The motion of the inner planets within the Solar System exhibits chaotic behavior, which precludes deterministic predictions of their future positions beyond a few tens of millions of years. Nonetheless, numerical simulations reveal a remarkable stability in their orbits over timescales comparable to the age of the Solar System. The likelihood of Mercury's eccentricity exceeding 0.7, potentially resulting in catastrophic events such as close encounters or collisions with either Venus or the Sun, is estimated to be only about 1% over the next 5 billion years.
In this talk, I will discuss recent advancements that highlight the hierarchical structure of chaos in the inner Solar System. I will show how certain symmetries characterize the strongest resonant interactions responsible for orbital chaos. These symmetries are weakly broken by secondary resonances, leading to the existence of quasi-conserved quantities that relate to the longest timescales of the dynamics, as notably evidenced by a principal component analysis of orbital solutions. By providing long-term memory of the current orbital configuration, these quasi-integrals of motion effectively constrain the chaotic diffusion of the inner orbits, playing a crucial role in their statistical stability over the lifetime of the Solar System.
Biological and bio-inspired locomotion is usually described by recognizing periodic patterns, or gaits, in the movement of limbs or other body parts. But is the evolution of the system actually periodic? Or more properly, relative-periodic, since, presumably, each cycle will propel the animal (or robot) a little bit forward? The answer is often no, due, for instance, to inertia or elasticity. However, we might expect the behaviour to converge asymptotically to a relative-periodic one. In this talk we will introduce this issue considering, as a case study, a dynamic model of rectilinear crawling locomotion.
We study the existence of a global periodic attractor for the reduced dynamics of the model, corresponding to an asymptotically relative-periodic motion of the crawler. The main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. Additional conditions and a counterexample for the existence of a bounded solution (and therefore of the attractor) will be briefly discussed. We conclude surveying the issue for some related models.