Digital Representation of a Nonlinear ODE with Small Differences in Initial Conditions

The graphic on the left overlays the plots of the solutions of a nonlinear ODE with 50 slightly different initial conditions.

The graphic on the right shows the digits of in base 2 for each of those solutions at . (The line is shown in red on the left.) Each row represents one of the 50 solutions.

Thus this Demonstration shows the change in the digital representation of the solutions to that set of nonlinear ODEs at a point as the ODE changes from periodic to chaotic with increasing .

Given a nonlinear ODE of the form , where , this Demonstration evolves the equation in time, assuming initial to be fixed.

With the advent of chaos (obtained by increasing the forcing parameter ), the digital representation loses its coherence, indicating a divergence between the solutions proportional to its Lyapunov measure.