CDS - Lecture 10

• For a negative value of $\beta$: $\beta <0$

• For a positive value of $\beta$: $\beta >0$
• This is a “real” example of a limit cycle.

• For a positive value of $\beta$: $\beta >0$, like before, but this time the initial conditions this time are inside the limit cycle.
• NOTE: the size and shape of the limit cycle is the same (even if it does not look like it), look at the scale of the graphs.

• The amplitude/size of the limit cycle depends on $\beta$.

• The limit cycle is part/called an attracting set.

• The same eigenvalues as for $\sigma =-1$

• In this case we have a divergence.

• If you choose inital condition outside of the unstable limit cycle (or better a “repuslive limit cycle”), then you have a divergence.
• We say that for positive value we have “full instability”, and for negative value we have “local stability” (inside the limit cycle).

• In the supercritical hopf bifurcation we found an attractive limit cycle.
• In the subcritical hopf bifurcation we found a repuslive limit cycle.

Let’s try with an example:

• Suppose we have an initial condition that given infinite time always stays in a substet $R$ $(R\in {\mathbb{R}}^2)$, and in this subset there are NO steadys states, then the Poincar’è-Bandixosn Theorem assures us that the system will converge to a limit cycle.
• NOTE: the hopf bifurcations both presents steady states (in $(0,0)$), however for those cases we can still use this theorem, we just “remove” a closed curve near the ss, like so:
  
  • But now is an open subset ????not-sure-about-this

Now let’s try to visualize better the meaning behind the bifurcation diagram:
Each point of the bifurcation diagram represents a phase plane/phase space (with infinite inital conditions), and the lines (dotted and continue) represents the stabilities/attracting set of the phase plane:

For the subcritical case:

Let’s see also the two hopf biffurcation side by side:

• NOTE: since the eigenvalues were the same also the “x-axis” is the same both for the “SUPercritcal” and “SUBcritical”, what chages is the limit cycle.
• NOTE: this graphs is 2D, it should be 3D!

We may find different cases, for different values, here’s another example:

• However if the limit cycles is stable when the ss is unstable, it is still considered a supercritical hopf bifurcation.
• And if the limit cycles is unstable when the ss is stable, it is still considered a subcritical hopf bifurcation.

---

• “undamped” = unstable

• First idea about chaos.

• Fix the sign of $b$, for simplicity.

• So the ss $(\frac{-b}{x-c})$ is an attractor.not-sure-about-this is it always an attractor???
• So the third equations acts as a stabilizer depending if $x\gtrless c$.

• If $a>\frac{c^2}{4b}$ or $a=0$, then the ss will be imaginary.
  Also, for $a=\frac{c^2}{4b}$ the two ss will coincide.
  ⇒ Then we can say that this is a saddle-node bifurcation.

• $S_1=⟨x_1,y_1,z_1⟩$
• $P_2=⟨x_2,y_2,z_2⟩$

• Here we can see, like we already said that for $a=\frac{c^2}{4b}$ the two ss will coincide.

• And like we said previously this is a saddle-node bifurcation.
• As we move $a$ the numerical values of the steady states will vary, this graph represents this movement/variation, remember that $b$, and $c$ are fixed.
  Just to be more clear, in this graph the x-axis represents the values of $a$.

• This is a zoom of the previous graph.
• This will give rise to a supercritical hopf bifurcation (in 3D)

• Here’s the limit cycle in 3D.
• This is calculated via simulation.
• NOTE: in 3D the limit cycle can be curved.

• Since as we have seen before, the limit cycle will increase its radius based on the parameter $a$, then in this case it will intersect the saddle, we will see what happens.

• When it reaches “a certain point” (the red bar), the it will “activate the mechanism” and “double itself”not-sure-about-this I understood this like i understood modern day yu-gi-oh.

• This is the “doubled limit cycle” after “the mechanism” “was activated”.
  This is formaly called a “2 period limit cycle”.
• This “mechanism” will be repeated many times, and every time the limit cycle “will double itself”.

• First figure: period 4 limit cycle
• First figure: period 8 limit cycle
• First figure: chaos.
  In this case no value of the phase space will be visited two times (remember that we are in 3D), so there will be no intersections in the phase space.
• The shape of the chaotic case in not compleatelly irregular as you can see, it is “similar” to the other ones, but it does not have a period, it is aperiod (or chaotic).

For $a$ less then a certain value we have a 1-period limit cycle:

If we increase $a$, then at a cartain point we will find a 2-period limit cycle:

• It remains in the same “region” as the previous 1-period limit cycle.
• The old 1-period limit cycle still exists, but now, it is unstable.

This is the dynamics representation of the 1-period limit cycle:

This is the dynamics representation of the 2-period limit cycle (in red):

If we increase $a$ again, then at a cartain point we will find a 4-period limit cycle:

• The green straight line is there to count the intersection it has with the limit cycle.

If we count the intersections of the green-straight-line with the 1-period LC and 2-period LC:

• 1 intersection with the 1-PLC (1-period limit cycle).
• 2 intersection with the 2-PLC.

Again remember that when $a$ is great enough to have a 4-PLC, then the previous 2-PLC and 1-PLC still exist, but now are unstable, similar to before:

Then, If we increase $a$ again, we will have an 8-period limit cycle. Then, If we increase $a$ again, we will have a chatic/aperiodic behaviour. If we represent the chaotic dynamics:

• It does not diverge, but it has NO period (aperiodic).

Now, consider another inital point, very very near the one we considered before, notice how, after a little time, the two trajectories are compleately different from one another:

• However all trajectories will be confined in the same “region”.
  Even if you start very far away:
  
• That’s way we still talk about attractors, since the trajectors will be attracted to this region.

The fact that two almost-identical initial condictions will have very different trajectories, is defined/called “sensitivity to inital conditions” A more formal definition: “Changing by a very tiny amount the inital condition, then you will not be able to predict the future value of the trajectory”. The two trajectories will be similar only for a small time.

Finally if we increase $a$ again , we will have instability.

What is deterministic chaos?IMPORTANT

• Deterministic means that the equations are deterministic, there are no stochastic inputs or variables in these equations.
• Chaos is an aperiodic dynamics/behaviour, that also shows sensitivity to intial conditions, and it characterized by what are called “strange attractors”.

Chaos is possible only in non-linear systems.