Parameter sensitivity analysis for nonlinear system

[Rahul Shenoy]

I am working on a problem, where I have arrived at the following nonlinear state space equation:

dx₁/dt = x₂;
dx₂/dt = c11 x₁ + c21x₂ + c31x₃ + c41x₄ + c51x₁x₂²;
dx₃/dt = x₄;
dx₄/dt = c12 x₁ + c22x₂ + c32x₃ + c42x₄ + c52x₁x₂²;

c11, c21, c31, c41, c51, c12, c22, c32, c42, c52 are all a function of a system parameter. I want to compute sensitivity of the system subject to parameter changes. The system has a stable equilibrium at (0,0,0,0) and I want to understand the affect of parameter changes on the stability. Can anyone suggest how I can approach this problem, as the system is nonlinear?

 

[pasmith]

I am working on a problem, where I have arrived at the following nonlinear state space equation:

dx₁/dt = x₂;
dx₂/dt = c11 x₁ + c21x₂ + c31x₃ + c41x₄ + c51x₁x₂²;
dx₃/dt = x₄;
dx₄/dt = c12 x₁ + c22x₂ + c32x₃ + c42x₄ + c52x₁x₂²;

c11, c21, c31, c41, c51, c12, c22, c32, c42, c52 are all a function of a system parameter. I want to compute sensitivity of the system subject to parameter changes. The system has a stable equilibrium at (0,0,0,0) and I want to understand the affect of parameter changes on the stability. Can anyone suggest how I can approach this problem, as the system is nonlinear?

The condition for existence of a fixed point is that $x_2 = x_4 = 0$ and $$c_{11}x_1 + c_{31}x_3 = 0 \\ c_{12}x_1 + c_{32}x_3 = 0.$$ Define $\mu = c_{11}c_{32} - c_{31}c_{12}$. If $\mu \neq 0$ then the unique solution is $(x_1, x_3) = (0,0)$, but if $\mu = 0$ then there are infinitely many solutions for $(x_1, x_3)$ and thus a line of degenerate fixed points.

You will have to look at the eigenvalues of the linearization about the origin to determine the stability of the fixed point. I would expect stability to change as $\mu$ passes through zero, but it may be that varying the system parameter won't make $\mu$ pass through zero.

However we do find that $\mu$ is the product of the eigenvalues, which are the roots of a quartic equation with real coefficients. Thus if $\mu < 0$ then there exist an odd number of strictly negative real roots, which implies the existence of an odd number of strictly positive real roots, ie. the fixed point is unstable.

 

[epenguin]

Yes, that is called 'local stability analysis'. You make the linear approximation and it is the same standard thing as usual lde's. That is the first thing you have to do. After that it gets interesting.

For linear systems the conclusions about stability apply over the whole x_{1, 2, 3, 4} space.

But for nonlinear ones they may not when you are far enough away from your s.p. at origin. The behaviour there might be opposite.

Also your equations seem to allow more than one s.p. Whether you get that would depend on your parameter and the way the constants c depend on that.

I have a feeling that this equation might be analytically solvable. I think I have solved it for what happens at large x₂.

For the case μ = 0 mentioned by pasmith x₁ and x₃ are in lockstep, in a constant ratio, so you can reduce the number of equations for that case.

Quite a fun problem.