Examples of ODEs with complex constant coefficients and applications to physics?

Question

This question is asked on stackexchange: Are there examples for ODEs with complex coefficients with applications in physics? but received no answers. I am reposting it here on the hope that it catches the eye of some expert.

Question:

I have stumbled upon this paper Jung and Roh - The linear differential equations with complex constant coefficients and Schrödinger equations.

It made me wonder:

Are there examples for ordinary differential equations with complex coefficients that have applications in physics? A reference or link would be very helpful.

Answer 1

A standard/classic example is a model for tippe top inversion. This model is a linear ODE with constant complex coefficients: $$ \ddot{z} + i \alpha \dot{z} + \beta \dot{z} + i \gamma z + \delta z = 0 $$ where $\alpha, \beta, \gamma, \delta$ are real constants. The complex $i \alpha \dot{z}$ term arises from Coriolis effects while the complex $i \gamma z$ term arises from damping in rotational variables.

Remarkably, these equations describe both (i) the stability/instability of the inverted/noninverted states of the tippe top; and (ii) the existence of a heteroclinic connection between these states —- as illustrated below.

To read more about this connection, see Section 2 of Dissipation-Induced Heteroclinic Orbits in Tippe Tops.

For an animation click here.