# Existence of divergent solutions to system of ODEs

• vega12
In summary, the conversation is about a research topic involving working with systems of ordinary differential equations (ODEs) to model ecological phenomena. The speaker is having trouble with a particular model that tends to diverge for certain initial conditions. They are seeking literature or techniques that can help determine when a system of ODEs will have divergent solutions. The speaker clarifies that by "diverge," they mean that the solution tends to infinity as time approaches infinity. They also mention that certain solutions, such as those that settle on a point or exhibit oscillatory behavior, would be considered non-divergent. The conversation also touches on the use of numerical integration and the stability of solutions.

#### vega12

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.

Last edited:
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.

vega12 said:
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.

vega12 said:
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.

## 1. What is the definition of a divergent solution to a system of ODEs?

A divergent solution to a system of ODEs is a solution that does not converge to a single value as the independent variable approaches a certain value or as the number of iterations increases. In other words, the solution does not approach a steady state and can become unbounded or oscillate indefinitely.

## 2. How can the existence of divergent solutions affect the accuracy of a numerical method for solving ODEs?

The existence of divergent solutions can greatly impact the accuracy of a numerical method for solving ODEs. If a numerical method does not account for the possibility of divergent solutions, it may produce inaccurate or even incorrect results. This is especially important to consider when using numerical methods for real-world applications, where small errors can have significant consequences.

## 3. What are some common causes of divergent solutions in a system of ODEs?

Divergent solutions in a system of ODEs can occur due to various reasons, such as incorrect initial conditions, errors in the formulation of the system, or instability in the numerical method used for solving the equations. Additionally, the presence of singularities or discontinuities in the system can also lead to divergent solutions.

## 4. Can divergent solutions be avoided in a system of ODEs?

In general, it is not always possible to avoid divergent solutions in a system of ODEs. However, certain techniques such as regularizing the equations or using alternative numerical methods that are more stable can help mitigate the occurrence of divergent solutions. It is also important to carefully choose initial conditions and ensure that the system is well-posed to reduce the likelihood of divergent solutions.

## 5. How can the existence of divergent solutions impact the interpretability of a system of ODEs?

The presence of divergent solutions can make it challenging to interpret the behavior of a system of ODEs. Since the solutions do not approach a steady state, it can be difficult to make predictions or conclusions about the dynamics of the system. This is why it is important to carefully analyze and consider the existence of divergent solutions when studying and interpreting a system of ODEs.