Sensitivity of the Duffing Equation
Explore the sensitivity of the Duffing equation to initial conditions.
To simplify the computations that follow, use ParametricNDSolveValue with only the initial conditions as parameters and other values fixed:
The derivatives of the ParametricFunction with respect to a and b are the sensitivities with respect to the initial values for x and its derivative x′, respectively:
The enormous sensitivities indicate that even a small change in either of these parameters will lead to a large deviation in the solution. Shown below are the solutions with a small perturbation from the origin in either direction:
Another way to visualize the sensitivity is to locally break it down into components parallel and perpendicular to the direction of the trajectory in the phase plane.