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(9 citation statements)

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“…Theorem 0.4.2. [Effective stable manifold theorem [37]] Assume thatỹ ∈ R n is a hyperbolic equilibrium point of (0.1). Let φ (t, y 0 ) (or φ t (y 0 )) denote the solution of (0.1) with the initial value y 0 at t = 0.…”

confidence: 99%

“…Theorem 0.4.2. [Effective stable manifold theorem [37]] Assume thatỹ ∈ R n is a hyperbolic equilibrium point of (0.1). Let φ (t, y 0 ) (or φ t (y 0 )) denote the solution of (0.1) with the initial value y 0 at t = 0.…”

confidence: 99%

“…Thus if one wishes to construct an algorithm that computes some S and U of (0.1) at y 0 , a different method is needed. In [37] an analytic, rather than algebraic, approach to the eigenvalue problem is used to allow the computation of S and U without calling for eigenvectors. The analytic approach is based on function-theoretical treatment of resolvents (see, e.g., [84], [46], [47], and [70]).…”

confidence: 99%

“…For that reason, qualitative results have been obtained in dynamical systems theory and computable version of several of these results exist. For example, in [GZB12] a computable version of the stable manifold is given, while [GZD12] provides a computable version of the Hartman-Grobman theorem. It is also shown in [GZ15] that the domain of attraction of an hyperbolic equilibrium point x 0 (i.e.…”

confidence: 99%

“…For example, it is possible to decide whether or not some (rational) initial point will converge to the fixed point of x = Ax at the origin, when A is an n × n hyperbolic (i.e. the real parts of the eigenvalues of A are nonzero) matrix [20].…”

confidence: 99%

“…Are these complex shapes computable? We have analyzed in [20] the case of Smale's horseshoe (see e.g. [14]), which was the first example of an hyperbolic invariant set which is neither an equilibrium point nor a periodic orbit.…”

confidence: 99%