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