# Existence of divergent solutions to system of ODEs

My question is in regards to systems of ordinary differential equations. One of my research topics right now involves working with some complicated coupled ODEs used to model ecological stuff. Without getting into the details, the model I am working on now has a bad tendency to diverge for some initial conditions. I have thought and searched for a while now, but have been unable to come up with some good ways of determining when a system of ODEs will have certain solutions that diverge. Does anyone know of some literature or techniques that might be useful for me? I would very much appreciate any help you could provide. Thank you very much.

EDIT:

To avoid further misunderstanding, the type of equations I am considering are of the form:

$$\frac{d X_i}{dt} = F_i(\mathbf{X)}$$

When I say that a solution diverges, I mean that given a set of initial conditions, the solution will diverge in time. As for what I mean by diverge, I mean that solutions tend to infinity as t approaches infinity. So solutions that settle in on a point, or go through some oscillatory behavior, even chaotic oscillatory behavior, would be considered non-divergent.

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HallsofIvy
Homework Helper
Strictly speaking, a divergent series does not define a function. So I am not at all sure what you mean by a divergent series being a solution. It sounds to me like you are asking for conditions in which the "series method" of solution is unstable.

Divergent series are solutions as formal series, and sometimes they define functions, but in sectors, of course not in discs. See Borel summability and multisummablity.

My apologizes for not making my question as clear as it should have been. All the systems of equations I am working with are of the form:

$$\frac{d X_i}{dt} = F_i(\mathbf{X)}$$

So when I say that a solution diverges, I mean that given a set of initial conditions, the solution will diverge in time. As for what I mean by diverge, I mean that solutions tend to infinity as t approaches infinity. So solutions that settle in on a point, or go through some oscillatory behavior, even chaotic oscillatory behavior, would be considered non-divergent.

I hope this clarifies what I mean. I'll append this to the opening post.

My apologizes for not making my question as clear as it should have been. All the systems of equations I am working with are of the form:

$$\frac{d X_i}{dt} = F_i(\mathbf{X)}$$

So when I say that a solution diverges, I mean that given a set of initial conditions, the solution will diverge in time. As for what I mean by diverge, I mean that solutions tend to infinity as t approaches infinity. So solutions that settle in on a point, or go through some oscillatory behavior, even chaotic oscillatory behavior, would be considered non-divergent.

I hope this clarifies what I mean. I'll append this to the opening post.

Well that just sounds like non-linear dynamics in general and the solution diverges as you described when it finds itself outside an attractor's basin of attraction.

Chestermiller
Mentor
My apologizes for not making my question as clear as it should have been. All the systems of equations I am working with are of the form:

$$\frac{d X_i}{dt} = F_i(\mathbf{X)}$$

So when I say that a solution diverges, I mean that given a set of initial conditions, the solution will diverge in time. As for what I mean by diverge, I mean that solutions tend to infinity as t approaches infinity. So solutions that settle in on a point, or go through some oscillatory behavior, even chaotic oscillatory behavior, would be considered non-divergent.

I hope this clarifies what I mean. I'll append this to the opening post.

Are you solving the equations numerically, using some kind of numerical integration package? If so, and you are using an explicit integrator, you may want to switch to an implicit integrator.

You can get an idea of the stability of the solution by looking at the eigenvalues of the Jacobian matrix, evaluated at the initial conditions. If any of the eigenvalues have a positive real part, the solution will be unstable, irrespective of the numerical integration method.