ordinary differential equation (ODE)
Rn∋(˙x1⋮˙xn)= ˙x_=f_(x_) =(f1(x1,…,xn)⋮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
\[ \begin{split} \ddot{x} \;+\; \omega^2 x \quad &= \quad 0 \\ m \; \ddot{x} \;+\; k \; x \quad &= \quad 0 \end{split} \]
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LARGE , Edward Wilson
200x
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2-D
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examples
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Pendelum
undamped, const torque
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... interacting in networks
(coupled)
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... references
References
keywords to do:
phase locking , phase locked loop
continuous synchronization of vast num of metronome
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non-
linear
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\[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \; cos(\phi) \\ ~ \\ r \; sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} x \\ \\ y \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows& \boxed{\boxed{ \quad \left( \begin{aligned} r \\ \\ \phi \end{aligned} \right) \quad }} \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ ~ \\ \text{atan2}(y,x) \end{aligned} \right) & ~ \end{align*} \]
\[ \begin{align*} ~ & ~~~~~~~~~~\left( \begin{aligned} r \cdot cos(\phi) \\ ~ \\ r \cdot sin(\phi) \end{aligned} \right) & ~ \\ ~\\ \boxed{\boxed{ \quad \left( \begin{aligned} \dot{x} = f_x(x,y)\\ \\ \\ \dot{y} = f_y(x,y) \end{aligned} \right) \quad }} &~~~~~~~~~~~~~~~~~~\leftrightarrows &\boxed{\boxed{ \quad \left( \begin{aligned} \dot{r} = f_r(r,\phi) \\ \\ \\ \dot{\phi} = f_{\phi}(r, \phi) \end{aligned} \right) \quad }} \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \frac{\partial \ \sqrt{ x^2 + y^2} }{\partial \ t} \quad = \quad \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ t} \quad = \quad \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot \frac{\partial \ x}{\partial t} \cdot \frac{\partial \ x}{\partial t} \end{aligned} \right) & ~ \\ ~\\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \\ \\ \\ \\ ~ & ~~~~~~~ \left( \begin{aligned} \sqrt{ x^2 + y^2} ~~ \\ \\ \\ \\ \frac{\partial \ \text{atan2}(y,x) }{\partial \ x} \cdot \frac{\partial \ \text{atan2}(y,x) }{\partial \ y} \cdot f_x(x,y) \cdot f_y(x,y) \end{aligned} \right) & ~ \end{align*} \]
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Interaction
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201x
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\[ \boxed{\boxed{ \begin{matrix} ~ \\ ~ \\ ~ \end{matrix} \tau_{[\text{i}]} = \frac{1}{f_{[\text{i}]}} \begin{matrix} ~ \\ ~ \\ ~ \end{matrix} }} \]
coupled oscillators
networks
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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
\[ \begin{align} \boxed{ \boxed{ \underline{\dot{x}} = \underline{f} \left( \underline{x} \right) } \quad conservative \quad } \quad & \iff \quad \exists \; E_{\left( \cdot \right)} \; \in \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) \quad \implies \quad \frac{d E_{(\cdot)}}{d t} = 0 \quad \left( ^{const} _{on trajectories} \right) \\ ~ \\ ~ \\ ~ \\ \; ^{conserved} _{~~quantity} \left( \mathbb{R}^n \right) & \quad = \quad \left\{ \quad E _{\left( \right)} : \ \mathbb{R}^n \to \mathbb{R} \begin{matrix} \quad \\ \quad \end{matrix} \middle| \quad \not \exists \; \mathbb{X} \; \in \; ^{open} _{~~set} \left( \mathbb{R}^n \right) \quad . \quad \forall \underline{x} , \underline{\tilde{x}} \in \mathbb{X} \quad . \quad E \left( \underline{x} \right) = E \left( \underline{\tilde{x}} \right) = const \quad \right\} \\ ~ \\ \; ^{open} _{~~set} \left( \mathbb{R}^n \right) & \quad = \quad \end{align} \]
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EXP.: Partical in a double well potential \[ V(x) = - \tfrac{1}{2} x^2 + \tfrac{1}{4} x^4 \]
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1-D, vectorfield on a line
\[ \dot{x} \quad = \quad f \left( x \right) \]
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1-D systems cannot oscillate
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setting to \( L := 1\)
\[ \begin{align} \tau_{i} \; \dot{x}_i \quad &=& \mathbf{f}_i \left( x_i \, , \; y_i , \lambda \right) \quad &+& \epsilon \cdot \mathbf{p}_i \left( x_{[1]} , y_{[1]} \ldots x_{[\text{i}-1]} , y_{[\text{i}-1]} \; , \; x_{[\text{i}+1]} , y_{[\text{i}-1]} \ldots x_n , y_n \ , \; \epsilon \right) \\ \\ \tau_i \; \dot{y}_i \quad &=& \mathbf{g}_i \left( x_i , y_i , \lambda \right) \quad &+& \epsilon \cdot \mathbf{q}_i \left( x_1 , y_1 \ldots , x_n , y_n \ , \; \epsilon \right) \\ \\ & \tiny x_i , y_i \in \mathbb{R} & \normalsize \end{align} \]
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different
time scales
weakly non-linear oscillator
\[ \begin{split} \ddot{x} \;+\; x \;+\; \varepsilon \; h \left( x , \dot{x} \right) \quad &= \quad 0 \\ & \boxed{ \small{0 \leq \varepsilon \ll 1}} \end{split} \]
\[ \boxed{\boxed{ \quad \begin{align} \mathcal{P} \left( \epsilon , z \right) \quad &=& z \; \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; z \, \right) ^{k} \quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; z \, } \right| \; < \; 1}& \frac{z}{1 - \sqrt{\epsilon} z} \\ \\ \mathcal{A} \left( \epsilon , \bar z \right) \quad &=& \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; \bar z \, \right) ^{k}\quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; \bar z \, } \right| \; < \; 1}& \frac{1}{1 - \sqrt{\epsilon} \bar z} \end{align} \quad }} \]
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relaxation oscillator
\[ \begin{split} \ddot{x} \;+\; x \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \quad &= \quad 0 \\ & \boxed{ \small{1 \ll \mu}} \end{split} \]
\[ \begin{split} & \ddot{x} \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \quad = \quad \ldots \\ \; \\ &= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, \left( \frac{1}{3} x^3 + x \right) \right) \\ \, \\&= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, F \left( x \right) \right) \qquad {with} \qquad F \left( x \right) = \frac{1}{3} x^3 + x \\ \, \\&= \quad \dot{w} \qquad \qquad \quad {with} \qquad w \;= \; \dot{x} + \mu \, F \left( x \right) \end{split} \]
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Gamma tone
filter bank
changing to \( \tilde \ell := \ell - (L-1)\)
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Hopf normal
(truncated) form
\[ \dot{z} \quad = \qquad z \; \cdot \; \underbrace{\left( \alpha + i \, \omega \right)}_{a} \qquad\qquad + \qquad\qquad s \]
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mass-spring-system
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\[\normalsize \frac{d y}{dt} = f \left( t , y \right) \; , \qquad y_\left(t_0 \right) = y_0 \]
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\[ {\hat y}_{n+1} \quad \leftarrow \quad {\hat y}_{n} \quad + \quad \boxed{ \Delta^{(s)} {\hat y}_{n} \;\; = \; \; h \cdot{\sum _{i=1}^{s}} b_i \cdot \boxed{ k_i \;\; = \; \; f\left( \;\; \underbrace{t_n}_{n \cdot h} + c_i \cdot h \quad , \quad y_n + h \sum_{j = 1}^{i-1} a_{ij} k_j\right)\quad } }\]
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\[ \tau_i \; \dot{z}_i \qquad=\qquad z_i \cdot \Large \sum_{\normalsize l = 1}^{\normalsize \infty} \normalsize a_{i l} \; | z_i |^{2 l} \qquad + \qquad \Large \sum_{\normalsize k \neq i}^{\normalsize n} \normalsize c_{i k} \;\; \mathcal{P} \left( \epsilon , z_k \right) \; \mathcal{A} \left( \epsilon , \bar z _i \right) \qquad+\qquad b_i \; s \]
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\[ \boxed{s} \]
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operator
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Shift
\[ \begin{align} e_{t+} \, u_{\left[ n \right]} &= \ u_{\left[ n+1 \right]}\\ \\ e_{t-} \, u_{\left[ n \right]} &= \ u_{\left[ n-1 \right]} \end{align} \]
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\[ \left. \begin{align} \delta_{t+} &:= \frac{1}{\Delta t}\ \left( e_{t+}-1 \right) \\ \\ \delta_{t-} &:= \frac{1}{\Delta t}\ \left( 1-e_{t-} \right) \\ \\ \delta_{t} &:= \frac{1}{2 \ \Delta t}\ \left( e_{t+}-e_{t-} \right) \end{align} \right\} \ \approx \ \frac{d}{dt} \]
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\[ \begin{align} \delta_{t+} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n \right]} \right) \\ \\ \delta_{t-} \, u_{\left[ n \right]} &= \ \frac{1}{\Delta t}\ \left( u_{\left[ n \right]} - u_{\left[ n-1 \right]} \right) \\ \\ \delta_{t} \; u_{\left[ n \right]} &= \ \frac{1}{2 \Delta t}\ \left( u_{\left[ n+1 \right]} - u_{\left[ n-1 \right]} \right) \end{align} \]
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Identity
\[ 1 \; u_{\left[ n \right]} = \ u_{\left[ n \right]} \]
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\[ \delta_{t t} \ := \ \delta_{t+} \ \delta_{t-} \ = \ \frac{1}{\Delta t^2} \left( e_{t+} - 2 + e_{t-} \right) \ \approx \ \frac{d^2}{dt^2} \]
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\[ \delta_{t t t t} \ := \ \delta_{t t} \ \delta_{t t} \ = \ \frac{1}{\Delta t^4} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
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\[ \delta_{t-} \delta_{x x} \ = \ \frac{1}{2 \ \Delta t \ \Delta x^2} \left( e_{t+}^2 - 4 e_{t+} + 6 - 4 e_{t-} + e_{t-}^2 \right) \ \approx \ \frac{d^4}{dt^4} \]
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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.
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\[ \left. \begin{align} \mu_{t+} &:= \frac{1}{2}\ \left( e_{t+}+1 \right) \\ \\ \mu_{t-} &:= \frac{1}{2}\ \left( 1+e_{t-} \right) \\ \\ \mu_{t} &:= \frac{1}{2}\ \left( e_{t+}+e_{t-} \right) \end{align} \right\} \ \approx \ 1 \]
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\[ \begin{align} \frac{1}{\Delta t^2} \left( y_{\left( x , \ t - \Delta t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x , \ t + \Delta t \right)} \right) \quad =& \quad \tfrac{T}{\mu} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad - \quad 2 \ \sigma_0 \frac{1}{2 \ \Delta t} \left( y_{\left( x , \ t + \Delta t \right)} \ - \ y_{\left( x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{1}{\Delta t} \frac{1}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x ,\ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{1}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \end{align} \]
\[ \begin{align} y_{\left( x , \ t + \Delta t \right)} \quad = \quad \frac{1}{1 + \sigma_0 \ \Delta t} & \Bigg( 2 \ y_{\left( x , \ t \right)} \ \ - \ \ y_{\left( x , \ t - \Delta t \right)} \quad \ldots \\ & \quad +\quad \tfrac{T}{\mu} \frac{\Delta t^2}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad \sigma_0 \ \Delta t \ y_{\left( x , \ t - \Delta t \right)} \qquad \ldots \\ \\ & \quad + \quad \sigma_1 \frac{\Delta t}{\Delta x^2} \left( y_{\left( x - \Delta x , \ t \right)} -\ 2 \ y_{\left( x , \ t \right)} +\ y_{\left( x + \Delta x , \ t \right)} \ - \ y_{\left( x - \Delta x , \ t - \Delta t \right)} \ + \ 2 \ y_{\left( x , \ t - \Delta t \right)} \ - \ y_{\left( x + \Delta x , \ t - \Delta t \right)} \right) \qquad \ldots \\ \\ & \quad + \quad E \ \ I \ \ \frac{\Delta t^2}{\Delta x^4} \left( y_{\left( x + 2 \Delta x , \ t \right)} -\ 4 \ y_{\left( x + \Delta x, \ t \right)} +\ 6 \ y_{\left( x , \ t \right)} \ - \ 4 \ y_{\left( x - \Delta x, \ t \right)} \ + \ y_{\left( x - 2 \Delta x , \ t \right)} \right) \Bigg) \end{align} \]
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Ideal String
- Vibrates only in one direction
- No friction and no other losses
- Perfectly flexible (rubber band)
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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.:
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modified Euler method [1]
\[ \begin{align*} k_1 \quad & = & h \quad \cdot \quad & f \big( t_i \ , & \; {\hat y}_i \big) \\ \\ k_2 \quad & = & h \quad \cdot \quad & f \big( t_i + \tfrac{h}{2} \ , & \; {\hat y}_i + \tfrac{k_1}{2} \big) \end{align*} \]
\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad \tfrac{1}{2} \left( k_1 + k_2 \right) \]
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4th-order Runge-Kutta step [1]
\[ \begin{align*} k_1 \quad & =& h \quad \cdot \quad & f \big( t_i \ ,& \; {\hat y}_i \big) \\ \\ k_2 \quad & =& h \quad \cdot \quad & f \big( t_i + tfrac{h}{2} \ ,& \; {\hat y}_i + \tfrac{k_1}{2} \big) \\ \\ k_3 \quad & =& h \quad \cdot \quad & f \big(t_i + \tfrac{h}{2} \ ,& \; {\hat y}_i + \tfrac{k_2}{2} \big) \\ \\ k_4 \quad & =& h \quad \cdot \quad &f \big( \underbrace{t_{i} + h}_{t_{i+1}} \ ,& \; {\hat y}_i + k_3 \big) \end{align*} \]
\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad \tfrac{1}{6} \left( k_1 + 2 k_2 + 2 k_3 + k_4 \right)\]
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\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad b \cdot | z |^{2} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
\[ \dot{z} \qquad=\qquad z \cdot \left( \alpha + \text{i} \, \omega \quad + \quad a_1 ~ | z |^{2} \;\; + \;\; a_2 ~ \varepsilon ~ | z |^{4} ~ \frac{1}{1 - \varepsilon |z|^2} \right) \qquad + \qquad c \;\; \mathcal{P} \left( \varepsilon , s \right) \; \mathcal{A} \left( \varepsilon , \bar z \right) \]
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setting \( \lambda_{o} := a \)
setting \( \forall \ell \in \{1 \ldots L\} ~~.~~ \lambda_{\ell} := b_{\ell} \)
unifying \( \forall \ell > L ~~.~~ \lambda_{\ell} := d \)
\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ \ell = L}^{\infty} \left( \varepsilon\cdot | z |^{2} \right)^{\ell-1} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
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\[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{\infty} ~ \lambda_{\ell+1} \cdot \varepsilon^{\ell} \cdot |z |^{2(\ell+1)} \right)\quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
\[ \dot{z} \quad = \qquad z \cdot \left( \underbrace{\left( \alpha_o + j \, \omega_o \right)}_{a_o} \quad + \quad \underbrace{\left( \alpha_1 + j \, \omega_1 \right)}_{a_1} | z |^2 \right) \quad\quad\quad+\quad\quad\quad h.o.t \qquad\qquad\qquad\qquad + \qquad\qquad s\]
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\[ \dot{z} \quad=\qquad z \cdot \left( a \qquad + \qquad \sum_{ \ell = 0}^{L-1} b_{\ell+1} ~ \varepsilon^{\ell} \cdot | z |^{2(\ell+1)} \qquad + \qquad d \cdot \varepsilon \cdot | z |^{2} \cdot \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\cdot | z |^{2} \right)^{\tilde \ell} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
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parameter rescaling
independent of \[ f = \frac{1}{\tau}\]
\[ \small \tfrac{1}{f_0} \cdot \dot{z} \quad=\quad z \left( \underbrace{\left( \beta_0 + j \, 2\pi \right)}_{b_0} \quad + \quad \underbrace{\left( \beta_1 + j \, \delta_1 \right)}_{b_1} \; | z |^2 \quad + \quad \sum_{\small k \ = 2}^{\infty} \underbrace{\left( \beta_k + j \, \delta_k \right)}_{b_k} \; ~ \varepsilon^{\, k-1} | z |^{2k} \right) \quad\quad\quad+\quad\quad\quad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
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splitting series at \( L \)
\[ \dot{z} \quad=\qquad z \cdot \left( \lambda_o \qquad + \qquad \sum_{ \ell = 0}^{L-1} \lambda_{\ell+1} \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \quad + \quad \sum_{ \ell = L}^{\infty} \lambda_{\ell+1} ~ \varepsilon^{\ell} ~ | z |^{2(\ell+1)} \right) \qquad+\qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
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\[ \frac{1}{f_i} \ \frac{d\boldsymbol{z} _i}{dt} \qquad = \qquad \Large \sum \left( \ldots \right) \]
\[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left(\qquad a_i \qquad + \qquad b_i \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad d_i \ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4} \sum_{ {\tilde \ell} = 0 }^{\infty} ~ \left( \varepsilon\ | \boldsymbol{z}_{\ i} |^{2} \right)^{\tilde \ell} \right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
\[ \boldsymbol{\dot{z}} _i \quad=\qquad \boldsymbol{z}_{\ i} \cdot \left( \underbrace{ \left( \alpha + j \, 2 \pi \right) \ f_i }_{a_i} \qquad + \qquad \underbrace{ \left( \beta + j \, \beta' \right)f_i }_{b_i} \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad \underbrace{ \left( \delta + j \, \delta' \right) f_i }_{d_i} \ \frac{ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4}}{1 - \varepsilon \ | \boldsymbol{z}_{\ i} |^{2}}\right) \qquad+ \qquad \boxed{^{resonance}_{~~~~term}} \qquad\qquad + \qquad\qquad s\]
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\[ \Large \sum_{\normalsize k = 1 }^{\normalsize n} \normalsize c_{k,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_k \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \qquad = \qquad \Large \sum \left( \ldots \right)\]
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\[ \boldsymbol{z}_{\ i} \cdot \left( \alpha_i + j \, 2 \pi \right) \ \]
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\[\boxed{k \neq i}\]
\[\boxed{k=i}\]
\[ \begin{split} \normalsize c_{i,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_i \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \normalsize \qquad =& \ \\ \\c_{i,i} \;\; \boldsymbol{z} _i \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \; \sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \normalsize \qquad =& \ \\ \\ c_{i,i} \;\; \boldsymbol{z} _i \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \; \sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \qquad =& \ \\ \\ c_{i,i} \boldsymbol{z} _i \Large \sum _{\ell\mathfrak{1} = \ell\mathfrak{2}} \normalsize \left( \ldots \right) \quad + \quad c_{i,i} \boldsymbol{z} _i \Large \sum_{\ell\mathfrak{1} \neq \ell\mathfrak{2}} \normalsize \left( \ldots \right) \end{split} \]
\[ \Large \sum_{\normalsize k \neq i }^{\normalsize 1 \leq k \leq n} \normalsize c_{k,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{z}_k \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \qquad = \qquad \Large \sum \left( \ldots \right)\]
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\[ \normalsize c_{s,i} \;\; \mathcal{P} \left( \varepsilon , \boldsymbol{s} \right) \; \mathcal{A} \left( \varepsilon , \boldsymbol{\bar z} _i \right) \]
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\[\boxed{\ell\mathfrak{1}=\ell\mathfrak{2}=: \ell} \]
\[ \begin{split} c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell } \qquad =& \\ c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell = 0}^{\infty} \left( \epsilon \; | \boldsymbol{z} _i |^2 \, \right) ^{\ell } \end{split} \]
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\[\boxed{\ell\mathfrak{1} \neq \ell\mathfrak{2}}\]
\[ c_{i,i} \;\; \boldsymbol{z} _i \; \; \sum_{\ell\mathfrak{1} = 0}^{\infty} \;\sum_{\ell\mathfrak{2} = 0}^{\infty} \left( \sqrt{\epsilon} \; \boldsymbol{z} _i \, \right) ^{\ell\mathfrak{1} } \left( \sqrt{\epsilon} \; \boldsymbol{\bar z} _i \, \right) ^{\ell\mathfrak{2} } \qquad = \quad 0\]
\[ \boxed{\boxed{ \quad \begin{align} \mathcal{P} \left( \epsilon , z \right) \quad &=& z \; \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; z \, \right) ^{k} \quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; z \, } \right| \; < \; 1}& \frac{z}{1 - \sqrt{\epsilon} z} \\ \\ \mathcal{A} \left( \epsilon , \bar z \right) \quad &=& \sum_{k = 0}^{\infty} \left( \sqrt{\epsilon} \; \bar z \, \right) ^{k}\quad &\mathop = \limits^{\left| { \sqrt{\epsilon} \; \bar z \, } \right| \; < \; 1}& \frac{1}{1 - \sqrt{\epsilon} \bar z} \end{align} \quad }} \]
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\[ \boldsymbol{z}_{\ i} \cdot \left( \left( \beta\mathfrak{1}_i + j \, \delta\mathfrak{1}_i \right) \ | \boldsymbol{z}_{\ i} |^{2} \qquad + \qquad \left( \beta\mathfrak{2} + j \, \delta\mathfrak{2} \right) \ \frac{ \varepsilon \ | \boldsymbol{z}_{\ i} |^{4}}{1 - \varepsilon \ | \boldsymbol{z}_{\ i} |^{2}}\right) \]
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\[ \begin{array}{c|cccc} c_1 & a_{11} & a_{12}& \cdots & a_{1s} \\c_2 & a_{21} & a_{22}& \cdots & a_{2s} \\ \vdots & \vdots & \vdots& \ddots& \vdots \\ c_s & a_{s1} & a_{s2}& \cdots & a_{ss} \\ \hline ~ & b_1 & b_2 & \cdots & b_s \\ \end{array} \]
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
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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
\[ \begin{split} -x \quad &= \quad \ddot{x} \;+\; \mu \, \left( x^2 - 1 \right) \, \dot{x} \\ \; \\ &= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, \left( \frac{1}{3} x^3 + x \right) \right) \\ \, \\&= \quad \frac{d}{dt} \left( \dot{x} \;+\; \mu \, F \left( x \right) \right) \qquad {with} \qquad F \left( x \right) = \frac{1}{3} x^3 + x \\ \, \\ -x \quad &= \quad \dot{w} \qquad \qquad \quad {with} \qquad w \;= \; \dot{x} + \mu \, F \left( x \right) \qquad \implies \qquad \dot{x} \; = \; w - \mu \, F \left( x \right) \\ \\ \\ \text{Let} \quad y \quad &= \quad \frac{w}{\mu} \\ \\ \\ \text{Then} \quad \dot{x} \quad &= \quad \mu \left( y - F(x) \right) \\ \\ \dot{y} \quad &= \quad - \tfrac{1}{\mu} x \end{split} \]
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Euler method [1]
\[ \begin{align*} k_1 \quad & = & h \quad \cdot \quad & f \big( t_i \ , & \; {\hat y}_i \big) \end{align*} \]
\[ {\hat y}_{i+1} \quad = \quad {\hat y}_i \quad + \quad k_1\]
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\[ y_{\left( x, t + \Delta t \right)} \quad = \quad \left( \frac{1}{1+\sigma_0 \Delta t} \right) \; \underbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} \dfrac{\Delta t^2}{\Delta x^2} && \sigma_0 \ \Delta t && \sigma_1 \ \dfrac{\Delta t}{\Delta x^2} && E \ I \ \dfrac{\Delta t^2}{\Delta x^4} \\ ~ \end{bmatrix} }_{\cdots} \; \begin{bmatrix} 2 & & -1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 0 & & 1 & & 1 & & 0 & & 0 & & 0 & & 0 \\ \\ 0 & & 1 & & 0 & & 0 & & 0 & & 0 & & 0 & & 0 \\ \\ -2 & & 2 & & 1 & & 1 & & -1 & & -1 & & 0 & & 0 \\ \\ 6 & & 0 & & -4 & & -4 & & 0 & & 0 & & 1 & & 1 \end{bmatrix} \; \begin{bmatrix} y_{\left( x \ , \ t \right)} \\ \\ y_{\left( x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t \right)} \\ \\ y_{\left( x - \Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x +\Delta x \ , \ t - \Delta t \right)} \\ \\ y_{\left( x - 2 \Delta x \ , \ t \right)} \\ \\ y_{\left( x + 2 \Delta x \ , \ t \right)} \end{bmatrix} \]
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\[\ldots \qquad \overbrace{ \begin{bmatrix} ~ \\ 1 && \dfrac{T}{\mu} && \sigma_0 && \sigma_1 && E \ I \\ ~ \end{bmatrix} \; \begin{bmatrix} 1 && 0 && 0 && 0 && 0 \\ \\ 0 && \dfrac{\Delta t^2}{\Delta x^2} && 0 && 0 && 0 \\ \\ 0 && 0 && \Delta t && 0 && 0 \\ \\ 0 && 0 && 0 && \dfrac{\Delta t}{\Delta x^2} && 0 \\ \\ 0 && 0 && 0 && 0 && \dfrac{\Delta t^2}{\Delta x^4} \end{bmatrix} }^{\cdots} \qquad \ldots \]
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} \\ ~ \\ ~ \\ \]
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