I'm currently studying period-doubling bifurcations, and being a physicist I want some physical intuition for what they are!
From what I've gathered, the system spontaneously undergoes a period-doubling transformation, obviously doubling the period of the dynamic system.
My question is does anybody have a physical example of a system that could undergo period-doubling bifurcation? Also, I am having trouble understanding the standard period-doubling bifurcation graph. It looks as if you have one branch, then it separates into two branches, then those separate...etc. I don't really understand how this corresponds to period doubling though. Can someone give me an idea of what these axis really represent and why two more branches corresponds to the system's period doubling? Thanks!
These result from an iterated system: $s_{n+1} = f(s_n)$. So suppose you have a system that is $p$-cyclic: $s_1\to s_2 \to \cdots \to s_p \to s_1$. Then each of the points in the $p$-cycle is a stable fixed point of $g(x) = f^p(x)$, where $f^p$ represents the $p$th iterate of $f$.
Now think about what happens as the parameter changes, and these fixed points of $g(x)$ become unstable. Since all these stable fixed points are iterates of each other, what happens at one fixed point will happen to them all.
Stability is determined by what happens to $g'(s_k)$ - it is stable if $|g'(s_k)| < 1$. So as it becomes unstable, it has two ways to go. Either $g'(s_k)$ changes from less than $1$ to more than $1$, or it changes from being larger than $-1$ to being smaller than $-1$.
Think about what happens to the fixed point if $g'(s_k)$ is close to $1$. TAny point close to the fixed point will have its iterates converge monotonically to the fixed point. So as $g'(s_k)$ becomes larger than $1$, then either the fixed point will merge with another fixed point, and they will switch stability status, or the fixed points will utterly disappear. Either way, it is rather boring.
Now suppose $g'(s_k)$ is close to $-1$. Then any point close to $s_k$ will have its iterates oscillate either side of $s_k$ as they converge to $s_k$. Thus in the generic case, when the fixed point becomes unstable, the oscillating convergence will become an oscillating divergence.
I am going to argue that there is a good chance that each alternate terms in the sequence will each converge to a different fixed point. (If they don't, then again the iterates will fly off, and all will be boring.)
Think about $h(x) = g(g(x))$. Let's think about its value when $x = s_k$, and the parameter is such $g'(s_k)$ is close to $-1$. Then $h'(s_k) = g'(g(s_k))g'(s_k) = (g'(s_k))^2$ is close to $1$. But also $h''(s_k) = g''(g(s_k))(g'(s_k))^2 + g'(g(s_k))g''(s_k) = g''(s_k) ((g'(s_k))^2 - g'(s_k))$ is close to zero. That is, $s_k$ is close to being a point of inflection for $h(x)$. But then as $h'(s_k)$ gets larger than one, there is a 50-50 chance (depending on the sign of the third derivative of $h$) that the single fixed point will spawn into three fixed points close to each other, which are alternatingly stable, unstable, and then stable.
So the one fixed point is reasonably likely to become a two cycle.
And a two cycle for $g(x)$ is a $2p$ cycle for $f(x)$.