CDS - Example of a Phase Space • Pendulum Example

Remember:

Given the following “pendulum” sytems, without friction:$\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-\frac{g}{L}\sin (x_1)\end{array}$We can draw its phase space:

• The system has two steady states, one unstable, when the pendulum is held with muss “up”, and one marginally stable ss when the mass in the “equilibrium position”, it is marginally stable because this system does not account for friction.
  • The marginally stable steady states are represented in the phase space with the “s” (look a the point inside the black lines)
  • While the unstable steady states are represeted with a “i” (look at the intersection of the blue lines with the $x_1$ axis)
• In the phase space:
  • The black circular lines indicate: when the pendulum oscillates (not enough initial velocity for a complete rotation)
  • The red lines indicate: when the pendulum compleately rotates, however notice that the velocity of the pendulum never reaches $0$.
  • The single blue line indicates: this is a special case in which the pendulum actually stops $(x_2=0)$ in the upper point, also note that for the exact initial conditions such that we begin with the pendulum in one of the “i” unstable ss, then it will not move (it is a ss after all).
• When you add friction the phase space changes such that the marginally steady states, now become stable steady states, and the flows will all reach this state, with enough time, except if the pendulum start exactly in the “up”-state with no velocity, in that case it will remain in that position (unstable ss)

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• Pendulum example, we do not account for friction, so it will continue to move endlessly.
  • For this system: $x_1$ represents the position, $x_2$ reperesents the velocity.not-sure-about-this
• The system has two steady states, one unstable, when the pendulum is held with muss “up”, and one marginally stable ss when the mass in the “equilibrium position”, it is marginally stable because this system does not account for friction.
  • The marginally stable steady states are represented in the phase space with the “s” (look a the point inside the black lines)
  • While the unstable steady states are represeted with a “i” (look at the intersection of the blue lines with the $x_1$ axis)
• In the phase space:
  • The black circular lines indicate: when the pendulum oscillates (not enough initial velocity for a complete rotation)
  • The red lines indicate: when the pendulum compleately rotates, however notice that the velocity of the pendulum never reaches $0$.
  • The single blue line indicates: this is a special case in which the pendulum actually stops $(x_2=0)$ in the upper point, also note that for the exact initial conditions such that we begin with the pendulum in one of the “i” unstable ss, then it will not move (it is a ss after all).
• When you add friction the phase space changes such that the marginally steady states, now become stable steady states, and the flows will all reach this state, with enough time.

• Remeber that $x$ is a vector, in the previous case: $x=⟨x_1,x_2⟩$, so we need to find the solutions $x^{\ast }=⟨x_1^{\ast },x_2^{\ast }⟩$ such that:$\{\begin{array}{l}{x}_2^{\ast }=0\\ -\frac{g}{L}\sin (x_1^{\ast })=0\end{array}$