An essential concept of economics (in many schools of thought) is equilibrium. The theoretical advantages of equilibrium is well known.

Of course there are numerous examples of economic equilibrium. Actually each mainstream model has an equilibrium point. Let's assume the most famous:

Supply equals Demand (a+ bP= c- dP)

Let's forget for a second how supply and demand curves are extracted (forget the logical fallacies of production function and utility function);

1) Note that the above system is linear. As a result there's a unique equilibrium point in each system. If we search for any mainstream model we'll find that it is always linear.

Now, let's assume that our systems/markets are not linear. Thus, they may have many possible equilibria: eg

Supply equals demand in 2 points: A(P1,Q1) and B(P2,Q2)

Supply curve is the same curve with Demand thus, we have infinite equilibria points (a+bP=a+bP, a=0 or for each a)

A system/market (eg some financial markets) may be chaotic, so it may have various strange attractors (dx/dt= P(y-x), dy/dt= Rx-y-xz, dz/dt=xy-By)

2) Also, in times where economy seems to be stabilized someone may assume that economy is in equilibrium. Thus, he may use linear systems to find out the equilibrium points. But, in cases where economy is destabilized (eg right now) is it justified to use models which assume that economy is always in equilibrium? eg Paul Krugman, has revived the old yet very famous IS-LM model. He even used it in his blog to describe what's going on right now, arguing that it's a great model for zero lower bound. Also, all DSGE models are as E indicates, equilibrium models. Should economists use equilibrium models when economy is clearly in inequilibrium?

So, my question is,1) should we use non-linear models? Theoretically, it sounds logical but, does non linearity give better predictions and explanations? If so, how neoclassical equilibrium (static or dynamic) may be solved?

2) Of course models are not equal to reality and many of their assumptions are not valid but, is it correct to assume equilibrium when it clearly doesn't exist? Should we explain crises by using equilibrium models or should we focus on non-equilibrium models?