I am having trouble understanding the concept of chaos in relation to stability:
In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no equilibrium? I read about steady and unsteady equilibrium but did not understand which gives rise to chaos. Does chaos indicate that the system is not stable?
Do chaotic systems become unstable? Do instability and sensitive dependence to initial conditions give rise to chaos?
Why does the strange attractor not collapse, even though it means that the chaotic system is losing energy?
An attractor is an orbit that is stable in the sense that orbits in its vicinity converge to it. Alternatively, it’s an orbit of the system for $t→∞$ for an open set of initial conditions. Attractors can be fixed points, periodic orbits, and chaotic dynamics:
I strongly recommend that you first look at limit cycles and try to understand those.
Whenever you invert the time of a dynamics with an attractor, you get a repellor, in which the former attractor is still an orbit, but it is now unstable: All orbits in its vicinity diverge from it. There are also saddles, which are unstable and stay unstable when time is inverted. For all types of dynamics listed above, there are also unstable counterparts (repellors and saddles). An unstable fixed-point dynamics is an unstable equilibrium. There are also unstable limit cycles and tori, and chaotic saddles and repellors.
Attractors only exist in dissipative dynamical systems, i.e., systems with $\nabla · f<0$ (on average), where $f$ is the phase-space flow. If you are looking at physical systems, this means that energy is not conserved, but dissipates (hence the name). Usually, energy is also fed into these systems; otherwise they can only exhibit a fixed-point dynamics. The attractor is reached when the dissipating energy balances the incoming energy. A typical example is the driven and damped pendulum. Dissipative systems are distinct from conservative systems, where energy is preserved, and which do not feature attractors. However, you can have chaos in conservative systems, e.g., the ideal double pendulum is chaotic and conservative.
To explicitly answer your questions:
The chaotic attractor is no equilibrium. However, most chaotic dynamical systems feature an equilibrium outside the attractor, which usually is unstable.
Chaos and stability are orthogonal concepts. Chaotic attractors are stable per definition. There are also unstable chaotic structures and there is chaos in conservative systems, where the notion of stability does not make sense. Mind that this is stable in the sense of dynamical-systems theory; e.g., ecological stability often only refers to equilibria.
Because the system is also supplied with energy.