Let $\rho : \mathbb{R} \times \mathbb{R}^{2n} \longrightarrow \mathbb{R}^{2n}$ be a hamiltonian flow, i.e. a 1-parameter family of symplectomorphisms with respect to $t$ obtained from integrating a hamiltonian vector field $X_H$ for which
$$ \imath_{X_H} \omega = dH ,$$
$\omega$ being the canonical 2-form over $\mathbb{R}^{2n}$ and $H \in C^\infty(\mathbb{R}^{2n})$. There are several definitions of chaos using measure theory, ergodicity, mixing, etc, but I would like a definition that is applicable to the orbit, not the system. This is usually done somewhat formally through Lyapunov exponents and affirmations such as "the classical orbits separate exponentially fast". This does not appear to be true in a chaotic sea bounded above and below by non-resonant KAM tori in a scenario of mixed phase space, since trajectories cannot separate for distances larger than the outer torus' diameter.
I was thinking about something like: If an orbit is regular there is an infinite number of $t$-values such that $\rho(t_1, x) = \rho(t_2, x) = \dots $, since the flow is periodic. It the orbit is chaotic, then there exists $\epsilon > 0$ such that $| \rho (t,x) - \rho(t',x)| > \epsilon$, so the orbit is not periodic. Since orbits that are not periodic are not symplectomorphic to a torus, the orbit is chaotic. Does this make any sense as a definition of a chaotic orbit?
I'm not sure I totally understand the question, but it sounds like you might be interested in symbolic dynamics where you divide up the phase space and show that there are orbits visting all states, which is one of the hallmarks of chaos in Hamiltonian systens. The horseshoe map possesses such dynamics and "looking for chaos" in Hamiltonian systems is often a euphemism for "looking for horseshoes."