Lecture 5 - Tue 2020 09 15 - Virtual
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From Lecture 4
Pitchfork bifurcation and forward. Skip hysteresis

STRUCTURAL STABILITY:
 Hand waving arguments [topology of bifurcation diagram].
 Analysis of the normal forms next: Imagine that you have
                      \dot{x} = f(x, r, h),
 where h is a "hidden" parameter you do not know about.  What happens if you
 have some bifurcation for h=0,  but it turns out h is NOT 0?

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Before proceeding: HOW TO (analytically) GET NORMAL FORMS.
Taylor series and asymptotic scalings [Can be extended to multi-D]
Do examples leading to
 \dot{x} = r+x^2              [x=O(eps), r=O(eps^2)]
 \dot{x} = h + r*x - x^3      [x=O(eps), r=O(eps^2), h=O(eps^3)]
 \dot{x} = r*x + h*x^2 - x^3  [x=O(eps), r=O(eps^2), h=O(eps)]
Expand for small perturbations near the critical values for the equilibrium
point and parameter, and collect dominant terms. Their balance gives the
asymptotic scalings. Then the implicit function theorem can be used to justify
the normal form [i.e: leading order reduction is actually exact in some region
near the bifurcation point].
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 Saddle node: ***********
 You know f(0, 0, 0) = 0, f_x(0, 0, 0) = 0, NOTHING else, so assume any other
 partial is, generically not zero. Thus: \dot{x} = a*h+b*r + c*x^2 + ...
 Normal form is then:
                     \dot{x} = h + r + x^2         ***S. STABLE***

 Trans-critical: ***********
 Same analysis yields
                     \dot{x} = h + r*x - x^2.      ***NOT S. STABLE ***
 BUT, if we KNOW x=0
 stays as solution, then \dot{x} = (r+h)*x - x^2; now s. stable.

 Pitchfork: ***********
                     \dot{x} = h + r*x \pm x^3     ***NOT S. STABLE***
 [DO PAGE 18-19 my notes].

 If we KNOW x = 0 stays as solution, only symmetry is broken. Then
 \dot{x} = r*x + h*x^2 \pm x^3. Still not s. stable.
 Show picture of the 2-D surface (with a cusp singularity) this yields.

 Describe perturbed bifurcation diagram [note that this describes what happens
 with, for example, a non-symmetric elastic blade under pressure]:
   The pitchfork breaks into a trans-critical and a nearby saddle-node, where
   the saddle-node is on the side opposite to the bias.
   Note: r*x + h*x^2 + x^3 = (r + h*x + x^2)*x, with zeros
         y = x   and   y = r + h*x + x^2 = (x+h/2)^2 + r - h^2/4.

   Think of a measuring tape blade: If you slowly increase the pressure it
   will always bend towards the side where the tape is concave. But a small
   tap can make it bend the other way, and this bent state exists and is
   stable *below* the pressure at which the tape bends in the biased direction.

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Example: PERTURBED pitchfork: "column buckling" [toy column] and
    "bead on a rotating bed". Add now a small asymmetry; still keeping zero
    as solution [e.g.: more resistance to bending in one direction than the
    other, or loop not quite circular].
    Argue then that: Pitchfork becomes transcritical + saddle node.
    Why? Because zero stays a solution, the bifurcation at zero [generically]
    will be trans-critical, while the other branch of solutions [column bent
    left or right] cannot disappear ... just move around.

    Note conservation of stability still applies!


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