scholarly journals Adapting the range of validity for the Carleman linearization

2016 ◽  
Vol 14 ◽  
pp. 51-54 ◽  
Author(s):  
Harry Weber ◽  
Wolfgang Mathis
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Nonlinear Differential Equation ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Liouville Problem ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential

Abstract. In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm–Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.

The Description of a Distributed Parameter Process through a Finite Discrete Dimensional System

2014 ◽  
Vol 657 ◽  
pp. 874-878
Author(s):  
Sever Şerban ◽  
Doina Corina Şerban
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Industrial Process ◽  
Time Discretization ◽  
Distributed Parameter ◽  
Geometrical Parameters ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Distributed Parameters

This article analyses the process of warming a metal by using a walking beam furnace. This process is meant to offer the technologist objective information that may allow him to produce eventual modifications of the temperature references from the furnaces zones. Thus making the metals temperature at the furnaces exit to have an imposed distribution, within precise limits, according to the technological requests. This industrial process has a geometrical parameters distribution, more precisely it can be described through a partial differential equation, by being attached to dynamic infinite dimensional systems (or with distributed parameters). Using a procedure called geometric-time discretization (in the condition of the solutions convergence), we have managed to obtain a representation under the form of a finite discrete dimensional linear system for a process with distributed parameters.

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

CHAOTIC BEHAVIOR OF ORBITS CLOSE TO A HETEROCLINIC CONTOUR

1996 ◽  
Vol 06 (01) ◽  
pp. 69-79 ◽  
Author(s):  
M. BLÁZQUEZ ◽  
E. TUMA
Keyword(s):  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Chua’S Circuit ◽  
Closed Contour ◽  
Chua's Circuit ◽  
Infinite Dimensional ◽  
Focus Type

We study the behavior of the solutions in a neighborhood of a closed contour formed by two heteroclinic connections to two equilibrium points of saddle-focus type. We consider both the three-dimensional case, as in the well-known Chua's circuit, as well as the infinite-dimensional case.

scholarly journals On entire solutions of a certain type of nonlinear differential equation

2001 ◽  
Vol 64 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Chung-Chun Yang
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Finite Order ◽  
Nonlinear Differential Equation ◽  
Entire Solution ◽  
Differential Polynomial ◽  
Entire Solutions ◽  
Value Distribution Theory ◽  
Linear Differential Polynomial ◽  
Linear Differential

In this note, we shall study, via Nevanlinna's value distribution theory, the uniqueness of transcendental entire solutions of the following type of nonlinear differential equation: (*) L (f (z)) – p (z) fn(z) = h (z), where L (f) denotes a linear differential polynomial in f with polynomials as its co-efficients, p (z) a polynomial (≢ 0), h an entire function, and n an integer ≥ 3. We show that if the equation (*) has a finite order transcendental entire solution, then it must be unique, unless L (f) ≡ 0.

scholarly journals Asymptotics for some discretizations of dynamical systems, application to second order systems with non-local nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Thierry Horsin ◽  
Mohamed Ali Jendoubi
Keyword(s):  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Equilibrium Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Finite Dimensional ◽  
Angle Condition ◽  
Non Local ◽  
Second Order Systems

<p style='text-indent:20px;'>In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.</p>

scholarly journals The initial value problem in Lagrangian drift kinetic theory

2016 ◽  
Vol 82 (3) ◽  
Author(s):  
J. W. Burby
Keyword(s):  
Phase Space ◽  
Kinetic Theory ◽  
Initial Value Problem ◽  
High Order ◽  
Dimensional System ◽  
Initial Value ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Basic Philosophy ◽  
System Phase

Existing high-order variational drift kinetic theories contain unphysical rapidly varying modes that are not seen at low orders. These unphysical modes, which may be rapidly oscillating, damped or growing, are ushered in by a failure of conventional high-order drift kinetic theory to preserve the structure of its parent model’s initial value problem. In short, the (infinite dimensional) system phase space is unphysically enlarged in conventional high-order variational drift kinetic theory. I present an alternative, ‘renormalized’ variational approach to drift kinetic theory that manifestly respects the parent model’s initial value problem. The basic philosophy underlying this alternate approach is that high-order drift kinetic theory ought to be derived by truncating the all-orders system phase-space Lagrangian instead of the usual ‘field$+$particle’ Lagrangian. For the sake of clarity, this story is told first through the lens of a finite-dimensional toy model of high-order variational drift kinetics; the analogous full-on drift kinetic story is discussed subsequently. The renormalized drift kinetic system, while variational and just as formally accurate as conventional formulations, does not support the troublesome rapidly varying modes.

Basic Predictor Feedback for Single-Input Systems

2020 ◽  
pp. 19-34
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Differential Equation ◽  
Target System ◽  
Alternative View ◽  
Cascade System ◽  
Actuator Dynamics ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Backstepping Approach ◽  
Plant Dynamics ◽  
Single Input

This chapter discusses the basic idea of a partial differential equation (PDE) backstepping approach for single-input LTI ordinary differential equation (ODE) systems with discrete input delay. The key point of the backstepping approach lies in it providing a systematic construction of an infinite-dimensional transformation of the actuator state, which yields a cascade system of transformed stable actuator dynamics and stabilized plant dynamics. The cascade system consisting of such infinite-dimensional stable actuator dynamics and finite-dimensional stabilized plant dynamics is referred to as the closed-loop “target system.” The chapter first presents an alternative view of the backstepping transformation based purely on standard ODE delay notation. Then the backstepping transformation is described in PDE and rescaled unity-interval transport PDE notation.

scholarly journals THE QUANTUM H3 INTEGRABLE SYSTEM

2010 ◽  
Vol 25 (30) ◽  
pp. 5567-5594 ◽  
Author(s):  
MARCOS A. G. GARCÍA ◽  
ALEXANDER V. TURBINER
Keyword(s):  
Integrable System ◽  
Differential Operators ◽  
Three Dimensional ◽  
Integrable Model ◽  
Dimensional System ◽  
Property A ◽  
Algebraic Form ◽  
Infinite Dimensional ◽  
Polynomial Coefficients ◽  
Finite Dimensional

The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattice in a space of H3-invariants "polynomially"-isospectral to the quantum H3 model is defined.

Predictor Feedback of Uncertain Multi-Input Systems

2020 ◽  
pp. 267-292
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Linear System ◽  
Output Feedback ◽  
State Feedback ◽  
System Matrix ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Unknown System ◽  
Distributed Actuator

This chapter discusses the predictor feedback for uncertain multi-input systems. This is based on the predictor feedback framework for uncertainty-free multi-input systems in the tenth chapter. The chapter addresses four combinations of the five uncertainties that come from a finite-dimensional multi-input linear system with distributed actuator delays. These uncertainties include the following types: unknown and distinct delays, unknown delay kernels, unknown system matrix, unmeasurable finite-dimensional plant state, and unmeasurable infinite-dimensional actuator state. The chapter then examines the adaptive state feedback under unknown as well as uncertain delays, delay kernels, and parameters. It also explores robust output feedback under unknown delays, delay kernels, and PDE or ODE states.

Center Manifold of Fractional Dynamical System

2015 ◽  
Vol 11 (2) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Dynamical System ◽  
Center Manifold ◽  
High Dimensional ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Reduction Methods ◽  
Fractional Dynamical System ◽  
Lower Dimensional

Dimension reduction of dynamical system is a significant issue for technical applications, as regards both finite dimensional system and infinite dimensional systems emerging from either science or engineering. Center manifold method is one of the main reduction methods for ordinary differential systems (ODSs). Does there exists a similar method for fractional ODSs (FODSs)? In other words, does there exists a method for reducing the high-dimensional FODS into a lower-dimensional FODS? In this study, we establish a local fractional center manifold for a finite dimensional FODS. Several examples are given to illustrate the theoretical analysis.

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