To allow for higher complexity, lets look at 2D systems

Definitions and examples

• A two-dimensional linear system is a system of the form

  $$ \begin{aligned} \dot x & = ax + by \\ \dot y & = cx + dy

  \end{aligned} $$

• Which we can neatly write as

  $$ \dot{\vec x} =A\vec x \quad \text{where} \quad A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \quad \vec x = \begin{pmatrix} x \\y \end{pmatrix} $$

🗒️ Notes:

• this system is linear, if $\vec x_1,\vec x_2$ are solutions then $c_1 \vec x_1+c_2\vec x_2$ is a solution
• $\dot { \vec x}= \vec 0$ when $\vec x= \vec 0$ so $\vec x^*= \vec 0$ is always a fixed point

💃 Example: Vibrations from a mass hanging from linear spring $m\ddot x + kx =0$

• We can represent the equation as

  $$ \begin{aligned} \dot x & = v \\ \dot v & = -\frac km x=-\omega^2 x

  \end{aligned} $$

• which a vector plot where

  $\vec f(x,v)=(v,-\omega^2 x)$

• Expand for physical correlation

Phase portrait of the system

🗒️ Note: origin is the only fixed point, starting anywhere else leads to a closed orbit

• This method was faster than solving it manually which would have given

Generalisation to 1st order differential system

$$ \begin{aligned} \dot x & = f_1(x,y) \\ \dot y & = f_2(x,y)

\end{aligned} $$

📖 Stability language

• Fixed point: $\dot {\vec x} = \vec0$
• Nullclines: curves defined by $\dot x = f_1 (x,y)=0$ or $\dot y =f_2(x,y)=0$
• Stable node: equilibrium where trajectories of the system are drawn towards the node in all direction
• Symmetrical node or star: all trajectories towards a stable node in a straight line
• Line of fixed points: all trajectories lead to a line
• Saddle point: where lines go towards in opposite directions and away in perpendicular
• Globally attracting: a point that attracts all trajectory in phase plane

Nonlinear 2D system

💼 Case: Pendulum described by $2$nd order ODE $\ddot \theta +\omega_0^2 \sin(\theta)=0$

• We can write it as where $x$ is angle

  $$ \begin{aligned} \dot x &= y \\ \dot y &= -\omega^2_0 \sin (x) \end{aligned} $$

We get the following phase space diagram

🗒️ Note: it is periodic in $x$ so $-\pi$ and $\pi$ represent the same physical setup

💎 Conclusion:

• Fixed points: at $(0,0)$ and $(\pi,0)$ are stable and unstable respectively
• Cycles: for moderate angles