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July 12th, 2018, 03:45 PM   #1
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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:

$\boldsymbol x\left(T\right)=\boldsymbol S\left(T-1\right)\boldsymbol S\left(T-2\right)\cdots\boldsymbol S\left(0\right)\boldsymbol x\left(0\right)+\boldsymbol k'\left(t\right)$ $\left(10\right)$

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 $c_k$.
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July 12th, 2018, 08:28 PM   #2
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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.
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July 12th, 2018, 08:37 PM   #3
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Quote:
Originally Posted by SDK View Post

I would strongly recommend avoiding Boyce/Diprima or any similar texts.
That book made me detest differential equations.
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