📂Dynamics

Quasiperiodic Orbit

Definition

If a system is bounded and its orbits aren’t asymptotically periodic, and it does not exhibit sensitive dependence on initial conditions, then the orbit is said to be quasiperiodic¹. Alternatively, if the flow $\phi (t)$ of a dynamical system represented by ordinary differential equations is a quasiperiodic function over time $t$, then any orbit passing through any point of $\phi (t)$ can also be termed as a quasiperiodic orbit².

Explanation

Quasiperiodic orbits are concepts needed to describe orbits that are neither periodic nor chaotic.

Module

$$f (x) = x + q \pmod{1}$$ Consider a map $f$ on an interval $[0, 1]$, defined via modulus as above. If $q$ is an irrational number, then all orbits in this system are non-periodic regardless of initial conditions, and their Lyapunov exponent is simply calculated as $0$. Indeed, these orbits are not sensitive to initial conditions, and all are quasiperiodic.

Torus ³

$$\begin{align*} \dot{\theta}_{1} =& \omega_{1} \\ \dot{\theta}_{2} =& \omega_{2} \end{align*}$$ Consider differential equations on a torus $T$, defined as above. The state of this system continuously moves along the torus at a constant rate $\omega_{1}$ and $\omega_{2}$. If the ratio $\omega_{1} / \omega_{2}$ is a rational number, the system is guaranteed to be periodic; if irrational, it wraps endlessly around the torus without closing. While there is no periodicity when the ratio is irrational, there is also no sensitivity to initial conditions, thus making the system quasiperiodic.