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I have a set of nonlinear differential equations with parameters.

$$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$

where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in \mathbb{R}$) and $F$ is a continuous real function with enough derivatives (these are just multivariate polynoms).

$\vec{x}(0)$ is known and solution $\vec{x}(t)$ converges very fast to $\vec{x}(\infty)$ (I use Runge-Kuta method for numerical evaluation)

I need to evaluate also $\frac{\partial{x_i(\infty)}}{\partial{\beta_j}}$.

Could someone please show me, how to expand the system of equations to obtain these derivatives?

V-X
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1 Answers1

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If it converges at $t=\infty$ it means that the system is stable such as $$\dot{\vec{x}} = F(\vec{x},\vec{\beta})=0 \ at\ t=\infty $$ For illustration I write down explicit form $$F_k(x_1,..,x_p,\beta_1,...,\beta_q)=0\ for \ k=1...p$$

If you want to see the effect of $\beta_j$ on $x_i$ then assume that other variables are fixed and take the total derivative $$d\bigg(F_k(x_1,..,x_p,\beta_1,...,\beta_q)\bigg)=\frac{\partial F_k}{\partial x_i}dx_i+\frac{\partial F_k}{\partial \beta_j}d\beta_j=0\ for \ k=1...p$$ and solve the system.

AnilB
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