Not from this distance. But a similar, easily mathematiically described chaotic system (the Lorenz equation) is done with three integrators at
In general to get chaoticity you need nonlinear terms.
Integration generally smooths things out... in the most general case an RC integrator is also a low-pass filter.
Your description has words that are related to reality but without a schematic I don't think you'll get anything definite.
Tim.