The phase portrait of the Hamiltonian system associated to a Pinchuk map

Artés, Joan C.; Braun, Francisco; Llibre, Jaume

doi:10.1590/0001-3765201820170829

Cited by 5 publications

(2 citation statements)

References 13 publications

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“…Itikawa et al [26] give two new classes of polynomial maps satisfying the real Jacobian conjecture in R 2 . A new proof of Pinchuk map which is a non-injective can be found in [2]. Applying the Puiseux solutions of differential equation, Giné et al provide a new sufficient condition for the two-dimensional real Jacobian conjecture, see [22].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

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We focus on investigating the real Jacobian conjecture. This conjecture claims that if F = (ƒ1 , . . . , ƒn) : Rn → Rn is a polynomial map such that det DF ≠ 0, then F is a global injective. In Euclidean space Rn, the Hadamard’s theorem asserts that the polynomial map F with det DF ≠ 0 is a global injective if and only if ∥ F (x) ∥ approaches to infinite as ∥ x ∥ → ∞. This paper consists of two parts. The first part is to study the two-dimensional real Jacobian conjecture via the method of the qualitative theory of dynamical systems. We provide some necessary and sufficient conditions such that the two-dimensional real Jacobian conjecture holds. By Bendixson compactification, an induced polynomial differential system can be obtained from the Hamiltonian system associated to polynomial map F. We prove that the following statements are equivalent: (A) F is a global injective; (B) the origin of induced system is a center; (C) the origin of induced system is a monodromic singular point; (D) the origin of induced system has no hyperbolic sectors; (E) induced system has a Ck first integral with an isolated minimun at the origin and k ∈ N+∪{∞}. The above conditions (B)-(D) are local dynamical conditions. Moreover, applying the above results we present a very elementary dynamical proof of the two-dimensional Hadamard’s theorem. In the second part, we give an alternate proof of the Cima’s result on the n-dimensional real Jacobian conjecture by the n-dimensional Hadamard’s theorem 2020 Math Subject Classification: Primary 34C05. Secondary 34C08. Tertiary 14R15

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian

Zhao

2022

Preprint

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“…Itikawa et al [21] give two new classes of polynomial maps satisfying the real Jacobian conjecture in R 2 . A new proof of Pinchuk map which is a non-injective can be found in [2]. In these works, their proofs rely only on the qualitative theory of planar differential systems, following ideas inspired by Theorem 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The necessary and sufficient conditions for the real Jacobian conjecture

Tian,

Zhao

2020

Preprint

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This paper is devoted to investigate the two-dimensional real Jacobian conjecture. This conjecture claims that if F = (f, g) : R 2 → R 2 is a polynomial map such that det DF (x, y) is nowhere zero and F (0, 0) = (0, 0), then F is a global injective.Firstly, we provide some new necessary and sufficient conditions such that the real Jacobian conjecture holds. By Bendixson compactification, an induced polynomial differential system can be obtained from the Hamiltonian system associated to polynomial map F . We prove that the following statements are equivalent: (A) F is a global injective; (B) the origin of induced system is a center; (C) the origin of induced system is a monodromic singular point; (D) the origin of induced system has no hyperbolic sectors; (E) induced system has a C k first integral with an isolated minimun at the origin and k ∈ N + ∪ {∞}. The above conditions (B)-(D) are local dynamical conditions.Secondly, applying the above results we present a sufficient condition for the validity of the real Jacobian conjecture. By definition a criterion function, when its limit at the origin of induced system exists, then F is a global injective. This analytical condition improves the main result of Braun et al [J. Differential Equations 260 (2016) 5250-5258]. Moreover, we use this analytical condition to give a new proof of the known algebraic sufficient condition. In this work, our all proofs are based on qualitative theory of dynamical systems.

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On Necessary and Sufficient Conditions for the Real Jacobian Conjecture

Tian,

Zhao

2023

Qual. Theory Dyn. Syst.

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The phase portrait of the Hamiltonian system associated to a Pinchuk map

Cited by 5 publications

References 13 publications

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

The necessary and sufficient conditions for the real Jacobian conjecture

On Necessary and Sufficient Conditions for the Real Jacobian Conjecture

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