An algorithm for computing a neighborhood includedin the attraction domain of an asymptotically stable point
Résumé
Consider a nonlinear dynamical system described by a differential equation _x ¼ f ðxÞ, where f : Rn ! Rn is a smooth vectorfield. The point x is an equilibrium point if f ðxÞ ¼ 0. To find the equilibrium points it suffices to solve n nonlinear equationswith n unknowns. This can be solved using elimination theory-based methods [18], or any local numerical algorithm [20]. Apoint x is asymptotically stable if for all neighborhoodMof x, there exists a neighborhood N of x such that all trajectoriesinitialized in N converge to x and remain inside M.From the theoretical point of view, the Hartman–Grobman theorem states that if f is sufficiently regular around a hyperbolicequilibrium state x then there exists a local homeomorphism between the solutions of the _x ¼ f ðxÞ and its linearization_x ¼ Df ðxÞðx xÞ. In other words, the qualitative behavior of the dynamical system f around x is the same that of Df ðxÞ.Therefore, the existence of N is usually provided by studying the eigenvalues of the Jacobian matrix of f at x. Interval basedmethods have already been used to study the stability of dynamical systems. In the case of linear system, a classical resultfrom control theory states that the origin (which is always an equilibrium state) is stable if and only if all roots of the characteristicpolynomial of f have a negative real part. Such a polynomial is said to be Hurwitz stable. In [16], Khraritonov gives anecessary and sufficient effective condition to the Hurwitz stability of a polynomial with interval coefficients. When f is linearwith unknown bounded coefficients (i.e. f can be represented by a matrix whose entries are intervals), the Khraritonov’scondition only offers a sufficient condition to check that the origin is stable. More recently, Wang and al [17] determine anecessary and sufficient effective condition to the Hurwitz stability of an interval matrix.The present paper deals with nonlinear dynamical system. Contrary to the linear case, the stability of an equilibrium stateis, most of the time, only local: the trajectories must be initialized sufficiently close to the equilibrium state x to converge tox. The set of initial states for which the trajectory converges to x is the attraction domain of x. The main contribution of thispaper is an algorithm which provides a neighborhood N of x included in the attraction domain of x.