My first thought would be problems involving human beings - economics or epidemiology. For instance, a person flying from South-Africa to the US might cause a (singular) step input in the population with a latent TB infection. In economics, a sudden outbreak of mad cow disease might cause a singular response in the agricultural market.

Dear Tobias, thanks for your thoughs. I think that the effects you mention can be modeled indeed by discrete time impulses (dirac deltas) in the corresponding equation of population evolution. Certainly, this is a kind of singularity because the solution is discontinuous.

To be more precise, the singular terms I'm looking for are on the state variable, the simplest example is

$$

x'(t)=1/x(t),

$$

which has a singularity at 0.

I do not know example from biology. But, may that one can be interested for You?

Let us consider the 2D flow field induced by two point vortex interacting with non stationary (psi_ex=xy(1+e*sin(wt)) external deformation velocity field. The vortex pair motion can be periodic or aperiodic due to parametric resonance. So induced velocity field will be singular due to pint vortices, but positions of singularities will changed in time.

Sorry some misprint.

Dear Konstantin, vortex dynamics is actually a topic of the highest interest for me. I have some modest contributions on this line:

A. Boscaggin, P.J. Torres, Periodic motions of fluid particles induced by a prescribed vortex path in a circular domain, Physica D: Nonlinear Phenomena, Volume 261, 15 October 2013, Pages 81-84.

P.J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, D.J. Frantzeskakis and P.G. Kevrekidis, Vortex Interaction Dynamics in Trapped Bose-Einstein Condensates, Communications on Pure and Applied Analysis 10 (2011) 1589-1615.

There are a lot of variants, but I would be interested to have a look to the concrete model you are mentioning, if you are so kind to send me a concrete reference. Thanks!!

I agree that mathematical models with singularities play a very importante role in applied sciences.

From my experience, I have dealt with singular differential and integral equations arising in such differente fields as: 1)hydrodynamics (formation of bubbles and droplets); 2) heat conduction; 3)electromagnetism, 4) oxygen diffusion in cells.

However in all these fields I have worked only with stationary problems, so I cannot give any example with periodicity in time.

I mean the stream function psi=ln[(x-x_1(t))^2+(y-y_1(t))^2]+ln[(x-x_2(t))^2+(y-y_2(t))^2], where x_i(t),y_i(t) are the positions of point vortices. oh! No external field needed.

So wi have dx/dt=1/(x-x_1(t)+...

I see it is singular nonlinear term.

For parametric instability You can see

https://www.researchgate.net/publication/236627339_Parametric_resonance_with_a_point-vortex_pair_in_a_nonstationary_deformation_flow?ev=prf_pub

Article Parametric resonance with a point-vortex pair in a nonstatio...

Oh! For point vortices You can use two vortices with same sine of rotation.

I worked on ode with oscillatory singular perturbation.

see :http://hal.archives-ouvertes.fr/hal-00133404.

May be it can help.

I am not a mathematical biologist, but your mind may be stimulated by a wonderful science fiction novel entitled "Blood Music," by Greg Bear, which manages to merge some very interesting ideas from biology and information theory in the context of a singularity. It might provide some ideas.

Thank you Ivan, I like science fiction but I didn't know about this novel, I'll look for it

Differential equations with time-periodic coefficients and a singularity in force (a concentrated force) in vibration of structures are well known. One typical kind of examples are all moving-load problems. Please read the following paper if you are interested in moving-load dynamic problems:

https://www.researchgate.net/publication/251600302_Moving-load_dynamic_problems_A_tutorial_(with_a_brief_overview)?ev=prf_pub

The concentrated force on the right-hand sdie of the differential equations is represented by a Dirac delta function.

Article Moving-load dynamic problems: A tutorial (with a brief overview)

Dear Prof Torres,

I juts browse through your list of publications and found you published a paper on jumping particle on a stairecase profile. I publsihed an oscillator jumping a beam so that there is a dynamic interaction between the bouncing oscililator and the vibration beam. Agian the differential equation has time-dependent coeffiicents and a singualr force with impact. Please see:

https://www.researchgate.net/publication/256594167_Dynamics_of_an_elastic_beam_and_a_jumping_oscillator_moving_in_the_longitudinal_direction_of_the_beam?ev=prf_pub

Article Dynamics of an elastic beam and a jumping oscillator moving ...

I developed the so-called "transition principle" that allows one to rigorously treat hamiltonian system with discontinious hamiltonians. This principle describes any kind of collisions, impacts, etc, in mechanics, reflection - refraction phenomena in geometrical optics and, generally, propagation of solution singularities of some kind PDEs. Details can be found in my joint papers with F.Puglese and J.Cortes. Exact ref. is on the cite www.diffiety.ac.ru

Hi

I have solved the nonlinear ODEs with singularity at orgin by Jacobi-Gauaa collocation method. This methods to solve any singular problem at a specifc point.

Article A Jacobi–Gauss collocation method for solving nonlinear Lane...

A trivial example is obtained from the Euler equation with singularity at the origin after dividing by the coefficient of the highest order derivative: x^n y^(n)(x)+a_{n-1}x^{n-1} y^(n-1)(x)+...+a_0y(x)=0.

Dear prof. Ouyang, I agree that particle collisions can be seen as a moving singularity. I thank you for the reference to the moving-load dynamic problem, I didn't know it. Perhaps the search of periodic oscillations for structures under a periodically moving load is meaningful.

Orbital resonance seems to be nice example. I found an understandable document from the Mathematica Journal (public access) http://www.mathematica-journal.com/2012/01/mathematical-exploration-of-kirkwood-gaps/

In population genetics, Kimura developed models with singularities (Kimura, M., 1955 Stochastic processes and distribution of gene frequencies under natural selection, pp. 33–53 in Cold Spring Harbor Symposia on Quantitative Biology, Vol. 20. Cold Spring

Harbor Laboratory Press, Cold Spring Harbor, NY). But not many have been working on similar things since then.

Possibly work of Tassilo Küpper, Markus Kunze and co-workers on non-smooth dynamical systems nd non-smooth mechanics might be of interest.

Dear Prof Torres: Moving-load problems (arguably including moving material problems) lead to time-varying coefficients. Asimpler problem is Mathieu equation or the more complicated Hill equation. They admit periodic solutions and aperiodic solutions

Dear Prof Ouyang: for Hill's equation my favourite monograph is the book of Magnus and Winkler. My main expertise area is on the existence of periodic solution of nonlinear differential equations with periodic coeffcients by using topological degree and related methods.

Nonsmooth mechanics is an area where there are a lot of issues with discontinuities. These discontinuities arise naturally and their location depends on the state of the system, and are not specified a priori. The two main kinds of problems where these arise are Coulomb friction problems and impact problems.

I should add "bang-bang" optimal controls for optimal control theory.

i have a nice example of such an equation - a physical object i built once that has a degree of motion freedom that can be described by a first-order differential equation with singularities in the derivative coefficients - it makes a good example for these methods, and i am curious about how well these methods solve the problem

the object is built from 6 identical PVC pipe elbows, connected in a simple closed curve (see diagram) - the connection between each adjacent pair of elbow ends is free to rotate, but the combined object seems rigid - it is not - there is still one degree of freedom, which appears to be a simple closed curve in a 9-dimensional parameter space representing corner coordinates, but i have not actually proved that assertion

in the diagram, the circles represent the joints between the elbows, and it is easily seen that the distances between the joints are constant, as are the distances between each joint and its adjacent corners - if you build one (with rubber bands to hold the joints together and allow them to rotate relative to each other), a little bit of manipulation will discover that there is a degree of motion freedom, and the constraints can be used to define the differential equation of motion

Dear Christopher, thank you very much for your detailed explanation. This type of model is pretty new to me, still I don't understand how appears the singularity in the derivative. In fact, I must confess myself unable to write down the equation ruling such device. Is it done anywhere? I will be glad to have a look on the problem

the way i remember it (and this was quite some years ago, sometime in the later part of the previous millenium), there were coefficients of the first order derivatives that were sometimes zero - the derivation is as follows: if you take three of the vertices in 3-space as defining the 9-dimensional real coordinate system of the configuration, and compute the derivatives of all of the distance invariants, you get a bunch of first order differential equations in those 9 coordinates (this derivation was an interesting and informative exercise, as i remember, so i'm not going to spoil the fun just yet; besides, i will have to work it out again before i can comment on it much further)

let me start with a better picture (i have labeled the corners) - if we define the origin to be where corner E is, and two axes where the arms of the elbow at E point, then corners D and F also have a fixed location, say E = (0,0), D=(1,0), F=(0,1) - the rotations at the joint EF do not affect the elbow at E, so they cannot move F, so they must move the segment FA, among others, and the rotations at the joint DE also do not affect the elbow at E, so they cannot move D, so they must move the segment DC, among others

thus the coordinates of A, B, C provide the 9 dimensions, and the distances FA, AB, BC, and CD provide a set of invariants (and their derivatives are the equations i mentioned above); there are also invariants for the distances EA, FB, AC, BD, CE - i suspect that these five constraints are dependent on the first four, since there are 9 equations in all and by observation of my old construction, there is still a degree of freedom of movement

i looked for a long time for some good symmetry arguments, but never found anything i liked

Dear Mariusz, thank you for your comment. In my opinion, every model has its own range of validity, and it is important to know it. On this basis, every model can be critizised. That being said, it is clear that simplifications like for instance considering a body as a point particle are helpful in the understanding of the underlying dynamics.

Dear Pedro,

one possible option may be to consider diffusive processes where the diffusivity scales as $D(r) \propto r^\gamma$, with $\gamma < 0$, which give rise to sub-diffusive phenomena. I can provide references on biological applications in this regard.

Thanks Alfonso. I'm interested. Perhaps it is related to Endem-Fowler equations, isn't it? Anyway, i will be grateful if you can send me the references.

Singular nonlinearities come up in a natural fashion in modeling actuators for MicroElectroMechanicalSystems. From the nonlinear PDE point of view the developing of singularity, in the equation, plays the role of boundedness of solutions in nonlinear (smooth) eigenvalue problems where extremal solutions stay bounded provided the underlying euclidean dimension does not exceed a critical value: this phenomenon appears also in stationary as well as dynamic MEMS type problems.

I guess you can arrive at differential equations with singularities if you consider an optimal contrl problem for some dinamical system with an appropriate nonlinear state equation. In this case the differential equations with singularities can appear in an natural way as an adjoint system in optimality conditions.

To understand singularities you should start for math. It does not depend what differential equation you considere and what dependent and independent variable result in singularities. For example, if y=f(x) and f is squire root then the derivative is inverse squire root and therefore function f is singular at x=0 (the derivatives are infinite). Practical it means that you could not use the Taylor series. In such a case you should use a new independent variable as squire root of x and consider function f in this variable which does not have singularities in mew variable and therefore Taylor series in this variable is applicable

I agree Abram, sometimes you can "regularize" the equation by using a suitable change of variables. Famous examples are found in Celestial Mechanics, where for instance the classical Levi-Civita regularization transforms the planar Kepler problem into a 2D harmonic oscillator.

Hi Pedro,

have you considered as an example the Dixon system?

https://www.researchgate.net/post/Do_we_have_chaotic_behaviour_in_two_dimensional_continuous_dynamical_systems

Maybe it would be also interesting to analyse how a periodic modulation of its singularity, as the ones you are suggesting, translate into the low-dimensional, pseudo-chaotic dynamics of the system.

Dear Alfonso, thanks for the suggestion. It seems that the Dixon system models the dynamical behavior of the magnetic field of a neutron star. Perhaps it is not totally nonsense to consider periodic fluctuations of the coefficients like in a pulsating star or so.

Yes, I agree. However, the Dixon model is a 2D reduction of a three dimensional ODE model. I'm not sure whether the singularities or the chaotic behaviour will be present or not in the full system.

I'm sorry to push my own work here and doubly sorry if this isn't relevant. We considered models that we called the fire-diffuse-fire model (or sometimes the skid-row-relay) in which there were a bunch of delta function sources in space that turned on for a certain time during which they released a fixed amount of calcium. If that time went to zero then the sources were delta functions in both space and time. We showed that the system could give a continuum release wave or a "saltatory" release wave.

Don't know if this is useful to you but here are links to the papers:

http://www.pnas.org/content/96/11/6060.full

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1299735/

http://www.sciencedirect.com/science/article/pii/S0378437198001368

Since you are looking for physical examples, I thought the dynamics of systems with Coulomb friction can also be a good example of singular ODE problems. The dynamic equation of motion of a rigid-block under base excitation is described by an ODE having a "signum" function in it. The sign(.) function appears because the friction force is always acting in a direction opposite to the sliding velocity. The sign(.) function introduces singularity into the ODE of motion making the solution problematic because conventional ODE solvers cannot solve the singular problems. The methods that are usually taken to solve this problem are good examples of how singularities are treated in such situations. The singular forces are usually approximated by some continuous functions. For example, an elastoplastic model can be used to approximate sign(.) function. Following article is introduces some other continuous functions that can be used and the implementation procedure

http://onlinelibrary.wiley.com/doi/10.1002/(SICI)1096-9845(199705)26:5%3C541::AID-EQE660%3E3.0.CO;2-W/abstract

Hope it helps,