Lecture 17 - Tue 2024 11 05
Local bifurcations in the plane. Same as 1-D plus Hopf.
% ===============================================================================
%
% ----------------------------------------------------------------------------- %
                                   Bifurcations of critical points in the plane.
Consider a stable critical point that becomes unstable, what can happen?
As usual, study by degree of generality.

CASE 1. Single real eigenvalue crosses from negative to positive.
        Explain how this gives rise (locally) to an attracting curve where
the dynamics happens. This gives rise then to the same bifurcations we saw
in 1-D. Write normal forms
--- Saddle Node:   \dot{x} = r - x^2, and \dot{y} = - lambda y (lambda > 0)
                   Do picture and show, saddle and node collide and vanish.
--- Transcritical: \dot{x} = r x - x^2, and \dot{y} = - lambda y
                   Do picture, saddle and node go through each other and
                               exchange type.
--- Pitchforks ... do the same.
Note: in 3-D and higher, \dot{y} = - lambda y gets replaced by
                   \dot{\vec{y}} = A y, where the eigenvalues of A have
      negative real parts.

CASE 2. Pair of complex eigenvalues, lambda and lambda*, cross imaginary axis.
        Then motion gets restricted to an attractive surface.
At crossing (bifurcation) linear terms give rise to a center at the fixed point,
   with slight spiral motions on each side of the bifurcation.
Argue that the leading order effect of the nonlinear terms is cubic on the
   (near elliptical) orbits. Why: because quadratic terms cancel upon reflection
   across the CP, while the cubic add up.
Explain how the balance of linear and nonlinear effects give rise to a limit
cycle:
Nonlinearity    stabilizes: supercritical Hopf bifurcation.
                            Stable limit cycle on re(lambda) > 0.
Nonlinearity de-stabilizes: subcritical   Hopf bifurcation.
                            Unstable limit cycle on re(lambda) < 0.
When cubic nonlinear terms is nonzero (generic case) limit cycle size goes like
     sqrt of the absolute value of the real part of the eigenvalue.

Do bifurcation diagrams and explain difference. Note analogy with pitchfork.

NORMAL FORM   \dot{r} = mu r + sigma r^3   and   \dot{theta} = nu,
where mu = re(lambda); nu = im(lambda).
Later we will see how to compute sigma.
Numerically you can detect if nonlinearity stabilizes or not by looking at
the behavior of the system (near the CP) for re(lambda) = 0.
Degenerate case: sigma = 0. Then must look at higher order nonlinearity.
%
% ----------------------------------------------------------------------------- %
                                   Stability of limit cycles and Poincar\'e Map
Work in 3-D here, to be able to draw.
Define Poincar\'e map and show how linear stability can be ascertained by
       looking at the linearization near critical point.  x_{n+1} = P x_n.

Numerical interlude: Computation of the matrix P requires computing derivatives
   with respect of initial data of the solutions. Show how this can be done by
   solving an additional set of ode.
Note: 3-D yields P a 2x2 matrix
      2-D yields P a constant, which must be positive because orbits
          stay on the same side of the limit cycle.

Stability:   eigenvalues of P inside  the unit circle (|z| < 1).
Instability: eigenvalues of P outside the unit circle (|z| > 1).
Three ways to loose stability:
  1) Real eigenvalue crosses though z =  1.
  2) Real eigenvalue crosses though z = -1 (not possible in 2-D).
  3) Pair of complex eigenvalues cross unit circle (not possible in 2-D).
We will get back to (1-2) later.
%
% ----------------------------------------------------------------------------- %
          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 [explain].
%
% ===============================================================================
EOF