My Math Forum Computational nonlinear system equations understanding.
 User Name Remember Me? Password

 Differential Equations Ordinary and Partial Differential Equations Math Forum

 August 11th, 2018, 08:29 PM #1 Newbie   Joined: Nov 2013 Posts: 17 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: $\boldsymbol x\left(T\right)=\boldsymbol S\left(T-1\right)\boldsymbol S\left(T-2\right)\cdots\boldsymbol S\left(0\right)\boldsymbol x\left(0\right)+\boldsymbol k'\left(t\right)$ $(10)$ 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: 471
Thanks: 262

Math Focus: Dynamical systems, analytic function theory, numerics
Quote:
 Originally Posted by rubis My questions are: Is here $\boldsymbol x$ a solution of $\boldsymbol z$? What does $\boldsymbol e^{i^T}$ mean?
No $x$ is a solution to the linearized dynamical system. This the matrix difference equation induced by the Jacobian. It is the discrete time version of the first variational equation which you have problem seen before. For more details see:

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

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post natt010 Real Analysis 2 November 17th, 2014 07:18 AM Degnemose Differential Equations 0 March 20th, 2012 11:53 AM brad66 Differential Equations 0 February 2nd, 2012 06:48 PM komaiteseba Math Software 0 January 29th, 2011 08:36 PM EoA Computer Science 5 April 19th, 2008 09:54 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top

Copyright © 2018 My Math Forum. All rights reserved.