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August 11th, 2018, 08:29 PM  #1 
Newbie Joined: Nov 2013 Posts: 19 Thanks: 0  Computational nonlinear system equations understanding.
Here is a snippet of some mathematical article: For a general, nonlinear dynamic system with parameters $\boldsymbol c$, running from time $0$ to time $T$ $\boldsymbol z\left(t+1\right)\boldsymbol=\boldsymbol s\left(\boldsymbol z\left(t\right),\boldsymbol u\left(t\right),\boldsymbol c\right)$ $\left(8\right)$ one may linearize a solution trajectory using the Jacobian of $\boldsymbol s$, $S$: $\boldsymbol x\left(t+1\right)\boldsymbol=\boldsymbol S\left(t\right)\boldsymbol x\left(t\right)+\boldsymbol k\left(t\right)$ $(9)$ The derivative of $\boldsymbol z\left(T\right)$ with respect to $z\left(0\right)$ are implicit in: $\boldsymbol x\left(T\right)=\boldsymbol S\left(T1\right)\boldsymbol S\left(T2\right)\cdots\boldsymbol S\left(0\right)\boldsymbol x\left(0\right)+\boldsymbol k'\left(t\right)$ $(10)$ and a similar formula may be derived (summing over $t$) for derivatives with respect to $\boldsymbol c$. From this, one can easily verify the validity of the following recursion formulas to determine the derivatives of a target variable $z_i\left(T\right)$ with respect to all of $z_j\left(0\right)$ (to appear in $x_j^{'}\left(0\right)$ ) and with respect to all the components of $\boldsymbol c$ (to appear in $\boldsymbol w\left(0\right)$): $\boldsymbol x'\left(\boldsymbol T\right)=\boldsymbol e^{i^T}$ $(11a)$ $\boldsymbol x'\left(t\right)=\boldsymbol x'\left(t+1\right)\boldsymbol S\left(t\right)$ $(11b)$ $\boldsymbol w\left(T\right)=\mathbf0$ $(11c)$ $\boldsymbol w\left(t\right)=\boldsymbol w\left(t+1\right)+\boldsymbol x{'}\left(t\right)s_c^{'}\left(t\right)$, $(11d)$ where $s_c^{'}$ refers to the matrix of derivatives of $s_i$ with respect to $c_k$. My questions are: Is here $\boldsymbol x$ a solution of $\boldsymbol z$? What does $\boldsymbol e^{i^T}$ mean? Last edited by rubis; August 11th, 2018 at 08:41 PM. 
August 13th, 2018, 04:22 PM  #2  
Senior Member Joined: Sep 2016 From: USA Posts: 598 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
https://www.encyclopediaofmath.org/i...onal_equations Presumably the second part should read $e^{Tx}$ and refers to the matrix exponential of $x$. This is defined analagously to the scalar exponential. For a matrix, $x$, the definition is \[ \exp(x) = \sum_{k = 0}^\infty \frac{x^k}{k!} \] where $x^k$ in this case denotes powers of the matrix under the usual multiplication. For more details see: https://en.wikipedia.org/wiki/Matrix_exponential  

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