Set Oriented Numerical Methods for Dynamical Systems

Controlling the Unsteady Analogue of Saddle Stagnation Points

“…We exploit these ideas in a numerical scheme by adapting set-oriented methods for the approximation of invariant sets [12,13]. For this we fix a certain time slice t 0 ∈ R, a small flow time T > 0, and a neighborhood of the saddle stagnation point strictly containing the ε-neighborhood of interest.…”

“…By a valid initial condition we mean one whose trajectory satisfies the determinant and boundedness criteria in both forwards (i.e., on [t 0 , t 0 + T ]) and backwards ([t 0 − T, t 0 ]) time. For the selection step (ii) appropriate test point strategies are employed, which could also be made rigorous following the ideas of Dellnitz and Junge [13]. We alternate steps (i) and (ii) while increasing T until we have obtained a tight bound on the initial conditions that give rise to hyperbolic trajectories.…”

On loops in the hyperbolic locus of the complex Hénon map and their monodromies

“…We begin with finding periodic points numerically. Since periodic points are of saddle type and hence are numerically unstable, we apply the subdivision algorithm [11] to find them. For each periodic orbit found numerically, we then construct a cubical index pair [17].…”

Recent development in rigorous computational methods in dynamical systems

“…Now we describe our algorithm to prove quasi-hyperbolicity. The algorithm involves the subdivision algorithm [16], that is, if it fails to prove quasi-hyperbolicity, then it subdivides cubes in K and L to have a better approximation of the invariant set, and repeats the whole step until it succeeds with the proof. (1) Find K such that R(f λ ) ⊂ int K holds for all λ ∈ L, and let N : This theorem implies that if L contains a non-quasi-hyperbolic parameter value, then Algorithm 5.7 never stops.…”