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July 12th, 2018, 02:45 PM  #1 
Newbie Joined: Nov 2013 Posts: 19 Thanks: 0  By exploring what knowledge of Maths can I understand this snippet of an article?
What field of Maths knowledge should I learn first i.e. by reading what books or articles can I understand the following snippet of a Maths 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 $\boldsymbol s$, $S$: $\boldsymbol x\left(t+1\right)\boldsymbol=\boldsymbol S\left(t\right)\boldsymbol x\left(t\right)+\boldsymbol k\left(t\right)$ $\left(9\right)$ The derivative of $\boldsymbol z\left(T\right)$ with respect to $z\left(0\right)$ are implicit in: $\boldsymbol x\left(T\right)=\boldsymbol S\left(T1\right)\boldsymbol S\left(T2\right)\cdots\boldsymbol S\left(0\right)\boldsymbol x\left(0\right)+\boldsymbol k'\left(t\right)$ $\left(10\right)$ 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 $\boldsymbol w\left(0\right)$): $\boldsymbol x'\left(\boldsymbol T\right)=\boldsymbol e^{i^T}$ $\left(11a\right)$ $\boldsymbol x'\left(t\right)=\boldsymbol x'\left(t+1\right)\boldsymbol S\left(t\right)$ $\left(11b\right)$ $\boldsymbol w\left(T\right)=\mathbf0$ $\left(11c\right)$ $\boldsymbol w\left(t\right)=\boldsymbol w\left(t+1\right)+\boldsymbol x'\left(t\right)s_{c}^{'}\left(t\right)$ $\left(11d\right)$ where $s_{c}^{'}$ refers to the matrix of derivatives of $s_i$ with respect to $c_k$. 
July 12th, 2018, 07:28 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics 
Any introductory text on (continuous) dynamical systems or differential equations should do. I would recommend one of: 1. Nonlinear dynamics and chaos  Strogatz 2. Differential equations, dynamical systems, and an introduction to chaos  Hirsh, Smale, and Devaney 3. Dynamical Systems  Robinson (be sure to get the version on continuous dynamical systems. He has another book which starts with discrete dynamics and is much more advanced.) I would strongly recommend avoiding Boyce/Diprima or any similar texts. 
July 12th, 2018, 07:37 PM  #3 
Senior Member Joined: Aug 2012 Posts: 2,306 Thanks: 706  

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