Review on contraction analysis and computation of contraction metrics

Abstract

<p style='text-indent:20px;'>Contraction analysis considers the distance between two adjacent trajectories. If this distance is contracting, then trajectories have the same long-term behavior. The main advantage of this analysis is that it is independent of the solutions under consideration. Using an appropriate metric, with respect to which the distance is contracting, one can show convergence to a unique equilibrium or, if attraction only occurs in certain directions, to a periodic orbit.</p><p style='text-indent:20px;'>Contraction analysis was originally considered for ordinary differential equations, but has been extended to discrete-time systems, control systems, delay equations and many other types of systems. Moreover, similar techniques can be applied for the estimation of the dimension of attractors and for the estimation of different notions of entropy (including topological entropy).</p><p style='text-indent:20px;'>This review attempts to link the references in both the mathematical and the engineering literature and, furthermore, point out the recent developments and algorithms in the computation of contraction metrics.</p>

Determination of limit cycles using stroboscopic set-valued maps

“…Along the lines of [3], we now synchronize the time between xi ptq and the corresponding solution ξ yi ptq of (1) with y i P Bpx i , δ i q X S i . More precisely, we consider the time-reparametrisation θ yi p¨q defined so that ξ yi pθ yi ptqq belongs to S i ptq.…”

“…This is possible due to the implicit function theorem if h and |y i ´xi | are sufficiently small (see [3], Section 2.1). Using ( 9), we define θ yi psq with θ yi p0q " y 0 , and y i`1 " ξ yi pθ i phqq.…”

“…The existence of a i`1 , b i`1 for all i P N follows from the assumptions that the 1st return times of x0 ptq and ξ y0 ptq to S 0 are finite (i.e., R1 P R ą0 and T y0 p1q P R ą0 for all y 0 P Bpx 0 , δ 0 q X S 0 ) together with conditions ( 8)-(18) (see Theorem 1). The bounds a i and b i can be computed using the implicit function theorem by time derivation of (9) (see [3], Section 2.1). The assumption that a i`1 ą 0 (i.e., inf sPr0,hs 9 θ yi psq ą 0) is true when h is sufficiently small and a lowerbound m ą 0 exists on the magnitude of vector field f (i.e., |f pxq| ą m for all x P Ω).…”

“…For a long time, methods based on the notion of (Jacobian) matrix measure (noted as µ P p¨q) have been used to show the convergence of a solution ξ x0 ptq to a stationary point (see e.g. [1], [2], [3]). These methods essentially assume that a bound c is known for the measure of the (transverse) Jacobian matrix Jpxq with respect to x for any x belonging to a Ω forward invariant space.…”