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Dynamics: is a subject that deals with change, with systems that evolve in time
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A system that evolves over continuous time, ie the system’s state changes smoothly over time
These systems are mainly described using differential equations. A general framework to study continuous dynamical system is:
$$ \begin{aligned} \dot x_1&=f_1(x_1,\ldots,x_n) \\ \dot x_2&=f_2(x_1,\ldots,x_n) \\ \vdots\;\,\,& \qquad \qquad \vdots \\ \dot x_n&=f_n(x_1,\ldots,x_n)
\end{aligned} $$
Properties:
💃 Example: We can convert the equation for a damped oscillator (autonomous system)
$$ m\frac{\text d^2 x}{\text dt^2}+b \frac{\text dx}{\text dt}+kx=0 $$
to the form above by setting $x_1=x$, $x_2=\dot x$ and thus $f_1=x_2$ so
$$ \begin{aligned} \dot x_1&=\dot x=x_2 \\ \dot x_2&=\ddot x=-\frac{b}{m} \dot x- \frac{k}{m}x=-\frac{b}{m} x_2- \frac{k}{m}x_1 \end{aligned} $$
🗒️ Note: this system is linear because all the powers on the RHS are $1$ ie no $x_n^2$ for example
💃 Example: While the the system is autonomous we can still solve nonautonomous equations, we convert the forced harmonic oscillator equation
$$ m\ddot x+b \dot x+kx=F\cos(t) $$
Here we can let $x_1=x$, $x_2=\dot x$, $x_3=t$ and thus $\dot x_3=1$ so
$$ \begin{aligned} \dot x_1 &= \dot x = x_2 \\ \dot x_2&=\ddot x= \frac{1}{m} [-k x-b \dot x+F\cos(t)]=\frac{1}{m}[-k x_1-b x_2+F\cos(x_3)] \\ \dot x_3&=\dot t=1
\end{aligned} $$
🗒️ Notes:
