New necessary and sufficient conditions in order that the real Jacobian conjecture in $\mathbb{R}^2$ holds.

This paper is devoted to investigate the two-dimensional real Jacobian conjecture. This conjecture claims that if $F=\left(f,g\right):\mathbb{R}^2\rightarrow \mathbb{R}^2$ is a polynomial map such that $\det DF\left(x,y\right)$ is nowhere zero and $F\left(0,0\right)=\left(0,0\right)$, 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 $k\in\mathbb{N}^{+}\cup\{\infty\}$. 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 {\bf 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.