How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that basically underlies on such a linear equation to describe highly nonlinear chaotic properties of classical systems?
The commenters on your question already dealt with the nonlinearity aspect, but the question's aspect that sticks out the most is actually how can any classical system, chaotic or not, be described by quantum mechanics, when they seem to describe two different worlds?
The short answer is "no one really knows", but we do know something. This quantum-classical relationship is often equated with decoherence and you can read a bit about it here, here, here, and here, besides of course Wikipedia.