- #1
sokrates
- 481
- 2
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:
<br /> \frac{dmy}{dt}={\it hs}\, \left( {{\it mx}}^{2}+{{\it mz}}^{2} \right) -\alpha\,{\it <br /> hp}\,{\it my}\,{{\it mz}}^{2}+{\it mz}\,{\it mx}+{\it hp}\,{\it mx}\,{<br /> \it mz}+\alpha\,{\it my}\,{{\it mx}}^{2}<br />
where anything other than mx, my, mz are constants. mx, my, mz are normalized quantities (between 0-1)
Thanks for any help.
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:
<br /> \frac{dmy}{dt}={\it hs}\, \left( {{\it mx}}^{2}+{{\it mz}}^{2} \right) -\alpha\,{\it <br /> hp}\,{\it my}\,{{\it mz}}^{2}+{\it mz}\,{\it mx}+{\it hp}\,{\it mx}\,{<br /> \it mz}+\alpha\,{\it my}\,{{\it mx}}^{2}<br />
where anything other than mx, my, mz are constants. mx, my, mz are normalized quantities (between 0-1)
Thanks for any help.