CDS - Logistic Equation • Stability Analysis for Discrete Time Systems

Remember:

Discrete Time Logistic Equation:$x_{t+1}=r{x}_t(1-x_t)$Where: $r\in \left(0,4\right]$, meaning $0$ is excluded. If we search for the steady states we will find:$\{\begin{array}{l}{x}_1^{\ast }=0\\ x_2^{\ast }=\frac{r-1}{r}\end{array}$

Stability Analysis: To study the stability we need first to calculate the derivative of $f$, so for the logistic equation, with $f(x)=rx(1-x)$ :$f^{\prime }=r-2rx$And from this we can say:

• If $\left|f^{\prime }(x^{\ast })\right|<1$ : then the steady state $x^{\ast }$ is asymptotically stable.
• If $\left|f^{\prime }(x^{\ast })\right|>1$ : then the steady state $x^{\ast }$ is unstable.
• If $\left|f^{\prime }(x^{\ast })\right|=1$ : then we cannot say anything about the stability of $x^{\ast }$.

~Ex.: stability of $x_1^{\ast }=0$: $f^{\prime }(x_1^{\ast })=r$Then:

• For: $0<r<1$ : $x_1^{\ast }$ is asymptotically stable.
• For: $1<r\le 4$ : $x_1^{\ast }$ is unstable.
• For $r=1$ then we could have a bifurcation point, notice that:${f^{\prime }(x_1^{\ast })|}_{r=1}=1$For discrete time system the $f=1$, is equivalent to the continus time case where $f=0$.
  • Then we can have a saddle-node bifurcation, a transcritical bifurcation or a pitchfork bifurcation.

~Ex.: stability of $x_2^{\ast }=\frac{r-1}{r}$: $f^{\prime }(x_2^{\ast })=2-r$Then:

• If $0\le r<1$ : $x_2^{\ast }$ is unstable.
• If $1<r<3$ : $x_2^{\ast }$ is asymptotically stable.
• If $3<r\le 4$ : $x_2^{\ast }$ is unstable.
• So for $r=1$ we could have a bifurcation point, since like for the $x_1^{\ast }$ example:${f^{\prime }(x_2^{\ast })|}_{r=1}=1$Like before, we can have a saddle-node bifurcation, a transcritical bifurcation or a pitchfork bifurcation.
• Also for $r=3$:${f^{\prime }(x_2^{\ast })|}_{r=3}=-1$We have another bifurcation point, but in this case we have a flip bifurcation, where the flip bifurcation is the equivalent of the hopf bifurcation for discrete time systems.
  NOTE: we need to verify that this is in fact a flip bifurcation via simulation.

Here’s the bifurcation diagram:
So as a recap we have that:

• $x_1^{\ast }=0$, and $x_2^{\ast }=\frac{r-1}{r}$
• For $0\le r<1$, we have $x_1^{\ast }$ asympt. stable, and $x_2^{\ast }$ unstable.
• For $1<r<3$, we have $x_1^{\ast }$ unstable, and $x_2^{\ast }$ asympt. stable.
• For $3<r<4$, we have $x_1^{\ast }$ unstable, and $x_2^{\ast }$ unstable.
• Here’s a drawing to better undestand:
  

Let’s see the cobweb graph:

1. We start by drawing the bisector line (representing $x_{t+1}=x_t$) and the function $f(x)$, and in this case $f(x_t)=r{x}_t(1-x_t)$, the logistic funciton.
   We choose an arbitrary parameter $r=0.6$:
   
2. We choose an arbitrary initial condition $x_0=0.4$, and start by calculating the next step, meaning $x_1=f(x_0)$:
   
3. We choose the last arbitrary value, the number of steps $(n)$ for drawin the cobweb graph, we have choosen $n=20$:
   And we iterate: $x_2=f(x_1)$, then $x_3=f(x_2)$ and so on…
   
   
4. This confirms that this is an asympotically stable bahaviour, we can also see it by plotting the progession of values, or solution of the discrete time system:
   

We can change the values: $r$ (parameter), $x_0$ (intial conditions) and $n$ (number of steps), to see how the system evolves. logistic_cobweb(r=1.2, x0=0.4, n=20):

logistic_cobweb(r=1.2, x0=0.01, n=20):

logistic_cobweb(r=2.0, x0=0.4, n=20):

logistic_cobweb(r=2.0, x0=0.4, n=20):

logistic_cobweb(r=2.8, x0=0.3, n=20):

logistic_cobweb(r=3, x0=0.3, n=20):

This is called transient dynamic.

logistic_cobweb(r=3.1, x0=0.3, n=100):

logistic_cobweb(r=3.5, x0=0.3, n=100):

• The trajectory now oscillates beetween $4$ points, instead before we could say that it oscillated between $2$ points.
• This is the equivalent of the period doubling in the continous time case.

logistic_cobweb(r=3.8, x0=0.3, n=100):

• We have reached the chaotic regime.

If we draw the bifurcation diagram:

• $r<1$ : the origin is an attractive state.
• $1<r<3$ : the positive state value begins to rise (the oringin is no longer a stable ss).
• $3<r<3.5$ : we have the flip bifurcation, as we have seen in this range we begin exepriencing the first “oscillation” of the system, see this cobweb plot.
• $\sim 3.5<r<\sim 3.6$ : each of the two branches has its onw flip bifurcation, we have seen it in this cobweb plot we have a period doubling, meaning that instead of having just one oscillation between $2$ points, here we have an oscillation between $4$ points.
• $\sim 3.6<r<???$ : one more time, each of the four branches has its onw flip bifurcation.
• Then we have chaos.
• ==As you can see there is a white space in this graph, at about $\sim 3.7$ this is the so called “periodic window”==.

Close up of the periodic windows:

• Here we can see that we have $3/4$ periodic windows, we can count the number of “points” in each of them and we can say that:
  • The first periodic window at $\sim 3.61$ is of period $6$.
  • The second periodic window at $\sim 3.74$ is of period $5$.
  • The first periodic window at $\sim 3.83$ is of period $3$.

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• There is an error in the slide, it should be $r\in \left(0,4\right]$, so $0$ is excluted.

• $f$ is defined as: $f(x)=rx(1-x)$.

• In discrete case we can have a new type of bifurcation: the “flip bifurcation”.

Some calculations (Lecture 20 - Part 2 @ 11:20 ~ 30:30):

• So the flip bifurcation is the equivalent of the hopf bifurcation for discrete time systems.

• Function logistic_cobweb(r, x0, n), where:
  • r : parameter $r$.
  • x0 : initial conditions.
  • n : number of iterations/steps.
• This are the results for logistic_cobweb(r=0.6, x0=0.4, n=20)

• Step 1.

• Step 2

• Step 20.
• This confirms that this is an asympotically stable bahaviour.

• Progression of values.

• logistic_cobweb(r=1.2, x0=0.4, n=20)
• Notice that this time there are two intersections between $f(x)$ and the bisector.

• So we have a steady state $\ne 0$.
• If we calculate it, it is at $\sim 0.1667$.

• logistic_cobweb(r=1.2, x0=0.1, n=20)

• Again it converges to the same ss as before.

• logistic_cobweb(r=2.0, x0=0.4, n=20)
• The ss has moved up, and it is still attractive.

• logistic_cobweb(r=2.0, x0=0.9, n=20)

• logistic_cobweb(r=2.8, x0=0.3, n=20)

• The system oscillate, but still converges to the ss.

• logistic_cobweb(r=3, x0=0.3, n=20)

• Transient dynamic.

• logistic_cobweb(r=3.1, x0=0.3, n=100)

• logistic_cobweb(r=3.3, x0=0.3, n=100)

• logistic_cobweb(r=3.5, x0=0.3, n=100)
• The trajectory now oscillates beetween $4$ points, instead before we could say that it oscillated between $2$ points.

• This is the equivalent of the period doubling.

• logistic_cobweb(r=3.8, x0=0.3, n=100)

• Chaos

• logistic_cobweb(r=3.8, x0=0.1, n=200)
• Like for the continous case, we can see that the trajectory will touch every single point in the phase space, without ever assuming the same value twice.

• Lecture 20 - Part 2 @ 53:10 ~ 54:50

• Lecture 20 - Part 2 @ 55:15 ~ 58:51

• $r<1$ : the origin is an attractive state.
• $1<r<3$ : the positive state value begins to rise (the oringin is no longer a stable ss).
• $3<r<3.5$ : we have the flip bifurcation, and in the graph we represent the two points of the oscillation.
• $\sim 3.5<r<\sim 3.6$ : each of the two branches has its onw flip bifurcation.
• $\sim 3.6<r<???$ : each of the four branches has its onw flip bifurcation.
• Then we have chaos.
• As you can see there is a white space in this graph, at about $\sim 3.7$ this is the so called “periodic window”.

• Close up of the periodic windows.

• Here we can see that we have $3/4$ periodic windows, we can count the number of “points” in each of them and we can say that:
  • The first periodic window at $\sim 3.61$ is of period $6$.
  • The second periodic window at $\sim 3.74$ is of period $5$.
  • The first periodic window at $\sim 3.83$ is of period $3$.