Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture

Abstract

The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were constructed in such a way that one locally stable limit cycle (hidden oscillation) co-exists with a locally stable equilibrium. In this paper we demonstrate a multistable configuration of three co-existing hidden oscillations (limit cycles) and a locally stable equilibrium in the phase space of the fourth-order system, which provides a new class of counterexamples to the Kalman conjecture.