
Applied Math Applied Math Forum 
 LinkBack  Thread Tools  Display Modes 
July 12th, 2018, 02:45 PM  #1 
Newbie Joined: Nov 2013 Posts: 16 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: 415 Thanks: 229 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: 1,971 Thanks: 550  

Tags 
article, exploring, jacobian, knowledge, linearize, maths, nonlinear system, snippet, understand 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Trying to understand integration with very basic knowledge of derivation  stipanrelix  Calculus  2  March 6th, 2018 12:41 PM 
How much Maths knowledge does one need to study and grasp geometry?  BachsMass  Math  7  December 15th, 2015 08:55 PM 
I can't understand how the answer to this gcse maths problem is correct  something  Math  4  May 29th, 2015 07:26 PM 
Exploring the values of a.  ZardoZ  Calculus  11  July 9th, 2012 02:46 AM 
Please help me understand maths!!  mathsphobia  Algebra  2  June 22nd, 2008 11:07 AM 