scholarly journals An extension to the planar Markus–Yamabe Jacobian conjecture

2021 ◽  
pp. 1-6
Author(s):  
Marco Sabatini
Keyword(s):  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Differential Systems

We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts.

scholarly journals Irreducibility Properties of Keller Maps

2016 ◽  
Vol 23 (04) ◽  
pp. 663-680 ◽  
Author(s):  
Michiel de Bondt ◽  
Dan Yan
Keyword(s):  
Linear Combination ◽  
Characteristic Zero ◽  
Jacobian Matrix ◽  
Jacobian Conjecture ◽  
Extra Condition ◽  
Irreducible Polynomials ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Keller Map

Jędrzejewicz showed that a polynomial map over a field of characteristic zero is invertible, if and only if the corresponding endomorphism maps irreducible polynomials to irreducible polynomials. Furthermore, he showed that a polynomial map over a field of characteristic zero is a Keller map, if and only if the corresponding endomorphism maps irreducible polynomials to square-free polynomials. We show that the latter endomorphism maps other square-free polynomials to square-free polynomials as well. In connection with the above classification of invertible polynomial maps and the Jacobian Conjecture, we study irreducibility properties of several types of Keller maps, to each of which the Jacobian Conjecture can be reduced. Herewith, we generalize the result of Bakalarski that the components of cubic homogeneous Keller maps with a symmetric Jacobian matrix (over ℂ and hence any field of characteristic zero) are irreducible. Furthermore, we show that the Jacobian Conjecture can even be reduced to any of these types with the extra condition that each affinely linear combination of the components of the polynomial map is irreducible. This is somewhat similar to reducing the planar Jacobian Conjecture to the so-called (planar) weak Jacobian Conjecture by Kaliman.

Codes for Approximating Lyapunov Exponents

2009 ◽  
Author(s):  
Luca Dieci ◽  
Michael S. Jolly ◽  
Erik S. Van Vleck
Keyword(s):  
Lyapunov Exponents ◽  
Jacobian Matrix ◽  
Alternative Strategy ◽  
Differential Systems ◽  
Small Systems ◽  
Large Systems ◽  
Linear Problems ◽  
Qr Methods ◽  
Nonlinear Differential Systems

We present a suite of codes for approximating Lyapunov exponents of nonlinear differential systems by so-called QR methods. The basic solvers perform integration of the trajectory and approximation of the Lyapunov exponents simultaneously. That is, they integrate for the trajectory at the same time, and with the same underlying schemes, as integration for the Lyapunov exponents is carried out. Separate codes solve small systems for which we can compute and store the Jacobian matrix, and for large systems for which the Jacobian matrix cannot be stored, and it may not even be explicitly known. If it is known, the user has the option to provide its action on a vector. An alternative strategy is also presented in which one may want to approximate the trajectory with a specialized solver, linearize around the computed trajectory, and then carry out the approximation of the Lyapunov exponents using codes for linear problems.

scholarly journals Numerical Techniques for Approximating Lyapunov Exponents and Their Implementation

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Luca Dieci ◽  
Michael S. Jolly ◽  
Erik S. Van Vleck
Keyword(s):  
Lyapunov Exponents ◽  
Jacobian Matrix ◽  
Alternative Strategy ◽  
Numerical Techniques ◽  
Differential Systems ◽  
Small Systems ◽  
Large Systems ◽  
Linear Problems ◽  
Computational Procedures ◽  
Qr Methods

The algorithms behind a toolbox for approximating Lyapunov exponents of nonlinear differential systems by QR methods are described. The basic solvers perform integration of the trajectory and approximation of the Lyapunov exponents simultaneously. That is, they integrate for the trajectory at the same time, and with the same underlying schemes, as is carried out for integration of the Lyapunov exponents. Separate computational procedures solve small systems for which the Jacobian matrix can be computed and stored, and for large systems for which the Jacobian cannot be stored, and may not even be explicitly known. If it is known, the user has the option to provide the action of the Jacobian on a vector. An alternative strategy is also presented in which one may want to approximate the trajectory with a specialized solver, linearize around the computed trajectory, and then carry out the approximation of the Lyapunov exponents using techniques for linear problems.

Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems with a Symmetric Jacobian Matrix

2018 ◽  
Vol 28 (03) ◽  
pp. 1830006 ◽  
Author(s):  
Marcelo Messias ◽  
Rafael Paulino Silva
Keyword(s):  
Jacobian Matrix ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Positive Answer ◽  
Algebraic Surfaces ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Criterion ◽  
Invariant Algebraic Surfaces

In this paper, we give an algebraic criterion to determine the nonchaotic behavior for polynomial differential systems defined in [Formula: see text] and, using this result, we give a partial positive answer for the conjecture about the nonchaotic dynamical behavior of quadratic three-dimensional differential systems having a symmetric Jacobian matrix. The algebraic criterion presented here is proved using some ideas from the Darboux theory of integrability, such as the existence of invariant algebraic surfaces and Darboux invariants, and is quite general, hence it can be used to study the nonchaotic behavior of other types of differential systems defined in [Formula: see text], including polynomial differential systems of any degree having (or not having) a symmetric Jacobian matrix.

DETERMINISTIC CHAOS SEEN IN TERMS OF FEEDBACK CIRCUITS: ANALYSIS, SYNTHESIS, "LABYRINTH CHAOS"

1999 ◽  
Vol 09 (10) ◽  
pp. 1889-1905 ◽  
Author(s):  
R. THOMAS
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Jacobian Matrix ◽  
Deterministic Chaos ◽  
Steady States ◽  
Nonlinear Dynamic Systems ◽  
Limit Case ◽  
Differential Systems ◽  
Feedback Circuits ◽  
Unstable Steady States

This paper aims to show how complex nonlinear dynamic systems can be classified, analyzed and synthesized in terms of feedback circuits. The Rössler equations for deterministic chaos are revisited and generalized in this perspective. It is shown that once a proper set of feedback circuits is present in the Jacobian matrix of the system, the chaotic character of trajectories is remarkably robust versus changes in the nature of the nonlinearities. "Labyrinth chaos", whereby simple differential systems generate large lattices of many unstable steady states embedded in a chaotic attractor, is constructed using this technique. In the limit case of a single three-element circuit without diagonal elements, one finds systems possessing an infinite lattice of unstable steady states between which trajectories percolate in a deterministic chaotic way.

LYAPUNOV VECTORS TREATMENT FOR BIFURCATIONS IN RETARDED DIFFERENTIAL SYSTEMS

1988 ◽  
Vol 49 (C2) ◽  
pp. C2-389-C2-392 ◽  
Author(s):  
M. LE BERRE ◽  
E. RESSAYRE ◽  
A. TALLET ◽  
J. J. ZONDY
Keyword(s):  
Differential Systems

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

Sign in / Sign up

Export Citation Format

Share Document