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I want to model a dynamical system of the form

$\frac{\text{d}x}{\text{d}t} = f(x)+nx\delta(\pi(t-0.2)). $

The problem is that I have a point source which is reoccurring at fixed time steps (say at 0.2,1.2,2.2...). How can I handle this numerically? I have tried to figure out solutions and it seems that there are two "easy" approaches:

1) Consider a grid with a fixed step size and define your delta function as 0 on all grid points except the relevant points. However, the influence of the delta function depends strongly on the step size solving this numerically which is why I think this is not correct.

2) Use a Gaussian to represent the Delta-function. This could also work for adaptive step sizes. However, in this case the results depend on the variance of the Gaussian which should be small. Is this a better approach? If this approach works, I could generate an array consisting of the Gaussians at different time steps which is zero elsewhere and use this with normal ODE-solvers, right?

The third approach is a little bit nasty in the sense that it cannot be used with "normal" ODE solvers. It would be to evaluate f(x) until we approach the point source and take $y(+\epsilon)=e^ny(-\epsilon)$ (Numerical way to deal with Dirac delta.). This does not depend on the variance or step size. Should I go with this one? I could use normal ODE-solvers until the fixed time step and then just take the exponential. And do you have any references on that?

Thank you very much for your help!

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If you have times $a_0<a_1<..$ and coefficients $c_k$ and want to solve $$ \dot x(t)=f(x(t))+\sum_{k=0}^N c_kδ(t−a_k) $$ then you can consider the function $u(t)=x(t)-\sum_{k=0}^N c_kH(t−a_k)$ where $H$ is the Heaviside or unit jump/ramp function. Then you get that $$ \dot u(t)=\dot x(t)-\sum_{k=0}^N c_kδ(t−a_k)=f\left(u(t)+\sum_{k=0}^N c_kH(t−a_k)\right) $$ which merely has a discontinuous right side. Numerical solvers of the usual sort with adaptive step size control will not be very happy with that situation, but usually can deal with it in a satisfying manner.


In the modified problem $$ \dot x(t)=f(x(t))+x(t)\sum_{k=0}^N c_kδ(t−a_k) $$ you can again try to combine $\dot x(t)-x(t)\sum_{k=0}^N c_kδ(t−a_k)$ into a derivative of some substitution variable. This is achieved in this case with the help of an integrating factor, resulting in setting $x(t)=\exp\left(\sum_{k=0}^N c_kH(t−a_k)\right)u(t)$ where now $u$ satisfies the ODE $$ \dot u(t)=\exp\left(-\sum_{k=0}^N c_kH(t−a_k)\right)f\left(\exp\left(\sum_{k=0}^N c_kH(t−a_k)\right)u(t)\right). $$ This has again a piecewise continuous right side, so that the solution $u$ is continuous and piecewise continuously differentiable. Again numerical methods can find this solution, in the post-processing the multiplication with the jumpy exponential introduces the jumps for $x$.

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  • $\begingroup$ First of all: Thank you for the answer. I just realized that I oversimplified the problem for the question. The magnitude of the pulse is state dependent. I changed that in the original question. Is this approach affected by this? As far as I understand it, this would make the derivative in the last line more complex, right? $\endgroup$ – gumpel Feb 21 at 23:55

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