Lecture18 Thu 2024 11 07
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          Poincar\'e map has a pair of complex eigenvalues crossing unit circle.
Show what this means geometrically: nearby orbits cork-screw around limit cycle.

Geometrical argument similar to the one for the Hopf shows that an "invariant"
torus enclosing the limit cycle arises (size sqrt of distance to bifurcation).
Hopf-on-a-Hopf. Can be super of sub-critical.

Orbits on this invariant Torii have two periods: one going around the limit
cycle (related to the argument of the pair of complex eigenvalues) and another
going along the limit cycle (bsically, same as limit cycle).

If the two periods are not rationally related, these orbits do not close and
end up being dense on the torus (typical case) ... quasi-periodic orbits. In
the exceptional case, they are periodic.

Map torus to rectangle and explain how this works there.
Another system where this happens: Lisanjous figures.
  Produced when, in the plane, x=x(t) and y=y(t) are sinusoidals, with
  (possibly) different frequencies. Easy to achieve on the screen of an
  oscillator (or by plotting the curve (x, y) numerically).
See file posted with links to videos next to this lecture.
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               Higher order bifurcations and Landau's hypothesis for turbulence.
In principle the process described above can continue, with further Hopfs
piling on each other [Hopf on a Hopf on a ...], giving rise to quasiperiodic
solutions with 2, 3, 4, ... periods.                                        [A]
This leads to increasingly complicated looking functions, which trace
torii of dimensions 2, 3, ... [Tor_n = (S_1)^n]

This lead to Landau's hypothesis for turbulence:                            [L]
     [A] continues to infinity, and turbulence occurs in the limit.
     Turbulence is an almost periodic process [infinite frequencies]

Why are these functions called almost periodic?
Defined by the property: for any 0 < epsilon, there is a sequence of "almost"
   periods T_n, with T_n \to \infty as n \to \infty, such that
   |f(t+T_n) - f(t)| < epsilon, for all t and n.
Any such function has a Fourier series  f(t) = \sum f_n e^{i w_n t}
for some real w_n.
If the w_n are "generated" by a finite set of frequencies
       w_n = m_1 omega_1 + ... + m_p omega_p   [m_j integers]
then f is called quasi-periodic of p-periods.

Explain why this hypothesis is now believed to be false.
   Dissipation shrinks volume argument ............... "weak".
   Failure to satisfy sensitivity to IC argument ..... "strong".

Ruelle and Takens (1971), limited version of [L]:  After 3 Hopfs the solutions
land on a Tor_3, which is 3-D. This is high enough for chaos, and a transition
to chaos happens there generically.  See the list of references posted next to
this lecture.
Let me point out that these papers are very hard reads, with "high level" math.
(a lot of it). The papers do not posit an actual mechanism^#1 for what happens
in turbulence. What they show is that, generically^#2 in 3-D, even if there is
no chaos,  small perturbations will produce it.   It is a structural stability
argument [same reason centers are rare].

#1 Note that papers often talk about the Ruelle and Takens mechanism when they
   find chaos after 3-Hopfs. But it is the wrong name.  What these papers show
   [say, experimentally] is that Ruelle and Takens are right: chaos is generic
   in 3-D.

#2 Employing high level versions of the sort of arguments that I have used in
   the lecture to explore "generic" bifurcations:  Do not look at things that
   that are structurally unstable [e.g. they require some coefficient to be 0,
   without any specific reason --- like a symmetry].

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            Back to limit cycles and stability via the Poincar\'e map P. Case 2.
Real eigenvalue crosses though z = -1 (not possible in 2-D).

Show what this means for nearby orbits.
Can lead to a period doubling bifurcation.
Again, a sequence of period doublings produces ever more complicated solutions.
The limit of infinite such things can lead to chaos [including sensitivity to
initial conditions].
We will look at this with more detail when we look at the Rossler attractor
   later on.
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Getting periods for periodic solutions with high accuracy [Pincar\'e-Lindstedt].
Weakly Nonlinear Things: Oscillators.
Do section 1 (intro) and subsection 1.2. Students should read subsection 1.1
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