CDS - Lecture 20

• CDS - Lecture 20 ~ Matlab Script ‘mandelbrot’

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• The solution to this system is a sequence of values $\left[x_0,x_1,\dots ,x_N\right]$.
• A discrete time system is also called a map.
• We will only see monodimensional maps, so $x\in {\mathbb{R}}^1$ ($x$ is a scalar and not a vector).
• We will see a logistic map, that shows deterministic chaos, and it is a monodimensional system.

• When we reach a steady state, the state will remain the same, so: $x_{t+1}^{\ast }=x_t^{\ast }$.
• $x_{t+1}=a{x}_t$ is called geometric progression.

• $x_0$ is the initial term.

• This plot is defined as the phase space of maps.
• On the bisector line we have that all $x_{t+1}=x_t$.

• Here we have reported (red line) the function: $x_{t+1}=a{x}_t$, for $a>1$.

• We can report how the system evolves with a positive inital condion $x_0$.

• And also for a negative inital condion.
• In both cases we have divergent progression.

• For $a=1$, any $x$ is a steady state.

• If we represent like before the line $x_{t+1}=x_t$ (which is the bisector), and in red the line representing the function: $x_{t+1}=a{x}_t$, as we can see, for $a=1$, the two lines coincide.

• For $0<a<1$, we have a convergent progression, toward $0$.

• For $a<0$ rember that we have an alternation of the sign, let’s see how.

• Again a convergent progession.

• For $a=-1$.

• For $a<-1$ instead we have a divergent progression, towards $\pm \infty$ (or unsigned infinity).
• These graphical representations are called “cobweb”, and it is similar to how we represent the flow for continous dynamical system.
• As you can see in the discrete case, we now have “jumps” from one value to another.

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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 dicrite 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 $3$.