# Parameter sensitivity analysis for nonlinear system

• Rahul Shenoy
In summary, you are working on a problem where you have arrived at a nonlinear state space equation. You are trying to determine the stability of a fixed point. There are some complications with this equation that make it nonlinear, so you will have to look at the eigenvalues of the linearization to determine if the fixed point is stable.
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?

Rahul Shenoy said:
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.

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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 x1, 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.

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## 1. What is parameter sensitivity analysis for nonlinear systems?

Parameter sensitivity analysis for nonlinear systems is a method used in scientific research to identify how changes in certain input parameters affect the behavior and output of a nonlinear system. It helps to understand the relationships between the inputs and outputs of a complex system and how small changes in the input parameters can lead to significant changes in the system's behavior.

## 2. Why is parameter sensitivity analysis important?

Parameter sensitivity analysis is important because it allows scientists to identify the most influential parameters in a system, which can help in making informed decisions and improving the overall understanding of the system. It also helps to identify which parameters can be controlled or manipulated to achieve a desired outcome.

## 3. How is parameter sensitivity analysis performed?

Parameter sensitivity analysis can be performed using various methods, such as local sensitivity analysis, global sensitivity analysis, and variance-based sensitivity analysis. These methods involve varying the input parameters of a system and measuring the resulting changes in the output, allowing for the identification of the most sensitive parameters.

## 4. What are the limitations of parameter sensitivity analysis?

One limitation of parameter sensitivity analysis is that it assumes a linear relationship between the input parameters and the system's output. In reality, many systems exhibit nonlinear behavior, which can lead to inaccuracies in the analysis. Additionally, the results of sensitivity analysis may vary depending on the chosen method and the assumptions made in the analysis.

## 5. How can parameter sensitivity analysis be used in practical applications?

Parameter sensitivity analysis has a wide range of practical applications, including in engineering, environmental science, and economics. It can be used to optimize the performance of complex systems, identify critical parameters for effective control, and evaluate the robustness of a system to changes in input parameters. It can also aid in decision-making processes, such as in risk analysis and cost-benefit analysis.

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