Abstract
We extend the planar Markus–Yamabe Jacobian conjecture to differential systems having Jacobian matrix with eigenvalues with negative or zero real parts.
Highlights
Let us denote by JF(X) the Jacobian matrix of F(X)
In [7] the question was raised, whether JF(X) having eigenvalues with negative real parts for every X ∈ IRn imply O to be globally asymptotically stable, i. e. whether all orbits in IRn tend asymptotically to O. Such a problem was named Markus–Yamabe Jacobian conjecture and several results were obtained under various additional hypotheses
A key step was made in [8], where it was proved that under Markus–Yamabe hypotheses, for planar systems the global asymptotic stability of O is equivalent to the injectivity of F(X)
Summary
If O is a critical point of (1.1) and the eigenvalues of JF(O) have negative real parts, O is asymptotically stable [2]. In [7] the question was raised, whether JF(X) having eigenvalues with negative real parts for every X ∈ IRn imply O to be globally asymptotically stable, i. This is likely due to the fact that accepting the possibility of eigenvalues with different real parts (positive, zero or negative) at different points of the plane does not allow to apply the procedure developed in [8] to establish the equivalence of injectiviy and global asymptotic stability.
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