# Are all aperiodic systems chaotic?

So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions.

So are all aperiodic systems chaotic? Are there counter-examples?

• Depending upon what exactly you mean by 'aperiodic' there might be trivial examples e.g. a damped harmonic oscillator's motion is a decaying spiral in phase space which isn't strictly periodic. – jacob1729 Feb 19 '19 at 10:09
• @jacob1729 or it decays to a stable infinite period :-) – Carl Witthoft Feb 19 '19 at 15:59
• What about translation on the plane? – MPW Feb 19 '19 at 21:51

No.

A system might be, for instance, stochastic, random - which is certainly not an example of deterministic chaos, but is aperiodic.

You can also have quasiperiodic behavior, where the system comes close (but not exactly) to previous states, for which the simplest example is the circle map: $$x \mapsto a+x$$, where $$a$$ is irrational and $$x$$ is angle-like (e.g., $$x\mapsto x\in[0,1)$$). This dynamics is aperiodic, but not sensitive to initial conditions.

Also a damped harmonic oscillator doesn't come back to exactly the same state, since energy is being lost and, thus, is strictly not periodic (nor sensitive to initial conditions).

These deterministic aperiodic but non-chaotic behaviors are often called regular, as in the classic book Regular and Chaotic Dynamics by Lichtenberg and Lieberman.

A related question is Conditions for periodic motions in classical mechanics.

• Even a particle with constant non-null speed is moving aperiodically and non-chaotically. – stafusa Feb 19 '19 at 15:45
• I think another simple example would be a system with two independent oscillators which do not interact in any way, but where the ratio between the two periods is an irrational number. – supercat Feb 19 '19 at 22:14
• @supercat Yes, that would be quasiperiodic. Considering that the energies of both oscillators are independently conserved, I wonder if there is a correspondence to the circle map... – stafusa Feb 19 '19 at 22:21