# Is instability + sensitivity to initial conditions = Chaos?

Please correct me where wrong. I am having trouble finding answers to these specific questions.

(1) In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no equilibrium? I read about steady & unsteady equilibrium but did not understand which gives rise to chaos. Does chaos indicate that the system is not stable?

(2) Do chaotic systems become unstable? Is instability + sensitive dependence to initial conditions give rise to chaos?

(3) Why does the strange attractor not collapse even though it means that the chaotic system is losing energy?

(1) In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no equilibrium?

It depends on how you define equilibrium. When starting from a given initial state, the system might display some sort of transient before settling down, so in the vague sense of the system having settled down on some behavior, an attractor would be an equilibrium state - though it's more likely to be called steady state, especially in the case of a strange attractor, which you expect to find in a system with both loss and injection of energy (think of a forced, damped pendulum). If by "equilibrium" you mean a constant state, then obviously chaotic behavior won't qualify.

It's also not uncommon for a system to display multistability, i.e., several final (steady) states, so that it can end up in any of them, depending on its initial state. In such a system, you can have both chaotic attractors and fixed points (but not both on one single orbit of the system, unless you have some other element, such as noise), so, regardless of the definitions one adopts for "equilibrium", the statement "a strange attractor indicate that there is no equilibrium" is in general false.

(1 cont.) I read about steady & unsteady equilibrium but did not understand which gives rise to chaos. Does chaos indicate that the system is not stable?

There are many concepts of stability. If you refer to the stability of a fixed point, then a book lying the table is stable and a pencil standing on its tip is unstable (there's plenty of material about that around, you can start with Wikipedia), in this sense, the link that exists between chaos and unstable orbits is that a chaotic attractor is associated with an infinitude of unstable periodic orbits (search word: horseshoe).

(2) Do chaotic systems become unstable?

The answer to that is given above.

(2 cont.) Is instability + sensitive dependence to initial conditions give rise to chaos?

Unstable periodic orbits and sensitivity to initial conditions are concurrent to chaos. For how it arises I'd refer again to the horseshoe mechanism.

(3) Why does the strange attractor not collapse eventhough it means that the chaotic system is losing energy?

Because it's also being supplied with energy. On a attractor, the system has reached a steady state where both equilibrate.

• Thank you for oyer answer. Just to clarify if I understood you correctly, the strange attractor indicates that the chaotic system is in order which technically means that it is in a stable equilibrium state. A stable equilibrium is also known as steady state. Outside the region of the strange attractor, there may be many unstable points that give rise to periodic orbits. Am I correct?
– Sm1
Commented Mar 11, 2020 at 14:25
• @Sm1 Yes, the attractor is a steady state. Outside the attractor there is a region of the phase space which leads to it - in the sense that initial states in this region have the attractor as their final state: that's his basin of attraction. Outside this region the system displays some other behavior, such as divergence, or convergence to a periodic attractor or another chaotic one. Embedded in a strange attractor you have an infinitude of unstable periodic orbits. Commented Mar 11, 2020 at 14:39
• thank you for answering the comment. A last clarification--does the presence of a chaotic attractor/strange attractor indicate the normal behavior for the system in chaos?
– Sm1
Commented Mar 11, 2020 at 15:49
• @Sm1 Not necessarily. If by "normal behavior" you mean "most common", then the decisive quantity is the fraction of the phase space occupied by the attractor's basin of attraction - the larger this fraction is, the more likely this behavior will be for randomly chosen initial states. Commented Mar 11, 2020 at 15:53