Dynamics

ordinary differential equation (ODE)


Rn∋(Ë™x1⋮˙xn)=  Ë™x_=f_(x_)  =(f1(x1,…,xn)â‹®fn(x1,…,xn))∈Rn

Dynamical Systems


A geometrical approach to study a system's dynamics, that is given in terms of differential equations.
By doing so, a qualitative analysis of the system's behavior throughout its state (phase) space can be gained, without having closed form solution.


Since every higher order differential equation with degree n,
that is to say the degree of the highest derivetive,
can be represented as an n dimensional ODE,
systems are presented as such.



linear

harmonic oscillator

\[ \begin{split} \ddot{x} \;+\; \omega^2 x \quad &= \quad 0 \\ m \; \ddot{x} \;+\; k \; x \quad &= \quad 0 \end{split} \]

click to edit

click to edit

click to edit

LARGE , Edward Wilson

hompage
GoogleScholar

200x

click to edit

click to edit

2-D

click to edit

click to edit

examples

click to edit

Pendelum

undamped, const torque

click to edit

click to edit

... interacting in networks
(coupled)

click to edit

... references

References

keywords to do:

phase locking , phase locked loop

continuous synchronization of vast num of metronome

click to edit

non-

linear

click to edit

click to edit

\[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \; cos(\phi) \\ ~ \\ r \; sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} x \\ \\ y \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows& \boxed{\boxed{ \quad \left( \begin{aligned} r \\ \\ \phi \end{aligned} \right) \quad }} \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ ~ \\ \text{atan2}(y,x) \end{aligned} \right) & ~ \end{align*} \]

\[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \cdot cos(\phi) \\ ~ \\ r \cdot sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} \dot{x} = f_x(x,y)\\ \\ \\ \dot{y} = f_y(x,y) \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows &\boxed{\boxed{ \quad \left( \begin{aligned} \dot{r} = f_r(r,\phi) \\ \\ \\ \dot{\phi} = f_{\phi}(r, \phi) \end{aligned} \right) \quad }} \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \frac{\partial \ \sqrt{ x^2 + y^2} }{\partial \ t} \quad = \quad \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ t} \quad = \quad \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot \frac{\partial \ x}{\partial t} \cdot \frac{\partial \ x}{\partial t} \end{aligned} \right) & ~ \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \\ \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \end{align*} \]

\newcommand{\xdownarrow}[1]{%

{\left\downarrow\vbox to #1{}\right.\kern-\nulldelimiterspace}

}

Interaction

click to edit

201x

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

\[ \boxed{\boxed{ \begin{matrix} ~ \\ ~ \\ ~ \end{matrix} \tau_{[\text{i}]} = \frac{1}{f_{[\text{i}]}} \begin{matrix} ~ \\ ~ \\ ~ \end{matrix} }} \]

coupled oscillators

networks

click to edit

alt text

click to edit

undamped

\[ \boxed{ \alpha \;\ddot{\mathsf{x}} + \beta \; \mathsf{x} = 0} \tiny \implies \left[ \begin{split} x = \mathsf{x} \\ y = \dot{\mathsf{x}} \end{split} \right] \implies \left[ \begin{split} \dot{x} &= \mathsf{\dot{x}} = y \;\; \\ \dot{y} &= \ddot{\mathsf{x}} = -\frac{\beta}{\alpha} \mathsf{x} \end{split} \right] \implies \normalsize \boxed{ \begin{split} \dot{x} &= &y \;\;\, \\ \dot{y} &= - \tfrac{\beta}{\alpha} &x \end{split} } \]



damped

\[ { \boxed{ \alpha \; \ddot{x} + \beta \; x + \gamma \; \dot{x} = 0 } \tiny \implies { \begin{split} u &= {x} \\ v &= \dot{x} \end{split} } \implies { \begin{split} \dot{u} &= x = v \\ \dot{v} &= \ddot{x} = - \frac{\beta}{\alpha} x \, - \, \frac{\gamma}{\alpha} \dot{x} \end{split} } \implies \normalsize \boxed{ \begin{split} \dot{u} &= v \\ \dot{v} &= - \frac{\beta}{\alpha} u \, - \, \frac{\gamma}{\alpha} v \end{split} }}\]

conservative systems


\[ \begin{align} \boxed{ \boxed{ \underline{\dot{x}} = \underline{f} \left( \underline{x} \right) } \quad conservative \quad } \quad & \iff \quad \exists \; E_{\left( \cdot \right)} \; \in \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) \quad \implies \quad \frac{d E_{(\cdot)}}{d t} = 0 \quad \left( ^{const} _{on trajectories} \right) \\ ~ \\ ~ \\ ~ \\ \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) & \quad = \quad \left\{ \quad E _{\left( \right)} : \ \mathbb{R}^n \to \mathbb{R} \begin{matrix} \quad \\ \quad \end{matrix} \middle| \quad \not \exists \; \mathbb{X} \; \in \; ^{open} _{~~set} \left( \mathbb{R}^n \right) \quad . \quad \forall \underline{x} , \underline{\tilde{x}} \in \mathbb{X} \quad . \quad E \left( \underline{x} \right) = E \left( \underline{\tilde{x}} \right) = const \quad \right\} \\ ~ \\ \; ^{open} _{~~set} \left( \mathbb{R}^n \right) & \quad = \quad \end{align} \]

click to edit

click to edit

click to edit

EXP.: Partical in a double well potential \[ V(x) = - \tfrac{1}{2} x^2 + \tfrac{1}{4} x^4 \]

click to edit

?
alt text

click to edit

?
alt text

1-D, vectorfield on a line

\[ \dot{x} \quad = \quad f \left( x \right) \]

click to edit

click to edit

click to edit

click to edit

1-D systems cannot oscillate

click to edit

setting to \( L := 1\)

\[ \begin{align} \tau_{i} \; \dot{x}_i \quad &=& \mathbf{f}_i \left( x_i \, , \; y_i , \lambda \right) \quad &+& \epsilon \cdot \mathbf{p}_i \left( x_{[1]} , y_{[1]} \ldots x_{[\text{i}-1]} , y_{[\text{i}-1]} \; , \; x_{[\text{i}+1]} , y_{[\text{i}-1]} \ldots x_n , y_n \ , \; \epsilon \right) \\ \\ \tau_i \; \dot{y}_i \quad &=& \mathbf{g}_i \left( x_i , y_i , \lambda \right) \quad &+& \epsilon \cdot \mathbf{q}_i \left( x_1 , y_1 \ldots , x_n , y_n \ , \; \epsilon \right) \\ \\ & \tiny x_i , y_i \in \mathbb{R} & \normalsize \end{align} \]

click to edit

alt text

alt text

different

time scales

weakly non-linear oscillator

\[ \begin{split} \ddot{x} \;+\; x \;+\; \varepsilon \; h \left( x , \dot{x} \right) \quad &= \quad 0 \\ & \boxed{ \small{0 \leq \varepsilon \ll 1}} \end{split} \]

\[ \boxed{\boxed{ \quad \begin{align} \mathcal{P} \left( \epsilon , z \right) \quad &=& z \; \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; z \, \right) ^{k} \quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; z \, } \right| \; < \; 1}& \frac{z}{1 - \sqrt{\epsilon} z} \\ \\ \mathcal{A} \left( \epsilon , \bar z \right) \quad &=& \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; \bar z \, \right) ^{k}\quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; \bar z \, } \right| \; < \; 1}& \frac{1}{1 - \sqrt{\epsilon} \bar z} \end{align} \quad }} \]

click to edit

relaxation oscillator

\[ \begin{split} \ddot{x} \;+\; x \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \quad &= \quad 0 \\ & \boxed{ \small{1 \ll \mu}} \end{split} \]

\[ \begin{split} & \ddot{x} \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \quad = \quad \ldots \\ \; \\ &= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, \left( \frac{1}{3} x^3 + x \right) \right) \\ \, \\&= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, F \left( x \right) \right) \qquad {with} \qquad F \left( x \right) = \frac{1}{3} x^3 + x \\ \, \\&= \quad \dot{w} \qquad \qquad \quad {with} \qquad w \;= \; \dot{x} + \mu \, F \left( x \right) \end{split} \]

click to edit

click to edit

Gamma tone

filter bank

changing to \( \tilde \ell := \ell - (L-1)\)

click to edit

click to edit

Hopf normal

(truncated) form

\[ \dot{z} \quad = \qquad z \; \cdot \; \underbrace{\left( \alpha + i \, \omega \right)}_{a} \qquad\qquad + \qquad\qquad s \]

click to edit

mass-spring-system

click to edit

?
alt text

click to edit

alt text

click to edit

click to edit

click to edit

click to edit

\[\normalsize \frac{d y}{dt} = f \left( t , y \right) \; , \qquad y_\left(t_0 \right) = y_0 \]

click to edit

\[ {\hat y}_{n+1} \quad \leftarrow \quad {\hat y}_{n} \quad + \quad \boxed{ \Delta^{(s)} {\hat y}_{n} \;\; = \; \; h \cdot{\sum _{i=1}^{s}} b_i \cdot \boxed{ k_i \;\; = \; \; f\left( \;\; \underbrace{t_n}_{n \cdot h} + c_i \cdot h \quad , \quad y_n + h \sum_{j = 1}^{i-1} a_{ij} k_j\right)\quad } }\]


click to edit

click to edit

\[ \tau_i \; \dot{z}_i \qquad=\qquad z_i \cdot \Large \sum_{\normalsize l = 1}^{\normalsize \infty} \normalsize a_{i l} \; | z_i |^{2 l} \qquad + \qquad \Large \sum_{\normalsize k \neq i}^{\normalsize n} \normalsize c_{i k} \;\; \mathcal{P} \left( \epsilon , z_k \right) \; \mathcal{A} \left( \epsilon , \bar z _i \right) \qquad+\qquad b_i \; s \]

click to edit

\[ \boxed{s} \]

click to edit

click to edit

click to edit

operator

click to edit

Shift

\[ \begin{align} e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\ e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]} \end{align} \]

click to edit

click to edit

\[ \left. \begin{align} \delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\ \delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\ \delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right) \end{align} \right\} \ \approx \ \frac{d}{dt} \]

click to edit

\[ \begin{align} \delta_{t+} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n \right]} \right) \\ \\ \delta_{t-} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n \right]} - u_{\left[ n-1 \right]} \right) \\ \\ \delta_{t} \; u_{\left[ n \right]} &= \ \frac{1}{2 \Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n-1 \right]} \right) \end{align} \]

click to edit

Identity

\[ 1 \; u_{\left[ n \right]} = \ u_{\left[ n \right]} \]

click to edit

click to edit

\[ \delta_{t t} \ := \ \delta_{t+} \ \delta_{t-} \ = \ \frac{1}{\Delta t^2} \left( e_{t+} - 2 + e_{t-} \right) \ \approx \ \frac{d^2}{dt^2} \]

click to edit

click to edit

\[ \delta_{t t t t} \ := \ \delta_{t t} \ \delta_{t t} \ = \ \frac{1}{\Delta t^4} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]

click to edit

click to edit

\[ \delta_{t-} \delta_{x x} \ = \ \frac{1}{2 \ \Delta t \ \Delta x^2} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]

click to edit

click to edit

click to edit

click to edit

Physical Modeling (Guitar)


\[ \overbrace{\underbrace{\frac{\partial^2 y}{\partial t^2}}_{\tiny\text{acceleration}} \quad =\quad \sigma_{\tiny{TLM}} \underbrace{\frac{\partial^2 y}{\partial x^2}}_{\tiny\text{curvature}} }^{\small{\text{basic wave equation}}}\ \overbrace{- \ \ 2 \sigma_{\tiny{0}} \frac{\partial y} {\partial t} \ + \ \sigma_{\tiny{1}} \frac{\partial}{\partial t} \frac{\partial^2 y}{\partial x^2} }^{\small{\text{damping}}}\ \overbrace{+ \ \ \sigma_{\tiny{E I}} \frac{\partial^4 y}{\partial x^4}}^{\small\text{stiffness}} \]


Martin Shuppius - Physical modelling of guitar strings
BILBAO, Stefan. Numerical sound synthesis: finite difference schemes and simulation in musical acoustics. John Wiley & Sons, 2009.

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

\[ \left. \begin{align} \mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\ \mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\ \mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right) \end{align} \right\} \ \approx \ 1 \]

+

click to edit

\[ \begin{align} \frac{1}{\Delta t^2} \left( y_{\left( x , \ t - \Delta t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x , \ t + \Delta t \right)} \right) \quad =& \quad \tfrac{T}{\mu} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad - \quad 2 \ \sigma_0 \frac{1}{2 \ \Delta t} \left( y_{\left( x , \ t + \Delta t \right)} \ - \ y_{\left( x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{1}{\Delta t} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x ,\ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{1}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \end{align} \]

\[ \begin{align} y_{\left( x , \ t + \Delta t \right)} \quad = \quad \frac{1}{1 + \sigma_0 \ \Delta t} & \Bigg( 2 \ y_{\left( x , \ t \right)} \ \ - \ \ y_{\left( x , \ t - \Delta t \right)} \quad \ldots \\ & \quad +\quad \tfrac{T}{\mu} \frac{\Delta t^2}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_0 \ \Delta t \ y_{\left( x , \ t - \Delta t \right)} \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{\Delta t}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t - \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{\Delta t^2}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \Bigg) \end{align} \]

eee

click to edit

Ideal String


  • Vibrates only in one direction
  • No friction and no other losses
  • Perfectly flexible (rubber band)

click to edit

Guitar String Properties


\(\sigma_{\tiny{TLM}} = \frac{T}{\mu} = \frac{T L}{M}\)


  • Tension   \(T\)
  • Mass     \(M\)
  • Length    \(L\)

Linear density   \(\mu = \frac{M}{L}\)

EXP.:

click to edit

click to edit

modified Euler method [1]


\[ \begin{align*} k_1 \quad & = & h \quad \cdot \quad & f \big( t_i \ , & \; {\hat y}_i \big) \\ \\ k_2 \quad & = & h \quad \cdot \quad & f \big( t_i + \tfrac{h}{2} \ , & \; {\hat y}_i + \tfrac{k_1}{2} \big) \end{align*} \]



\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad \tfrac{1}{2} \left( k_1 + k_2 \right) \]

click to edit

4th-order Runge-Kutta step [1]



\[ \begin{align*} k_1 \quad & =& h \quad \cdot \quad & f \big( t_i \ ,& \; {\hat y}_i \big) \\ \\ k_2 \quad & =& h \quad \cdot \quad & f \big( t_i + tfrac{h}{2} \ ,& \; {\hat y}_i + \tfrac{k_1}{2} \big) \\ \\ k_3 \quad & =& h \quad \cdot \quad & f \big(t_i + \tfrac{h}{2} \ ,& \; {\hat y}_i + \tfrac{k_2}{2} \big) \\ \\ k_4 \quad & =& h \quad \cdot \quad &f \big( \underbrace{t_{i} + h}_{t_{i+1}} \ ,& \; {\hat y}_i + k_3 \big) \end{align*} \]



\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad \tfrac{1}{6} \left( k_1 + 2 k_2 + 2 k_3 + k_4 \right)\]



click to edit

click to edit

click to edit

click to edit

\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad b \cdot | z |^{2} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

\[ \dot{z} \qquad=\qquad z \cdot \left( \alpha + \text{i} \, \omega \quad + \quad a_1 ~ | z |^{2} \;\; + \;\; a_2 ~ \varepsilon ~ | z |^{4} ~ \frac{1}{1 - \varepsilon |z|^2} \right) \qquad + \qquad c \;\; \mathcal{P} \left( \varepsilon , s \right) \; \mathcal{A} \left( \varepsilon , \bar z \right) \]

click to edit

setting \( \lambda_{o} := a \)


setting \( \forall \ell \in \{1 \ldots L\} ~~.~~ \lambda_{\ell} := b_{\ell} \)


unifying \( \forall \ell > L ~~.~~ \lambda_{\ell} := d \)

\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ \ell = L}^{\infty} \left( \varepsilon\cdot | z |^{2} \right)^{\ell-1} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

click to edit

...

\[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{\infty} ~ \lambda_{\ell+1} \cdot \varepsilon^{\ell} \cdot |z |^{2(\ell+1)} \right)\quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

\[ \dot{z} \quad = \qquad z \cdot \left( \underbrace{\left( \alpha_o + j \, \omega_o \right)}_{a_o} \quad + \quad \underbrace{\left( \alpha_1 + j \, \omega_1 \right)}_{a_1} | z |^2 \right) \quad\quad\quad+\quad\quad\quad h.o.t \qquad\qquad\qquad\qquad + \qquad\qquad s\]

click to edit

click to edit

\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

click to edit

click to edit

parameter rescaling

independent of \[ f = \frac{1}{\tau}\]

\[ \small \tfrac{1}{f_0} \cdot \dot{z} \quad=\quad z \left( \underbrace{\left( \beta_0 + j \, 2\pi \right)}_{b_0} \quad + \quad \underbrace{\left( \beta_1 + j \, \delta_1 \right)}_{b_1} \; | z |^2 \quad + \quad \sum_{\small k \ = 2}^{\infty} \underbrace{\left( \beta_k + j \, \delta_k \right)}_{b_k} \; ~ \varepsilon^{\, k-1} | z |^{2k} \right) \quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

splitting series at \( L \)

\[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{L-1} \lambda_{\ell+1} \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \quad + \quad \sum_{ \ell = L}^{\infty} \lambda_{\ell+1} ~ \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

click to edit

\[ \frac{1}{f_i} \ \frac{d\boldsymbol{z} _i}{dt} \qquad = \qquad \Large \sum \left( \ldots \right) \]

\[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left(\qquad a_i \qquad + \qquad b_i \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad d_i \ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4} \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\ | \boldsymbol{z}_{\ i} |^{2} \right)^{\tilde \ell} \right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

\[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left( \underbrace{ \left( \alpha + j \, 2 \pi \right) \ f_i }_{a_i} \qquad + \qquad \underbrace{ \left( \beta + j \, \beta' \right)f_i }_{b_i} \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad \underbrace{ \left( \delta + j \, \delta' \right) f_i }_{d_i} \ \frac{ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4}}{1 - \varepsilon \ | \boldsymbol{z}_{\ i} |^{2}}\right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]

click to edit

click to edit

click to edit

\[ \Large \sum_{\normalsize k = 1 }^{\normalsize n} \normalsize c_{k,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_k \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \qquad = \qquad \Large \sum \left( \ldots \right)\]

click to edit

\[ \boldsymbol{z}_{\ i} \cdot \left( \alpha_i + j \, 2 \pi \right) \ \]

click to edit

\[\boxed{k \neq i}\]

\[\boxed{k=i}\]

\[ \begin{split} \normalsize c_{i,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_i \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \normalsize \qquad =& \ \\ \\c_{i,i} \;\; \boldsymbol{z} _i \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \; \sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \normalsize \qquad =& \ \\ \\ c_{i,i} \;\; \boldsymbol{z} _i \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \; \sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \qquad =& \ \\ \\ c_{i,i} \boldsymbol{z} _i \Large \sum _{\ell\mathfrak{1} = \ell\mathfrak{2}} \normalsize \left( \ldots \right) \quad + \quad c_{i,i} \boldsymbol{z} _i \Large \sum_{\ell\mathfrak{1} \neq \ell\mathfrak{2}} \normalsize \left( \ldots \right) \end{split} \]

\[ \Large \sum_{\normalsize k \neq i }^{\normalsize 1 \leq k \leq n} \normalsize c_{k,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_k \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \qquad = \qquad \Large \sum \left( \ldots \right)\]

click to edit

click to edit

click to edit

\[ \normalsize c_{s,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{s} \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \]

click to edit

click to edit

\[\boxed{\ell\mathfrak{1}=\ell\mathfrak{2}=: \ell} \]

\[ \begin{split} c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell } \qquad =& \\ c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell = 0}^{\infty} \left( \epsilon \; | \boldsymbol{z} _i |^2 \, \right) ^{\ell } \end{split} \]

click to edit

\[\boxed{\ell\mathfrak{1} \neq \ell\mathfrak{2}}\]

\[ c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \;\sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \qquad = \quad 0\]

\[ \boxed{\boxed{ \quad \begin{align} \mathcal{P} \left( \epsilon , z \right) \quad &=& z \; \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; z \, \right) ^{k} \quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; z \, } \right| \; < \; 1}& \frac{z}{1 - \sqrt{\epsilon} z} \\ \\ \mathcal{A} \left( \epsilon , \bar z \right) \quad &=& \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; \bar z \, \right) ^{k}\quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; \bar z \, } \right| \; < \; 1}& \frac{1}{1 - \sqrt{\epsilon} \bar z} \end{align} \quad }} \]

click to edit

\[ \boldsymbol{z}_{\ i} \cdot \left( \left( \beta\mathfrak{1}_i + j \, \delta\mathfrak{1}_i \right) \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad \left( \beta\mathfrak{2} + j \, \delta\mathfrak{2} \right) \ \frac{ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4}}{1 - \varepsilon \ | \boldsymbol{z}_{\ i} |^{2}}\right) \]

click to edit

\[ \begin{array}{c|cccc} c_1 & a_{11} & a_{12}& \cdots & a_{1s} \\c_2 & a_{21} & a_{22}& \cdots & a_{2s} \\ \vdots & \vdots & \vdots& \ddots& \vdots \\ c_s & a_{s1} & a_{s2}& \cdots & a_{ss} \\ \hline ~ & b_1 & b_2 & \cdots & b_s \\ \end{array} \]

testing for

closed orbits

Poincare - Bendixon - Theorem

prove existence of a closed orbit


\[ \underbrace{\begin{matrix} \boxed{\; \text{i} \; } & \mathcal{R} = \text{closed, bounded region in } \mathbb{R}^2 \\ \boxed{\, \text{ii} \, } & \underline{\boldsymbol{\dot{x}}} = \underline{\boldsymbol{f}} (\underline{\boldsymbol{x}}) \quad \text{is smooth} \qquad \qquad \quad \\ \boxed{\text{iii}} & \text{no fixed points in } \mathcal{R} \qquad \qquad \qquad \\ \boxed{\text{iv}} & \exists \text{ a trapped trajectory } \mathcal{T} = \underline{\boldsymbol{x}}_{(t)} ^{\star} \quad \\ \text{i.e.: } & \quad \exists t_0 . \forall t \; . \; t \leq t_0 \implies \underline{\boldsymbol{x}}^{\star}_{(t)} \in \mathcal{R} \end{matrix} }_{\implies \mathcal{T} \text{ is itself closed, or spirals towards one, as } t \to \infty } \]

trick, find \( \;\; \boxed{\text{iv}}_{~b} \implies \boxed{\text{iv}}\)

\[ \underbrace{annulus \quad s.t. \quad \left\{ \begin{matrix} \text{the vector field } \underline{\boldsymbol{\dot{x}}} \\ \text{points into the } annulus \\ \text{on its boundaries} \end{matrix} \right.}_{\boxed{\text{iv}}_{~b} }\]

examples

click to edit

click to edit

alt text alt text alt text

Van der Pol

\[ \begin{split} \ddot{x} \;+\; x \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \quad &= \quad 0 & \qquad \boxed{ \small{1 \ll \mu}} \\ \ddot{x} \;+\; x \;+\; \varepsilon \, \left( x^2 - 1 \right) \, \dot{x} \quad &= \quad 0 & \qquad \boxed{ \small{0 \leq \varepsilon \ll 1}} \end{split} \]

Relaxation Oscillator


alt text

\[ \begin{split} -x \quad &= \quad \ddot{x} \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \\ \; \\ &= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, \left( \frac{1}{3} x^3 + x \right) \right) \\ \, \\&= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, F \left( x \right) \right) \qquad {with} \qquad F \left( x \right) = \frac{1}{3} x^3 + x \\ \, \\ -x \quad &= \quad \dot{w} \qquad \qquad \quad {with} \qquad w \;= \; \dot{x} + \mu \, F \left( x \right) \qquad \implies \qquad \dot{x} \; = \; w - \mu \, F \left( x \right) \\ \\ \\ \text{Let} \quad y \quad &= \quad \frac{w}{\mu} \\ \\ \\ \text{Then} \quad \dot{x} \quad &= \quad \mu \left( y - F(x) \right) \\ \\ \dot{y} \quad &= \quad - \tfrac{1}{\mu} x \end{split} \]

click to edit

click to edit

click to edit

?
alt text

click to edit

click to edit

Euler method [1]


\[ \begin{align*} k_1 \quad & = & h \quad \cdot \quad & f \big( t_i \ , & \; {\hat y}_i \big) \end{align*} \]



\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad k_1\]

click to edit

click to edit

click to edit

click to edit

click to edit

click to edit

\[ y_{\left( x, t + \Delta t \right)} \quad = \quad \left( \frac{1}{1+\sigma_0 \Delta t} \right) \; \underbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} \dfrac{\Delta t^2}{\Delta x^2} && \sigma_0 \ \Delta t && \sigma_1 \ \dfrac{\Delta t}{\Delta x^2} && E \ I \ \dfrac{\Delta t^2}{\Delta x^4} \\ ~ \end{bmatrix} }_{\cdots} \; \begin{bmatrix} 2 & & -1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 0 & & 1 & & 1 & & 0 & & 0 & & 0 & & 0 \\ \\ 0 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 2 & & 1 & & 1 & & -1 & & -1 & & 0 & & 0 \\ \\ 6 & & 0 & & -4 & & -4 & & 0 & & 0 & & 1 & & 1 \end{bmatrix} \; \begin{bmatrix} y_{\left( x \ , \ t \right)} \\ \\ y_{\left( x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - 2 \Delta x \ , \ t \right)} \\ \\ y_{\left( x + 2 \Delta x \ , \ t \right)} \end{bmatrix} \]

click to edit

\[\ldots \qquad \overbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} && \sigma_0 && \sigma_1 && E \ I \\ ~ \end{bmatrix} \; \begin{bmatrix} 1 && 0 && 0 && 0 && 0 \\ \\ 0 && \dfrac{\Delta t^2}{\Delta x^2} && 0 && 0 && 0 \\ \\ 0 && 0 && \Delta t && 0 && 0 \\ \\ 0 && 0 && 0 && \dfrac{\Delta t}{\Delta x^2} && 0 \\ \\ 0 && 0 && 0 && 0 && \dfrac{\Delta t^2}{\Delta x^4} \end{bmatrix} }^{\cdots} \qquad \ldots \]

generalised for 6 strings



\[ \underline{y}^{(1 \ldots 6)}_{\left( x, t + \Delta t \right)} \quad = \quad \overbrace{ \; \begin{bmatrix} \\ \dfrac{1}{1+\sigma_0^{(1)} \ \Delta t} && 0 && 0 && 0 && 0 && 0 \\ \\ 0 && \dfrac{1}{1+\sigma_0^{(2)} \ \Delta t} && 0 && 0 && 0 && 0 \\ \\ 0 && 0 && \dfrac{1}{1+\sigma_0^{(3)} \ \Delta t} && 0 && 0 && 0 \\ \\ 0 && 0 && 0 && \dfrac{1}{1+\sigma_0^{(4)} \ \Delta t} && 0 && 0 \\ \\ 0 && 0 && 0 && 0 && \dfrac{1}{1+\sigma_0^{(5)} \ \Delta t} && 0 \\ \\ 0 && 0 && 0 && 0 && 0 && \dfrac{1}{1+\sigma_0^{(6)} \ \Delta t} \\ ~ \end{bmatrix} \; \begin{bmatrix} 1 && \sigma_{T / \mu} ^{(1)} && \sigma_0^{(1)} && \sigma_1^{(1)} && \sigma_{E I}^{(1)} \\ \\ 1 && \sigma_{T / \mu} ^{(2)} && \sigma_0^{(2)} && \sigma_1^{(2)} && \sigma_{E I}^{(2)} \\ \\ 1 && \sigma_{T / \mu} ^{(3)} && \sigma_0^{(3)} && \sigma_1^{(3)} && \sigma_{E I}^{(3)} \\ \\ 1 && \sigma_{T / \mu} ^{(4)} && \sigma_0^{(4)} && \sigma_1^{(4)} && \sigma_{E I}^{(4)} \\ \\ 1 && \sigma_{T / \mu} ^{(5)} && \sigma_0^{(5)} && \sigma_1^{(5)} && \sigma_{E I}^{(5)} \\ \\ 1 && \sigma_{T / \mu} ^{(6)} && \sigma_0^{(6)} && \sigma_1^{(6)} && \sigma_{E I}^{(6)} \end{bmatrix} \; \begin{bmatrix} 1 && 0 && 0 && 0 && 0 \\ \\ 0 && \dfrac{\Delta t^2}{\Delta x^2} && 0 && 0 && 0 \\ \\ 0 && 0 && \Delta t && 0 && 0 \\ \\ 0 && 0 && 0 && \dfrac{\Delta t}{\Delta x^2} && 0 \\ \\ 0 && 0 && 0 && 0 && \dfrac{\Delta t^2}{\Delta x^4} \end{bmatrix} \; \begin{bmatrix} 2 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 0 & & 1 & & 1 & & 0 & & 0 & & 0 & & 0 \\ \\ 0 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 2 & & 1 & & 1 & & -1 & & -1 & & 0 & & 0 \\ \\ 6 & & 0 & & -4 & & -4 & & 0 & & 0 & & 1 & & 1 \end{bmatrix} }^{=const, \quad \text{is known before}} \; \begin{bmatrix} y_{\left( x \ , \ t \right)}^{(1)} && \cdots && y_{\left( x \ , \ t \right)}^{(6)} \\ \\ y_{\left( x \ , \ t - \Delta t \right)}^{(1)} && \cdots && y_{\left( x \ , \ t - \Delta t \right)}^{(6)} \\ \\ y_{\left( x - \Delta x \ , \ t \right)}^{(1)} && \cdots && y_{\left( x - \Delta x \ , \ t \right)}^{(6)} \\ \\ y_{\left( x +\Delta x \ , \ t \right)}^{(1)} && \cdots && y_{\left( x +\Delta x \ , \ t \right)}^{(6)} \\ \\ y_{\left( x - \Delta x \ , \ t - \Delta t \right)}^{(1)} && \cdots && y_{\left( x - \Delta x \ , \ t - \Delta t \right)}^{(6)} \\ \\ y_{\left( x +\Delta x \ , \ t - \Delta t \right)}^{(1)} && \cdots && y_{\left( x +\Delta x \ , \ t - \Delta t \right)}^{(6)} \\ \\ y_{\left( x - 2 \Delta x \ , \ t \right)}^{(1)} && \cdots && y_{\left( x - 2 \Delta x \ , \ t \right)}^{(6)} \\ \\ y_{\left( x + 2 \Delta x \ , \ t \right)}^{(1)} && \cdots && y_{\left( x + 2 \Delta x \ , \ t \right)}^{(6)} \end{bmatrix} \\ ~ \\ ~ \\ \]

click to edit