Symbolic dynamics in a binary asteroid system

Euler integral as a source of chaos in the three-body problem

“…Our interest in H C is indeed precisely related to the Euler integral ( 16): we aim to find zones in the phase space of H C where E affords slow variations and, simultaneously, chaos is detected. A similar point of view has been taken up, on the other hand, in the published papers [22,3,9].…”

“…For general information on chaotic phenomena, the reader may consult [6,18,7]. Closely related papers to the current one are the aforementioned [22,3,9]. Specifically, in [22] Conjecture 1.3 has been proved in the case (c), while in [3] it has been proved in the case (a), with r 1 and for motions very close to S 0 (r).…”

“…The full three-body (in general, many-body) problem inherits much of its reputation -especially in Hamiltonian mechanicsafter the breakthrough paper by V. I. Arnold [1] which will be recalled below. In fact, this paper is motivated by previous research [20,9,22,21], which here we briefly recall, in order to keep the paper self-contained.…”

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In this paper we address, from a purely numerical point of view, the question, raised in [20,21], and partly considered in [22,9,3], whether a certain function, referred to as "Euler Integral", is a quasi-integral along the trajectories of the three-body problem. Differently from our previous investigations, here we focus on the region of the "unperturbed separatrix", which turns to be complicated by a collision singularity.

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Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

“…Another problem is that even if some periodic orbits are calculated, it is still uncertain whether all the short periodic orbits are found. A key step for solving this problem is to build appropriate symbolic dynamics, which can encode all the possible orbits and their topological layout [33][34][35]. Symbolic dynamics are thoroughly investigated and usually used for 1D unimodal maps.…”