CDS - Linear 2D System • Example of Vector Fields

Remember:

This is the system: $\{\begin{array}{l}\dot{x}=ax\\ \dot{y}=-y\end{array}$We will see how it changes, and how we can represent the vector field, at different values of $a$.

To have a better understanding on how you can calculate the vector field, and nullclines aslo refer to this other example.

First we represent $\dot{y}=-y$, this is its vector field:
$y=0$ is a steady state

We can represent the flow of $\dot{y}=-y$ in its vector field:

• On the left of the steady state, we have that $\dot{y}>0$, then the flow will point to the right.
• On the right of the steady state, we have that $\dot{y}<0$, then the flow will point to the left.
• We can conclude that $y=0$ is a stable steady state.

Now we can study $\dot{x}=ax$, for $a<-1$:

• Like before the vector field and so the flow are very similar.

Let’s combine this two solutions, and let’s draw the nullclines, so let’s find for $\dot{x}=0$ and $\dot{y}=0$, so:$\begin{array}{l}\dot{x}=0\Rightarrow x=0\\ \dot{y}=0\Rightarrow y=0\end{array}$For this case preatty simple, let’s draw the nullclines:
Then we can we can represent the flow (for $a<0$):
Note that:

• For $x\to \infty$ we have that the flow will be parallel to the $x$ axis.
• For $x\to 0$ (the ss) we have that the flow will be parallel to the $y$ axis.

If we represent the “motion of a family of particles” (the flow ${\varphi }_t$):

• This form is becasue $a<-1$, so the velocity along $x$ is higher than the velocity along $y$.
• The velocities we said can be seen as $\dot{x}$ and $\dot{y}$.

If we take $-1<a<0$:

• The form slightly changes since now, the velocity along $x$ is lower than the velocity along $y$.

Special case for $a=-1$:

• In all of these cases this steady state is called a “node”.
• In this particular case, it is called a “node star”

Instead for $a>0$:
This is an unstable steady state, since:

• $\dot{x}\to 0^{+}>0$
• $\dot{x}\to 0^{-}<0$

The nullclines changes:

If we represent ${\varphi }_t$ for $a>0$:

• All initial condition will diverge, except and initial condition such that it lies along the $y$ axis, so for $x=0$.
  • For $x_0=0$ the system will converge to $0$ $(\forall y_0)$.
  • For $x_0\ne 0$ the system will diverge $(\forall y_0)$.
• So one component converges $(y)$ and the other $(x)$ diverges, when that is the case we obtain this graph, called a “saddle”, name taken from the form of the vector field.

Another special case: for $a=0$:
We cannot represent any arrow, and it said to be marginally stable. And the flow ${\varphi }_t$ for $a=0$:

• Called “line of steady states”.

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• This is a special linear system, $\dot{x}$ depends only on $x$ and $\dot{y}$ depends only on $y$.

• General case.

If we solve analytically we find:$x(t)=c{e}^{at}$We can define:$x_0=c{e}^{a\cdot 0}=c$And we can find:$\begin{array}{l}x(t)=x_0{e}^{at}\\ y(t)=y_0{e}^{-t}\end{array}$However remeber, that we will not find analytic solutions.

We can represent the graph $\dot{y}=-y$:

• $y=0$ is a steady state.

Let’s represent the flow:
On the right of the ss (steady state) the derivitave over time ($y$) is positive.
While on the left $\dot{y}<0$, so:

• As we can see $y=0$ is a stable steady state.

Now we can study $\dot{x}=ax$, notice that $a$ can assume different values, mainly we will focus on $a>0$ and for $a<0$. For $a<-1$, basically the graph is the same as before (for $\dot{y}=y$):

Let’s combine this two solutions, and let’s draw the nullclines, so let’s find for $\dot{x}=0$ and $\dot{y}=0$, so:$\begin{array}{l}\dot{x}=0\Rightarrow x=0\\ \dot{y}=0\Rightarrow y=0\end{array}$For this case preatty simple, let’s draw the nullclines:

Now we can represent the flow:

• For $x\to \infty$ we have that the flow will be parallel to the $x$ axis.
• For $x\to 0$ (the ss) we have that the flow will be parallel to the $y$ axis.

If we represent the “motion of a family of particles” (the flow ${\varphi }_t$):

• This form is becasue $a<-1$, so the velocity along $x$ is higher than the velocity along $y$.
• The velocities we said can be seen as $\dot{x}$ and $\dot{y}$.

If we take $-1<a<0$:

Special case for $a=-1$:

• In all of these cases the node is called “node”.
• In this particular case, it is called a “node star”

Let’s see what happens for $a>0$:

• This is an unstable steady state, since:
  • $\dot{x}\to 0^{+}>0$
  • $\dot{x}\to 0^{-}<0$

The nullclines changes:

If we represent ${\varphi }_t$ :

• All initial condition will diverge, except and initial condition such that it lies along the $y$ axis, so for $x=0$.
  • For $x_0=0$ the system will converge to $0$.
  • For $x_0\ne 0$ the system will diverge.
• So one component converges $(y)$ and another $(x)$ diverges, for this reason this graph is called a “saddle”, sincenot-sure-about-this

For $a=0$:

• We cannot represent any arrow, and it said to be marginally stable.

And the flow ${\varphi }_t$ for $a=0$:

• Called “line of steady states”.

As a sneak-peak for the future, in this system we can define the eigenvalues as:${\lambda }_1=a$ and ${\lambda }_1=-1$, the sign of the eigenvalues is strictly correleted to the stability of the steady states.