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:

Show a plot of the solution starting at {0,0} and log plots of the sensitivities:

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.

Define a function that gives a vector perpendicular to the trajectory as a function of time:

Now define functions that give the magnitude of the components of the sensitivity with respect to the initial value of x in the parallel and perpendicular directions:

For this equation, the two components grow at comparable rates.

The component breakdown allows a nice visualization in the phase plane. Changing the scaling of the sensitivity makes it possible to see it for different time intervals: