Discrete-time Dynamical Systems
Dr Matthew Science University of Auckland
Limit Sets
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A limit set is a set of one or more states that the system would be in if we could ‘fast-forward’ time, and look at the infinite future…
lim t→∞ xt
Fixed points, periodic orbits, and chaotic sets are all examples of limit sets. We’ve seen examples of each of these in the logistic map.
We’ve also seen that a single system can have multiple attractors—which one the system falls into depends upon the initial state of the system.
Qualitative change
In the last section, we saw that the behaviour of a system changes qualitatively as a parameter increases.
Specifically, as we increased r, the system went from
– having a single non-zero fixed point ‘attractor’ to (r=2.5)
– having a periodic attractor (r=3.2), to
– having a chaotic attractor (r=3.6)
– (always in addition to the x = 0
fixed-point)
How do these qualitative changes in the system’s dynamics occur? One way to gain some insight into these transitions is to generate a bifurcation diagram.
For r < ~3.0, the system approaches a For ~3,9 < r < ~3.4, the system For various values r > 3.5, the single value, close to 0.6. oscillates between two values. system’s trajectory is chaotic.
To generate this plot…
I simulated 10,000 trajectories, each with a different value of r (between 2.4 and 3.9). i.e. one for each ‘column’ of the figure.
Half of these started with an initial condition of x0 = 0.5 (black dots) and half with x0 = 0.3 (red dots).
For each trajectory I simulated 10,000 iterations. I then plotted the state of the system during the final 2500 iterations of that run against the value of r.
Put another way, this is a scatter plot, where each point plotted is
(r, xt for t ∈[7500,10000] ).
Bifurcation points (values of r) where the system’s dynamics qualitatively change.
Interestingly, for certain values of r > 3.6, the system returns to having a periodic attractor. You could write a program to explore some of these regions (you should be able to write a bifurcation diagram-generating program like this for the exam).
Looking at the code…
logistic_map_bif.py
Cobweb Plot
What is another way that we can come to understand what is going on here?
Cobweb plots are a useful tool for visualizing 1-dimensional iterated maps.
Let’s look at an example. To start, we can plot xt as a function of xt-1
..this is an inverted parabola with roots at 1 and 0, and a height which is influenced by the parameter r, which we’ll set to 3.2 for now.
Cobweb Plot
We then add the diagonal line
… and with these lines in place, it is possible to trace out a trajectory of this system.
Cobweb Plot
We then add the diagonal line
… and with these lines in place, it is possible to trace out a trajectory of this system.
x1 = f(0.1) ≅ 0.3
Cobweb Plot
We then add the diagonal line
… and with these lines in place, it is possible to trace out a trajectory of this system.
x1 = f(0.1) ≅ 0.3
x2 = f(0.3) ≅ ?
Cobweb Plot
We then add the diagonal line
… and with these lines in place, it is possible to trace out a trajectory of this system.
x1 = f(0.1) ≅ 0.3
x2 = f(0.3) ≅ 0.65
x3 = f(0.65) ≅ ?
Cobweb Plot
We then add the diagonal line
… and with these lines in place, it is possible to trace out a trajectory of this system.
x1 = f(0.1) ≅ 0.3
x2 = f(0.3) ≅ 0.65
x3 = f(0.65) ≅ 0.72
Here is the cobweb plot for r=3.6, and following this link, we can see how the cobweb plot changes as r changes.
https://en.wikipedia.org/wiki/Cobweb_plot# /media/File:LogisticCobwebChaos.gif
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