What are the *necessary* conditions to deterministic chaos?

What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos?

We collected yet:

1. A positive feedback loop
2. Non-linearity
3. Minimum of three instable eigenmodes
4. ...???
• Related question by OP: physics.stackexchange.com/q/68226/2451 Commented Jun 19, 2013 at 13:26
• Hi al-Hwarizmi, if you are not satisfied by the answers on your earlier question, it would be better to leave a comment there, asking for clarification or expansion. Commented Jun 23, 2013 at 20:34
• @Qmechanic: I think it is more of a duplicate than just "related", actually. Commented Jun 24, 2013 at 3:26

The dimension should be 3 or larger. If the dimension is smaller then 3 the existence and uniqueness theorem for differential equations will tell you that functions can't intersect (since you want them to be continuous and differentiable).

In 1 dimension this means you can only have movement in one direction

In 2 dimensions this means that your value either goes to infinity or to a perticular point.

You need 3 dimensions or more to get these strange attractors and that weird chaotic behaviour.

• This is true only for continuous-time dynamical systems. In discrete time, even one-dimensional systems (such as the logistic map) can behave chaotically. Also, even in continuous time, two-dimensional systems can still exhibit (non-chaotic) limit cycles in addition to divergence and point attractors. Commented Jun 23, 2013 at 22:10
• Yes of course, the theorem I used to give the condition was also one for differential equations of continuous variables. My bad :(.
– Nick
Commented Jun 23, 2013 at 23:28
• how would time dilatation effect? and how the degree of memory? Commented Jun 24, 2013 at 9:37
• I think if you're thinking about relativity the best way to parametrize is to use the proper time since it's invariant for all observers. And considering memory-effects, I believe they are more statistical which breaks down your "deterministic" in deterministic chaos.
– Nick
Commented Jun 24, 2013 at 10:26
• Also, the chaotic manifold doesn't need to be an attractor, it can aslo be a chaotic saddle (leading to transient chaos). Commented Jul 9, 2013 at 15:04

According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are:

1. Deterministic system (only one future for each state)

2. Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature)

3. A positive maximum Lyapunov exponent.

3) is pretty much the quantitative standard in journals of chaos, assuming you meet the conditions of 1). 2) is subjective and there's things like "stable chaos" and there can be periodic behavior that appears irregular but just has a really long period before it repeats itself, so you have to be careful with 2).