Chaos (and logistic functions); what is it and is it truly chaotic?

Question

I'm currently studying discrete dynamic models and I am now reading about the logistic function $x_{n+1} = ax_n(1-x_n)$. Below there is a picture what happens with different values of a:

original image

These are for the points $a = (2.25 ; 3,25 ; 3.50 ; 3.56 ; 3.569 ; 3.75)$

The first picture shows as just a normal, stable fixed point. The second one shows a stable period 2 solution. The third one has a non-stable (?) period 2 solution, but a stable period 4 solution. The one after that has a stable period 8 solution, the one after that has a period 16 solution and the one after that doesn't have a period 32 solution, but chaos!

Now I've only heard about chaos from the jurassic park novel, so I don't know what it entails. What does it mean?

If I've understood it correctly; for values below $a<3.75$, you get solutions of a certain period; whether stable or instable. However, with chaos; your solutions don't have a period?

This explanation probably isn't correct, but it gives me a couple of questions;

• What does chaos (in this example) mean? Can you globally predict when there is chaos?

• Purely mathematically, without using graphing calculators or pictures like this one; how could you know you have chaos for certain values of a?

• If you have chaos for a certain point $b$, does that mean all points $>b$ will also have chaos?

• Another question; what does stability and instability imply? I know how to find out if a solution is stable or not, but what does it mean? I used to think stable meant it converges; but how can for example a period 2 solution converge/diverge? It makes a rotation, so how can it converge?

Answer 1

For a rigorous answer, you will need to consult chaos theory and calculate e.g. the Lyapounov exponent for the system to show that it is chaotic. There are different aspects and definitions to chaos, e.g., ergodicity, strong K-property, average negative curvature, KAM, etc. Intuititively, but not very accurately, a system is defined as chaotic if it moves away from its starting state at an exponential rate in an unpredictable (quasi-random) manner. If trajectories diverge, you get chaos. This may occur even after long lingering in some apparently 'stable' regime. For example, with concave mirror, the rays may first focus towards each other, but after passing through the focal point, they diverge (NB: a regular concave mirror is not chaotic; I am only using it as a rough analogy here. For it to be chaotic, it would need to have a rough surface so that rays are diffusely reflected.) There is a nice paperback by Gleich(?) on Chaos that explains this topic in fairly accessible manner.

Answer 2

"However, with chaos; your solutions don't have a period?" Not at all. First, let me call them trajectories, rather than solutions. One popular definition of chaos insists upon having periodic trajectories, in fact, it insists that every open interval, no matter how small, contains points that have periodic trajectories. It's just that those periodic trajectories are all unstable.

"If you have chaos for a certain point $b$, does that mean all points $\gt b$ will also have chaos?" I take it by $b$, you mean the parameter $a$. The answer is no. For example, there are values of $a\gt3.75$ where there are stable periodic points of period 3. As $a$ increases, chaos and order come and go, in an intensely complex way that is (to the best of my knowledge) not all that well-understood. The are many images on the web of what's called the bifurcation diagram for the logistic map, where you can see these transitions from order to chaos and back again. There are also images of what's called the Lyapunov exponent for the logistic map; there's chaos where it's positive, order where it's negative, and again the images show the complex behavior as $a$ increases.

"I used to think stable meant it converges; but how can for example a period 2 solution converge/diverge?" To say a period $2$ trajectory (also known as a $2$-cycle) is stable is to say that any trajectory that starts near the $2$-cycle will converge to the $2$-cycle; that is to say, the even-numbered terms in the trajectory will converge to one point in the $2$-cycle, the odd-numbered terms to the other. To say a periodic trajectory is unstable is to say that even trajectories that start arbitrarily near it will diverge from it.

Answer 3

Let me record a slightly more formal description of the known properties of the logistic family $f_a (x)= a x(1 - x), a \in [0,4]$ on the domain $[-1, 1]$.

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Coexistence phenomena for the logistic map family.

Let us call a parameter $a \in [0,4]$ regular if the associated logistic map $f_a(x) = a x (1 - x)$ has the property that the domain $[-1, 1]$ is covered Lebesgue-almost surely by the basins of a periodic sink; we say that $a \in [0,4]$ is stochastic if the logistic map $f_a$ has the following ``chaotic properties'':

• $f_a$ possesses an ergodic, absolutely continuous invariant measure $\mu$.
• The Lyapunov exponent $\lambda := \lim_{n \to \infty} \frac1n \log |(f^n)'(x)|$ exists for $\mu$-a.e. $x \in [-1,1]$ and is positive (note: by Birkhoff we have that $\lambda = \lambda(x)$ is $\mu$-a.e. constant).

The following results are known:

• (Lyubich and many others): Almost every $a \in [0,4]$ is either stochastic or regular.
• (Jakobson, Benedicks and Carleson; see also Rychlik): The set of stochastic parameters has positive Lebesgue measure in $[0,4]$.
• Exercise for the reader: the set of regular parameters is open and dense in $[0,4]$.

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Let me now answer your question:

Purely mathematically, without using graphing calculators or pictures like this one; how could you know you have chaos for certain values of a?

From the results listed above, the set of stochastic parameters for the logistic map is a positive measure set with a complicated Cantor-like structure. In particular, the 'stochastic' property is extremely structurally unstable: any stochastic parameter $a \in [0,4]$ can be approximated by a sequence $\{ a_n \}$ of regular parameters converging to $a$.

So, unless you get lucky (as you do for example when $a = 4$, so that $f_a$ is conjugate to the tent map), for a given parameter $a \in [0,4]$ it is typically impossible to rigorously verify the stochasticity property, even if you simulate $f_a$ on extremely long timescales and it `looks' like it should be stochastic. All we know is that the set of stochastic parameters has positive Lebesgue measure (and a rigorous lower bound on this Lebesgue measure-- look up the work of Luzzatto and others).

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Postscript: Avila and Moreira showed that for almost every stochastic $a \in [0,4]$, the map $f_a$ has exponential decay of correlations in addition to a positive Lyapunov exponent and an absolutely continuous invariant measure.

Answer 4

The logistic map is one of the more complex systems exhibiting chaos. A simpler system is the iterated Beta Transform (aka the "tent map"). This system separates out more clearly the effects of attractors vs. the effects of ergodic uniformity. There are many tracts on the Beta Transform, but the one that I like the best is the one that I wrote, here: https://linas.org/math/solvit.pdf The first chapter places the logistic eqn and beta transform side-by-side, making it clear which parts correspond to which; this helps establish a intuitive feeling for "what's really going on" and "why". There are other, far more mathematical tracts on the Beta transform; I tried to write a much simpler and more approachable treatment.