## Introduction to set theory by Monk J.D.

By Monk J.D.

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The matrix M (T0 ) is called a monodromy matrix of the cycle L0 . The following Liouville formula expresses the determinant of the monodromy matrix in terms of the matrix A(t): det M (T0 ) = exp T0 0 tr A(t) dt . 6 The monodromy matrix M (T0 ) has eigenvalues 1, µ1 , µ2 , . . , µn−1 , where µi are the multipliers of the Poincar´e map associated with the cycle L0 . 30 1. 14) near the cycle L0 . Consider the map ϕT0 : Rn → Rn . Clearly, ϕT0 x0 = x0 , where x0 is an initial point on the cycle, which we assume to be located at the origin, x0 = 0.

A) Lyapunov versus (b) asymptotic stability. If x0 is a ﬁxed point of a ﬁnite-dimensional, smooth, discrete-time dynamical system, then suﬃcient conditions for its stability can be formulated in terms of the Jacobian matrix evaluated at x0 . 2 Consider a discrete-time dynamical system x → f (x), x ∈ Rn , where f is a smooth map. Suppose it has a ﬁxed point x0 , namely f (x0 ) = x0 , and denote by A the Jacobian matrix of f (x) evaluated at x0 , A = fx (x0 ). Then the ﬁxed point is stable if all eigenvalues µ1 , µ2 , .

Most points leave the square S under iteration of f or f −1 . Forget about such points, and instead consider a set composed of all points in the plane V 12 1. 9. Location of the invariant set. that remain in the square S under all iterations of f and f −1 : Λ = {x ∈ S : f k (x) ∈ S, for all k ∈ Z}. Clearly, if the set Λ is nonempty, it is an invariant set of the discrete-time dynamical system deﬁned by f . This set can be alternatively presented as an inﬁnite intersection, Λ = · · · ∩ f −k (S) ∩ · · · ∩ f −2 (S) ∩ f −1 (S) ∩ S ∩ f (S) ∩ f 2 (S) ∩ · · · f k (S) ∩ · · · (any point x ∈ Λ must belong to each of the involved sets).