One of the most fascinating things about complex analysis is that it provides insights into real analysis.

Here are two pictures from Needham's book Visual Complex Analysis (p. 65):

Picture 1 (real). Consider the following two functions. real

Both have the radius of convergence $R = 1$. But while it's clear what is the problem with [a] (its behavior at $x =\pm 1$), it's not clear what's stopping [b] from having a larger radius of convergence.

Picture 2 (complex). The mystery is revealed when we realize that from the point of view of the complex analysis, both [a] and [b] are slices of the same function and the radius of convergence is just the distance to the nearest pole.


Is there something similar (not necessarily with pictures) regarding complex dynamics? Does complex dynamics offer any insights into real dynamics?

Edit: by "real dynamics" I mean dynamics of real maps.

  • $\begingroup$ What do you mean by "real dynamics"? Are you looking for examples that generalize real analysis in a nice way, or are you looking for real-world applications of complex analysis? $\endgroup$ Sep 8, 2016 at 11:51
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    $\begingroup$ I think OP means dynamics of real maps $\endgroup$
    – Albert
    Sep 8, 2016 at 12:11
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    $\begingroup$ @Lovsovs: as Glougloubarbaki pointed out, I meant dynamics of real maps. I should have made it more clear. It's sometimes easy to forget that the world out there is also referred to as "real". $\endgroup$
    – Leo
    Sep 8, 2016 at 12:31
  • $\begingroup$ I don't have time for more than a comment (the graphics in the question and the answer by @Mark McClure are beautiful), but the Andronov-Hopf bifurcation is understood by movement of eigenvalues off the complex circle, and it's usefulness is in the onset of turbulence in fluids, which is a real dynamics problem. $\endgroup$
    – hkr
    Sep 8, 2016 at 13:30

1 Answer 1


Two reasons (among others) that an understanding of complex variables adds to our understanding of real variables are

  • The real axis fits nicely along the real axis of the complex plane and
  • The complex numbers are algebraically complete - i.e. every polynomial of degree $n$ has exactly $n$ roots.

Example 1

Suppose we apply Newton's method to the polynomial $f(x)=x^2+1$. We find the Newton's method iteration function $N(x) = x/2 - 1/(2x)$ and if we iterate it from a random starting point, we'll see chaotic behavior. Here's a pretty standard cobweb plot for this function:

enter image description here

From the point of view of complex dynamics, the real axis is right between two attractive fixed points at $\pm i$ thus the chaos makes perfect sense. In fact, $N(x)$ is conjugate to the square function via the Mobius transformation $\varphi(z) = (z-i)/(z+i)$ since $$\varphi(z)^2 = \frac{(z-i)^2}{(z+i)^2} = N(\varphi(z).$$ Thus, the dynamics of $N$ on the real line are isomorphic to the dynamics of $z^2$ on the unit circle, which are well known to be chaotic.

Example 2

On the left of the following animation, we see the graph of $f_c(x)=x^2+c$ for $0 \leq c \leq 1/2$ together with the line $y=x$. This clearly shows the collision of two fixed points - one attractive and the other repulsive. They briefly merge into one neutral fixed point and then disappear all together. On the right, we move to the complex plane and plot those same fixed points in the Julia sets for those polynomials. From this perspective, they never really disappeared at all.

enter image description here

I guess this is a rather special case of the fact that every complex polynomial has periodic points of every order while real polynomials do not.

Example 3

The image below shows the relationship between the bifurcation diagram for the real family $f_c(x)=x^2+c$ and the Mandelbrot set. Specifically, you can see how the periodic windows in the bifurcation diagram align with stable regions in the Mandelbrot set.

enter image description here

Example 4

This answer describes the nature of the real iterates of $f(x)=x^2-x-1$. It's not too hard to show that there is an attractive three-cycle. Of course, the famous Period Three Theorem implies that there is chaotic behavior, though, it's difficult to see. The reason is that the chaotic set is a Cantor set, when restricted to the real line. It's much more clear to see, when look at the whole Julia set:

enter image description here

The large white regions contain the intervals complementary to the Cantor set where the chaos lies.

Example 5

Finally, one of the most important theoretical results in complex dynamics is the following:

For a rational map of degree at least 2, every attractive orbit contains at least one critical point in its basin of attraction.

The corresponding result for real dynamics is quite a bit more complicated - we must add the assumption that the function under consideration has negative Schwarzian derivative.

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    $\begingroup$ while it is very interesting, I don't think the first example really sheds light on the real case. I mean, clearly the real situation is much easier to understand than parabolic implosion =) $\endgroup$
    – Albert
    Sep 8, 2016 at 13:12
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    $\begingroup$ @Glougloubarbaki hmm... Taken to it's logical conclusion, your comment seems to suggest that the fundamental theorem of algebra sheds no light on the roots of real polynomials. Certainly, it's easy to say that $x^2+1$ has no roots, but... That example is also closely related to the recent comment of hkr to the original question. I really think it's quite natural. $\endgroup$ Sep 8, 2016 at 13:42
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    $\begingroup$ I would also mention that all known proofs of density of hyperbolicity in real one-dimensional dynamics goes through the complex plane in one way or another. This is one of the central questions of one-dimensional dynamics. In the real quadratic family, it means that the "period windows" are dense in the diagram, and is a celebrated theorem of Lyubich and (independently) Graczyk and Swiatek. $\endgroup$ Nov 28, 2019 at 12:02
  • $\begingroup$ to example 2 see also 3d image commons.wikimedia.org/wiki/… $\endgroup$
    – Adam
    Apr 30, 2022 at 7:51

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