On the numerical solution of chaotic dynamical systems using extend precision floating point arithmetic and very high order numerical methods

Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step independent solutions have been impossible to compute, no matter how accurate the numerical method. Taking the well-known Lorenz equations as an example, we examine the numerical solution of chaotic dynamical systems using very high order methods as well as extended precision floating point number systems. Time step independent solutions are obtained over a finite period of time. However even with a sixteenth order numerical method and with quad-double floating point numbers, there is a limit to this approach.