A Computer-Assisted Proof of $\Sigma_3$-Chaos in the Forced Damped Pendulum Equation

Abstract

The present paper is devoted to studying Hubbard's pendulum equation x + 10 −1ẋ + sin(x) = cos(t). By rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is lifted from the level of experimentally observed facts to the level of a theorem completely proved. A distinguished family of solutions is shown to be chaotic in the sense that on consecutive time intervals (2kπ, 2(k+1)π) (k ∈ Z) individual members of the family can freely "choose" between the following possibilities: the pendulum either crosses the bottom position exactly once clockwise or does not cross the bottom position at all or crosses the bottom position exactly once counterclockwise. The proof follows the topological index/degree approach by Mischaikow, Mrozek, and Zgliczynski. The novelty is a definition of the transition graph for which the periodic orbit lemma, the key technical result of the approach aforementioned, turns out to be a consequence of Brouwer's fixed point theorem. The role of wholly automatic versus 'trial and error with human overhead' computer procedures in detecting chaos is also discussed.

Optimal circle covering problems and their applications

“…Using the branch-and-bound strategy to search for an optimal cover seemed promising again as this approach was studied in Cambini and Sodini (2008) and applied successfully in similar situations Bánhelyi et al (2008). We add only the sets which potentially contain optimal configurations based on Theorem 5.4.…”

Reduction Methods and Chaos for Quadratic Systems of Differential Equations

“…Even such classical system, as the Lorenz system [33], studied in a number of works, is not understood completely (where n = 3!). A computer assisted proof of chaos existence for the Lorenz system, see [34], and for the forced damped pendulum equation, see [35].…”

Tinkerbell Is Chaotic

“…Consider the function f : 4]) is the optimal enclosure of the image (it is depicted in Figure 2). The image enclosure can be pessimistic if other expressions of the function are used; e.g., x 2 1 + 2x 1 x 2 + x 2 2 , (x 2 1 − 2x 1 + 1) cos x 2 evaluated for the same box gives rise to ([−2, 4], [−1, 4]) (also depicted in Figure 2).…”

“…Corollary 3.8 also generalizes Theorem 1 of [41] to continuous maps that are not diffeomorphisms and to bi-infinite pseudo-orbits. 4 We say that a sequence (x i ) i∈Z hits a sequence of sets (E i ) i∈Z if and only if x i ∈ E i holds for all i ∈ Z.…”

“…It consists of computing with intervals of reals instead of reals, providing a framework for handling uncertainties and verified computations (see [27,1,28,17,20] for a survey). Among other applications, interval analysis was used to provide rigorous numerical proofs of mathematical statements (resolution of the 14th Smale problem [38], properties of manifolds [6,7], and properties of chaotic dynamical systems [3,18,4,39,33]).…”