Linearizability Problem Research Articles

Invariants of Linear Control Systems with Analytic Matrices and the Linearizability Problem

The paper continues the authors’ study of the linearizability problem for nonlinear control systems. In the recent work (Sklyar, Syst Control Lett 134:104572, 2019), conditions on mappability of a nonlinear control system to a preassigned linear system with analytic matrices were obtained. In the present paper, we solve more general problem on linearizability conditions without indicating a target linear system. To this end, we give a description of invariants for linear nonautonomous single-input controllable systems with analytic matrices, which allow classifying such systems up to transformations of coordinates. This study leads to one problem from the theory of linear ordinary differential equations with meromorphic coefficients. As a result, we obtain a criterion for mappability of nonlinear control systems to linear control systems with analytic matrices.

On mappability of control systems to linear systems with analytic matrices

The paper deals with the problem of analytic linearizability for nonlinear non-autonomous control systems, i.e., the problem of mapping of a nonlinear non-autonomous control system to a linear system with analytic matrices by a transformation of coordinates. We introduce a concept of a driftless form of a non-autonomous system, which allows simplifying our analysis. We give linearizability conditions and find invariants for nonlinear systems that are mappable to a preassigned linear system.

Linearizable boundary value problems for the nonlinear Schrödinger equation in laboratory coordinates

Boundary value problems for the nonlinear Schrödinger equation on the half line in laboratory coordinates are considered. A class of boundary conditions that lead to linearizable problems is identified by introducing appropriate extensions to initial-value problems on the infinite line, either explicitly or by constructing a suitable Bäcklund transformation. Various soliton solutions are explicitly constructed and studied.

HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

Linearizability problem of persistent centers

Linearizability problem of persistent centers

Linearizability and critical period bifurcations of a generalized Riccati system

In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

Isochronicity and linearizability of a planar cubic system

In this paper we investigate the problem of linearizability for a family of cubic complex planar systems of ordinary differential equations. We give a classification of linearizable systems in the family obtaining conditions for linearizability in terms of parameters. We also discuss coexistence of isochronous centers in the systems.

Integrability and Linearizability of Three Dimensional Vector Fields

This paper deals with the integrability and linearizability problem of three dimensional systems $$\begin{aligned} \dot{x}&= x(1 +ax+by+cz),\\ \dot{y}&= -y+ dx^2 +exy + fxz+gyz+hy^2+kz^2,\\ \dot{z}&= z(1 + \ell x + my +pz). \end{aligned}$$ More precisely, we give a complete set of necessary conditions for integrability and linearizability and then prove their sufficiency using Darboux method and Darboux inverse Jacobi multiplier, power series argument and a solution of a Riccati equation.

Linearizability of homogeneous quartic polynomial systems with 1:−2 resonance

In this paper, we consider the linearizability problem of complex planar polynomial systems of the form ẋ=x+P4, ẏ=−2y−Q4, where P4 and Q4 are homogeneous quartic polynomials. We obtain the necessary and sufficient conditions for linearizing the systems.

Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

Linearizability Problem of Resonant Degenerate Singular Point for Polynomial Differential Systems

The linearizability (or isochronicity) problem is one of the open problems for polynomial differential systems which is far to be solved in general. A progressive way to find necessary conditions for linearizability is to compute period constants. In this paper, we are interested in the linearizability problem of p : −q resonant degenerate singular point for polynomial differential systems. Firstly, we transform degenerate singular point into the origin via a homeomorphism. Moreover, we establish a new recursive algorithm to compute the so‐called generalized period constants for the origin of the transformed system. Finally, to illustrate the effectiveness of our algorithm, we discuss the linearizability problems of 1 : −1 resonant degenerate singular point for a septic system. We stress that similar results are hardly seen in published literatures up till now. Our work is completely new and extends existing ones.

Linearizability conditions of quasi-cubic systems

In this paper we study the linearizability problem of the two-dimensional complex quasi-cubic system ˙

Linearization of two-dimensional systems of ODEs without conditions on small denominators

We study the problem of linearizability for two-dimensional systems of ODEs in a neighborhood of the saddle type singular point with rationally incommensurable eigenvalues. It is shown that if the linearizing transformation is convergent in one of the variables, then it is absolutely convergent.

An approach to solving systems of polynomials via modular arithmetics with applications

The objective of this paper is twofold. First, we describe a method to solve large systems of polynomial equations using modular arithmetics. Then, we apply the approach to the study of the problem of linearizability for a quadratic system of ordinary differential equations.

Linearizability conditions of a time-reversible quartic-like system

In this paper we study the linearizability problem of polynomial-like complex differential systems. We give a reduction of linearizability problem of such non-polynomial systems to the problem of polynomial systems. Applying this reduction, we find some linearizability conditions for a time-reversible quartic-like complex system and derive from them conditions of isochronous center for the corresponding real system.

Linearizability conditions for Lotka–Volterra planar complex quartic systems having homogeneous nonlinearities

In this paper we investigate the linearizability problem for the two-dimensional Lotka–Volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials, i.e., we consider systems of the form x ̇ = x ( 1 − a 30 x 3 − a 21 x 2 y − a 12 x y 2 − a 03 y 3 ) , y ̇ = − y ( 1 − b 30 x 3 − b 21 x 2 y − b 12 x y 2 − b 03 y 3 ) . The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.

Feedback Linearizable Feedforward Systems: A Special Class

The problem of feedback linearizability of systems in feedforward form is addressed and an algorithm providing explicit coordinates change and feedback given. At each step, the algorithm replaces the involutive conditions of feedback linearization by some, easily checkable. We also reconsider type II subclass of linearizable strict feedforward systems introduced by Krstic and we show that it constitutes the only linearizable among the class of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">quasilinear</i> strict feedforward systems. Our results allow an easy computation of the linearizing coordinates and thus provide a stabilizing feedback controller for the original system among others. We illustrate by few examples including the VTOL.

Linearizability conditions for Lotka–Volterra planar complex cubic systems

In this paper, we investigate the linearizability problem for the two-dimensional planar complex system . The necessary and sufficient conditions for the linearizability of this system are found. From them the conditions for isochronicity of the corresponding real system can be derived.

Linearizability conditions of time-reversible quartic systems having homogeneous nonlinearities

In this paper we investigate the linearizability problem for the two-dimensional time-reversible complex system x ̇ = x + P ( x , y ) , y ̇ = − y + Q ( x , y ) , where P and Q are homogeneous quartic polynomials. The necessary and sufficient conditions for the linearizability of this system are found. As a corollary, the necessary and sufficient conditions for the origin to be an isochronous center of the planar time-reversible real system u ̇ = − v + F ( u , v ) , v ̇ = u + G ( u , v ) , where F and G are homogeneous quartic polynomials, are obtained.

Linearizable 3-webs and the Gronwall conjecture

In the article On the linearizability of (Nonlinear analysis 47, (2001) pp.2643-2654), published in 2001, we studied the linearizability problem for 3-webs on a 2-dimensional manifold. Four years after the publication of our article, V.V.Goldberg and V.V.Lychagin in the paper On linearization of planar three-webs and Blaschke's conjecture (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)) obtained similar results by a different method and criticized our article by qualifying the proofs incomplete. However, they obtained false result on the linearizability of a certain web. We present here the complete version of our work with computations and explicit formulas, because we deem that their opinion concerning our work is unjustified.