I got puzzled when I thought of two concepts that is 1) Chaotic Systems 2)Well posed and ill-posed systems.
Chaotic systems are those dynamical systems which have the sensitive dependence to the initial conditions if the initial conditions change slightly then the dynamics of the systems change drastically.
Well posed systems have unique solutions and every system possesses the solution and the solution behavior changes continuously with the initial conditions. and if the system is not well-posed then we say that the system is ill-posed.
How are the two systems different?
You probably are thinking of this, but the dynamics do not change; rather the ending state at some time in the future does. The behavior of the system looks different, but the dynamics are the same. Also, the solutions of chaotic systems can change continuously with initial conditions. Which by your definition, is not the case for ill-posed problems, hence you can have well-posed chaotic systems.