# Stability of a vector differential equation

1. Nov 29, 2009

### sokrates

I am trying to analyze the stability of a particular component of a non-linear differential equation system.

It is a dynamical equation and the criterion for instability of a particular axis will help me determine whether there's a drastic switching event in the solution.

My solution is a time dependent vector in the unit sphere, so there are two independent variables: theta, and phi (spherical coordinates). But the same equation typically is written in Cartesian coordinates, in terms of three (not linearly independent) vectors.

My problem is to determine the instability in -one- direction only - the other components (say dz/dt and dy/dt) are of no interest to me - but of course since this is a system of equations, y and z components appear in the dx/dt equation. What I'd like to know is whether there's a known method or procedure to go about doing this? I can make some assumptions based on physical knowledge of the system, but I don't know what's necessary and what's not. Averaging over one variable (say phi) leads to very trivial solutions which do not capture the essential physics. So I wanted to know the most general stability criterion, for a single component if there exists any.

My equation is this:

$$\frac{dmy}{dt}={\it hs}\, \left( {{\it mx}}^{2}+{{\it mz}}^{2} \right) -\alpha\,{\it hp}\,{\it my}\,{{\it mz}}^{2}+{\it mz}\,{\it mx}+{\it hp}\,{\it mx}\,{ \it mz}+\alpha\,{\it my}\,{{\it mx}}^{2}$$

where anything other than mx, my, mz are constants. mx, my, mz are normalized quantities (between 0-1)

Thanks for any help.

2. Nov 29, 2009

### gato_

The stability involves all three components. Include the remaining 2 equations

3. Nov 29, 2009

### sokrates

I will post them.

but I am pretty sure I have seen different analysis (of the same problem) where only one component is considered and a threshold is derived.

For instance, if one component's time average is zero (i.e it's precessing) it won't have any effect on stability.

4. Dec 1, 2009

### trambolin

You are looking for bifurcations I suppose. Why don't you just start finding the equilibrium points by adding the other two equations?

If you are trying the singular perturbation method, then you need to equate the other two equations to and solve everything with respect to "my" plug the solutions in your equation and check if the coefficient of "my" is negative or positive.