Stability of a vector differential equation

In summary: If it is negative, then the system is unstable in the -x-direction. If it is positive, then the system is stable in the -x-direction.
  • #1
sokrates
483
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:

[tex]
\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}
[/tex]

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

Thanks for any help.
 
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  • #2
The stability involves all three components. Include the remaining 2 equations
 
  • #3
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
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.
 

FAQ: Stability of a vector differential equation

1. What is a vector differential equation?

A vector differential equation is a mathematical equation that describes the relationship between a vector function and its derivatives. It is often used to model physical systems in physics, engineering, and other sciences.

2. How is stability defined in a vector differential equation?

In a vector differential equation, stability refers to the behavior of the system over time. A stable system will maintain its initial state or return to it after small perturbations, while an unstable system will diverge from its initial state.

3. What are the different types of stability in vector differential equations?

There are three main types of stability in vector differential equations: asymptotic stability, where the system returns to its initial state; exponential stability, where the system oscillates around its initial state; and unstable, where the system diverges from its initial state.

4. How is the stability of a vector differential equation determined?

The stability of a vector differential equation is determined by analyzing the eigenvalues of the system's Jacobian matrix. If all eigenvalues have negative real parts, the system is asymptotically stable. If all eigenvalues have negative real parts and at least one has a zero real part, the system is exponentially stable. If any eigenvalue has a positive real part, the system is unstable.

5. What are some real-world applications of vector differential equations and stability analysis?

Vector differential equations and stability analysis are used in a variety of fields, including physics, engineering, biology, and economics. They are used to model systems such as electrical circuits, chemical reactions, population dynamics, and economic growth. Stability analysis is also important in control theory, where it is used to design controllers that can stabilize unstable systems.

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