Failure of Kalman's conjecture

Counterexample

Kalman's conjecture concerns systems of the form $$\dot x = P x + q\, f(e), \qquad e = r^{T} x.$$ Here $x \in \R^n$, $P$ is a constant real $n \times n$ matrix, and $q, r \in \R^n$ are constant vectors. The function $f \colon \R \to \R$ is differentiable with $f(0) = 0$. Assume that $k_1 < f'(e) < k_2$ for all $e$, and that for every $k \in (k_1, k_2)$ the linear system $\dot x = (P + k\, q r^{T}) x$ is asymptotically stable. The conjecture claims that then the equilibrium $x = 0$ attracts every solution.

The claim is false in general. For $n \le 3$ it is true. But for some $n > 3$ there are $P$, $q$, $r$ and $f$ that satisfy all the hypotheses, while the system has a non-constant periodic orbit. A solution on that orbit does not tend to $0$, so $0$ is not a global attractor.