A Polynomial Counterexample to the Markus–Yamabe Conjecture

Stability in generic mitochondrial models

“…Since the Markus-Yamabe conjecture does not hold in dimensions greater than 2 [21], global stability does not follow automatically from local stability, and requires independent proof.…”

On the global asymptotic stability of difference equations satisfying a Markus-Yamabe condition

“…Conditions I and II are also known as Markus-Yamabe type conditions because they are similar to the condition σ(DF (x)) ⊂ {z ∈ C : Re(z) < 0} proposed for ordinary differential equations, see [3,11] and the references therein. In [4] it is proved that condition I implies GAS for planar polynomial maps and that there are planar rational maps satisfying it having other periodic points.…”

“…Nevertheless, assuming also these additional conditions it turns out that it is possible to obtain dynamical systems for which the origin is not GAS, see [2]. In [3,6] there are examples of polynomial maps defined in R n , n ≥ 3, satisfying the condition and having unbounded orbits. On the other hand, in [4] it is proved that, when F is polynomial, condition II implies GAS.…”

A Note on the Divergence-Free Jacobian Conjecture in R2

“…A recent result, which is the object of this paper, proves the global invertibility of jacobian maps of the form Φ(x, y) = (x + p(x, y), y + q(x, y)), p(x, y) and q(x, y) without terms of degree 1, under the additional assumptions that = 0. In higher dimensions, a striking result states that, in order to prove the n-dimensional Jacobian conjecture, it is sufficient to prove it for maps of the form Φ = L + C, L linear, C cubic, [1], or even for maps of the form Φ(X) = X + (AX) 3 , where A is a nilpotent matrix [4].…”

“…Thanks also to such a new approach, such a question was positively settled in dimension 2 in [5], [6], [7]. In higher dimensions it is known to be false [3], unless some additional hypotheses hold.…”