonlinear Systems ◽  
Multiple Delays ◽  
A Chain

A Chain Observer for Nonlinear Systems with Multiple Time-Varying Measurement Delays

2014 ◽  
Vol 52 (3) ◽  
pp. 1862-1885 ◽  
Author(s):  
Filippo Cacace ◽  
Alfredo Germani ◽  
Costanzo Manes
Keyword(s):  
Nonlinear Systems ◽  
Multiple Time ◽  
Time Varying ◽  
A Chain

SPATIAL MODELING OF WIND SOIL EROSION IN TAIZ GOVERNORATE USING REMOTE SENSING AND GEOGRAPHIC INFORMATION SYSTEMS

2020 ◽  
pp. 67-97
Author(s):  
Ibrahim Darwish
Keyword(s):  
Remote Sensing ◽  
Soil Erosion ◽  
Spatial Modeling ◽  
Wind Erosion ◽  
Spatial Variable ◽  
Spatial Variables ◽  
Building Model ◽  
Raster Calculator ◽  
Mathematical Algorithms ◽  
The Impact

The variation Wind erosion of soil from place to another, cause the variation of variables affecting its activity, and the research aims to quantify the variables of wind erosion soil in governorate of Taiz, to reveal effect of each of them on wind erosion in governorate, was followed by revealing effect of all these spatial variables combined on wind erosion. , By building model by which all these variables were Merge according their weights. The research methodology was to collect data on each spatial variable related to wind erosion of soil, And processing this data, and deriving the informational layers for each spatial variable using a number of mathematical algorithms in the raster calculator for Arc Map 10.5, Highlighting effect of each spatial variable separately on wind erosion activity in governorate, In preparation for integrating all these spatial variables together by means of a model that shows suitability of soil for wind erosion in governorate according to the impact of all these spatial variables. The results of research found that area of district that was Described a high suitability for wind erosion is 4129.2 km2, or 41.22%, and the area of district that was Described moderate suitability for wind erosion is 2267.4 km2, or 22.63%, and area of district that was Described low suitability for wind erosion 2267.4 km2, or 22.63%, and the area of district that was Described by strong suitability for wind erosion is 461.9 km2, or 4.61%, and the area of district that was Described poorly suitability for wind erosion 275.2 km2, or 2.74% of total area governorate.

Robust asymptotic tracking control of a chain of integrator nonlinear systems with input constraint and hysteresis nonlinearity

2016 ◽  
Vol 38 (12) ◽  
pp. 1500-1508 ◽  
Author(s):  
Zhenle Dong ◽  
Jianyong Yao ◽  
Dawei Ma
Keyword(s):  
Nonlinear Systems ◽  
Tracking Control ◽  
Controller Design ◽  
Input Constraint ◽  
Hysteresis Nonlinearity ◽  
Asymptotic Tracking ◽  
Disturbance Term ◽  
A Chain ◽  
Constant Slope ◽  
Bounded Disturbance

This paper focuses on the problem of tracking control of a chain of integrator nonlinear systems with input constraint and hysteresis nonlinearity. Input constraint, always existing in physical systems, has been proved a source of performance degradation. To handle this issue, an effective hyperbolic saturation function is employed, which is bounded no matter how the disturbances and error signals change. Furthermore, hysteresis nonlinearity, which may also limit the system performance, is modelled as a combination of a linear term with constant slope and a bounded disturbance term, which makes it possible to be integrated in the model based controller design. The robust integral of the sign of error (RISE) control is synthesized to guarantee the asymptotic tracking performance in the presence of parametric uncertainties and unmodelled nonlinearities such as external disturbances and unmodelled hysteresis nonlinearity. The closed-loop stability is proved via Lyapunov analysis. Some simulations are carried out to verify the effectiveness of the proposed controller.

scholarly journals Reservoir computing based on a silicon microring and time multiplexing for binary and analog operations

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Massimo Borghi ◽  
Stefano Biasi ◽  
Lorenzo Pavesi
Keyword(s):  
Time Series Prediction ◽  
Free Carrier ◽  
Disruptive Technology ◽  
Reservoir Computing ◽  
Integrated Optical ◽  
Spatially Distributed ◽  
High Bandwidth ◽  
Spatio Temporal ◽  
A Chain

AbstractPhotonic implementations of reservoir computing (RC) promise to reach ultra-high bandwidth of operation with moderate training efforts. Several optoelectronic demonstrations reported state of the art performances for hard tasks as speech recognition, object classification and time series prediction. Scaling these systems in space and time faces challenges in control complexity, size and power demand, which can be relieved by integrated optical solutions. Silicon photonics can be the disruptive technology to achieve this goal. However, the experimental demonstrations have been so far focused on spatially distributed reservoirs, where the massive use of splitters/combiners and the interconnection loss limits the number of nodes. Here, we propose and validate an all optical RC scheme based on a silicon microring (MR) and time multiplexing. The input layer is encoded in the intensity of a pump beam, which is nonlinearly transferred to the free carrier concentration in the MR and imprinted on a secondary probe. We harness the free carrier dynamics to create a chain-like reservoir topology with 50 virtual nodes. We give proof of concept demonstrations of RC by solving two nontrivial tasks: the delayed XOR and the classification of Iris flowers. This forms the basic building block from which larger hybrid spatio-temporal reservoirs with thousands of nodes can be realized with a limited set of resources.

scholarly journals Linear Differential Operators Invertible in the Integro-differential Sense

2019 ◽  
pp. 20-33
Author(s):  
V. N. Chetverikov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Differential Operators ◽  
Spatial Variable ◽  
Diagonal Form ◽  
Operator Equations ◽  
Evolutionary Systems ◽  
Linear Differential Operators ◽  
The Matrix ◽  
Linear Differential

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

scholarly journals Generalized conformable operators: Application to the design of nonlinear observers

2021 ◽  
Vol 6 (11) ◽  
pp. 12952-12975
Author(s):  
Fidel Meléndez-Vázquez ◽  
◽  
Guillermo Fernández-Anaya ◽  
Aldo Jonathan Muñóz-Vázquez ◽  
Eduardo Gamaliel Hernández-Martínez ◽  
...  
Keyword(s):  
Differential Operator ◽  
Nonlinear Systems ◽  
Exponential Stability ◽  
Numerical Simulations ◽  
Stability Function ◽  
Nonlinear Observers ◽  
Conformable Derivatives ◽  
Generalized Differential ◽  
The Stability ◽  
Comparison Of The Results

<abstract><p>In this work, a pair of observers are proposed for a class of nonlinear systems whose dynamics involve a generalized differential operator that encompasses the conformable derivatives. A generalized conformable exponential stability function, based on this derivative, is introduced in order to prove some Lyapunov-like theorems. These theorems help to verify the stability of the observers proposed, which is exponential in a generalized sense. The performance of the observation scheme is evaluated by means of numerical simulations. Moreover, a comparison of the results obtained with integer, fractional, and generalized conformable derivatives is made.</p></abstract>

scholarly journals Existence and Relaxation Results for Second Order Multivalued Systems

2021 ◽  
Vol 173 (1) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro
Keyword(s):  
Differential Operator ◽  
Nonlinear Systems ◽  
Variational Inequalities ◽  
Control Systems ◽  
Maximal Monotone ◽  
Combined Effects ◽  
Nonhomogeneous Differential Operator ◽  
And Control ◽  
Differential Variational Inequalities ◽  
Multivalued Perturbation

AbstractWe consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term $A(x)$ A ( x ) and of a multivalued perturbation $F(t,x,y)$ F ( t , x , y ) which can be convex or nonconvex valued. We consider the cases where $D(A)\neq \mathbb{R}^{N}$ D ( A ) ≠ R N and $D(A)= \mathbb{R}^{N}$ D ( A ) = R N and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

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