Timeseries of Determinisic Dynamic Systems

Starting from Renaissance rationality has dominated in the science. W. G. Leibnitz and I. Newton based foundations to mathematical and physical doctrine of determinism—everything in the nature is defined by few deterministic laws, and thus can be explained (computed) automatically from initial conditions. However, the 20th century made quite serious amendments to the core idea of determinism. It turned out that an adequate mathematical model of the process and initial conditions are not sufficient for determining the state of the process in many cases. Small disturbances in initial conditions or changes in model will yield to totally different behaviour. The concept of such ill-posed problems was noted already in 1902 by J. Hadamard. The three body problem (solved by H. Poincaré and K. F. Sundman) gave a second warning: under some initial conditions orbits of three bodies under the gravitational forces can reach a state of chaos—all orbits are non-periodic and do not approach any limiting trajectory. The similar effects were noted for many physical phenomenas, but still there was a lack of mathematical insight why such mysterious fluctuations happen. In a certain sense, the the chaos theory got really popular after publications of Edward Lorenz, who discovered that a simple mathematical model for weather forecasting is unstable and found a simple explanation of the phenomenon. Nowadays it is well established fact that even simple deterministic nonlinear systems can have truly chaotic behaviour. Hence, if we assume that some phenomenon is governed by (simple) deterministic rules there are three possible behavioural patterns. The system can head to a catastrophe—trajectory in the phase space grows without limit. Recall that phase space is a vector space R that incorporates all parameters which determine the state of system. In case of three body problem, the phase space might consist of three dimensional coordinates of all bodies and their velocities. In case of catastrophe, one of the bodies departures from the others without ever returning. Systems can also evolve in a stable way—the trajectory in the phase space is either periodic or quasi-periodic (oscillates around stable trajectory). The third alternative is chaos—a trajectory in the phase space jumps seemingly