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An Algorithm for Computing a New Normal Form for Dynamical Systems

Guoting Chen, Jean Della Dora
2000 Journal of symbolic computation  
We propose in this paper a new normal form for dynamical systems or vector fields which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms.  ...  As a particular case, if the matrix of the linear part is a companion matrix then we reduce the dynamical system to a single differential equation.  ...  Using the Carleman linearization procedure and a Frobenius basis in H k we introduced in Chen and Della Dora (1999a) a rational method for the normal form of any dynamical system.  ... 
doi:10.1006/jsco.1999.0305 fatcat:szginflozzchrddcsrdvma2ueq

Page 8025 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
Summary: “In this paper, motivated by the restrictive conditions required to obtain an exact chained form, we propose a quadratic normal form around a one-dimensional equilibrium submanifold for systems  ...  Poincaré normal form for a class of driftless systems in a one-dimensional submanifold neighborhood. (English summary) Math. Control Signals Systems 15 (2002), no. 3, 256-274.  ... 

Page 3490 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
and restricted smooth state feedback, a gen- eralized normal form called the p-normal form, which includes the Brunovsky canonical form and feedback linearizable systems in a lower-triangular form as  ...  Summary: “For a class of dynamical systems, with uncertain non- linear terms considered as unknown inputs, we give sufficient conditions for observability.  ... 

Page 981 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
Xue Min Li (PRC-SHDN; Jinan) 2002b:34010 34A25 37G05 Chen, Guoting (F-LILL; Villeneuve d’Ascq); Della Dora, Jean (F-GREN-LM; Grenoble) Rational normal form for dynamical systems by Carleman linearization  ...  They recall the so called Carleman Linearization, that is they consider the action of the derivative D in the infinite-dimensional space generated by monomials of the form x® = x}"'x;?...xf".  ... 

Page 2133 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
They also provide Carleman estimates and global For the web version of Mathematical Reviews, see http: //www.ams .org/mathscinet  ...  The authors study controllability properties of a class of linear control systems of the form x(t) = Ax(t)+ Bu(t), 1>0, x(t)eE X, u(t) € U, where X, U are non-reflexive Banach spaces.  ... 

Exact Solutions of Discrete Kinetic Models and Stationary Problems for the Plane Broadwell Model

A. V. Bobylev
1996 Mathematical methods in the applied sciences  
for a non-empty class of solutions p(x,t) and u(x,t) of the linear system (5) .  ...  The approach based on the Poincare normal form method (Carleman model) We consider here the Carleman equations in dimensionless form I a a -+- at ax where f,(x,t) a a f= 1 !  ... 
doi:10.1002/(sici)1099-1476(19960710)19:10<825::aid-mma799>3.0.co;2-1 fatcat:coxfxxokqfanjcbrv44nebdu3m

Further Reductions of Normal Forms for Dynamical Systems

Guoting Chen, Jean Della Dora
2000 Journal of Differential Equations  
For dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal (or unique) with respect to equivalence by formal near identity transformations  ...  We propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields, with concrete and interesting applications.  ...  We take v=w=0 in the following, since we are looking for rational normal forms. We study further reductions of systems with the given linear part by near identity transformations.  ... 
doi:10.1006/jdeq.2000.3783 fatcat:5esm3xsmyrchlbsrivg2c4zdhy

Page 6923 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
Hartje Kriete (D-GTN; Gottingen) 98k:30032 30D05 58F23 Gong, Zhimin (PRC-FUDAN-IM; Shanghai); Ren, Fuyao (PRC-FUDAN-IM; Shanghai) A random dynamical system formed by infinitely many functions.  ...  This result, which the author proves for rational functions R(z), was proved for polynomials P(z) by A. Eremenko and G. M. Levin [Ukrain. Mat.  ... 

Reducing nonlinear dynamical systems to canonical forms

Léon Brenig
2018 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
The drawback of this diversity is that it has prevented the construction of a theoretical apparatus as powerful as the one developed from linear algebra for linear dynamical systems of form (2.1).  ...  as a comparison, let us write down the corresponding abstract Lie algebra for linear systems of form (2.1).  ... 
doi:10.1098/rsta.2017.0384 pmid:29891502 fatcat:43yov5yu7fazfmvvidorgurali

Page 1917 of Mathematical Reviews Vol. 57, Issue 5 [page]

1979 Mathematical Reviews  
For a system of linear equations characterizing the initial mixed boundary-value problems of thermoelastic shells, a uniqueness theorem is obtained without the use of a definiteness assumption for the  ...  The Laplace trans- form is used and the general solution of the equation of motion corresponding to general dynamic loads is obtained.  ... 

Page 8 of Mathematical Reviews Vol. , Issue 2004i [page]

2004 Mathematical Reviews  
Gaiko], Limit cycle bifurcations in polynomial models of dynamical sys- tems (525-534); Katrin Gelfert, Lower estimates of the topological entropy for dynamical systems on Riemannian manifolds (535 543  ...  Vasil’ev], Obtaining the self-similar asymptotics of solutions to the Navier-Stokes equations by power geometry (93-101); Shuji Watanabe, A generalized Fourier trans- form (103-113); Anatoly A.  ... 

Data-driven modeling and control of large-scale dynamical systems in the Loewner framework [article]

Ion Victor Gosea, Charles Poussot-Vassal, Athanasios C. Antoulas
2021 arXiv   pre-print
This is a data-driven approach, applicable to large-scale systems, which was originally developed for applications to linear time-invariant systems.  ...  In recent years, this method has been extended to a number of additional more complex scenarios, including linear parametric or nonlinear dynamical systems.  ...  For recent tutorial papers on LF applied to linear dynamical systems, we refer the reader to [7, 33] .  ... 
arXiv:2108.11870v1 fatcat:o2anvfh4kncjravkvto2gxfw4y

Rational function approximations in the numerical solution of Cauchy-type singular integral equations

Michael A. Driscoll, R.P. Srivastav
1985 Computers and Mathematics with Applications  
Rational function approximations in the solution of the dommant equation results in a linear algebraic system which possesses blockdiagonal structure.  ...  Hence. approximations by rational functions IS feasible.  ...  Section 3 outlines the basic strategy of the rational function method for the dominant equation and brings out the desirable block-diagonal structure of the linear system of equations.  ... 
doi:10.1016/0898-1221(85)90099-9 fatcat:x43ayy3r7vapjg6qon6pq6z4iy

A modified particle method for semilinear hyperbolic systems with oscillatory solutions

R. C. Fetecau, T. Y. Hou
2004 Methods and Applications of Analysis  
We introduce a modified particle method for semi-linear hyperbolic systems with highly oscillatory solutions.  ...  We prove the convergence of the modified particle method essentially independent of the small scale for the variable coefficient Carleman model.  ...  Hou was supported in part by NSF under the NSF FRG grant DMS-0353838 and ITR Grant ACI-0204932.  ... 
doi:10.4310/maa.2004.v11.n4.a8 fatcat:wjzmd7f43rce7jserh3pcgwh6m

Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems

Zhaojun Bai
2002 Applied Numerical Mathematics  
The surge of interest was triggered by the pressing need for efficient numerical techniques for simulations of extremely large-scale dynamical systems arising from circuit simulation, structural dynamics  ...  In this paper, we begin with a tutorial of a Lanczos process based Krylov subspace technique for reduced-order modeling of linear dynamical systems, and then give an overview of the recent progress in  ...  Thanks to Louis Komzsik and Tom Kowalski for providing case studies from structural dynamics.  ... 
doi:10.1016/s0168-9274(02)00116-2 fatcat:kxudva7urvelnbogve5a27uywy
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