Chaos and Fractals

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Chaos and Fractals

• Chapter 6
• Bifurcations

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Chapter 6

• Family of functions Qc(x) x2 c (with c
  constant)
• It looks simple but..
• Complicated dynamics
• Behavior not understood for some c
• C is a parameter each c ? different system

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Goal

• Understand how dynamics of
• Qc(x) x2 c change when c varies
• ?
• introduce 2 important bifurcations

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Fixed points (FP) solve x2 c x
FP should be real (1-4c) 0 ? c ¼ 1) c gt
¼ no FP 2) c ¼ 1 FP (p p-) 3) c lt ¼ 2 FP
(distinct)
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Case c gt ¼ (does not meet yx) Case c
¼ 1 FP (at x ½) x2
¼ is tangent to line yx Case c lt ¼ 2 FP p
and p- attracting/repelling/neutral ??
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• Look at Qc(x) in the FP p and p-
• Qc(x) 2x

Case for p

• If c ¼ Qc(p) 1 ? neutral FP
• If c lt ¼ Qc(p) gt 1 ? repelling FP

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Case for p- slightly more complicated

• If c ¼ Qc(p-) 1 ? one neutral FP
• If c lt ¼ Qc(p-) lt 1 ? attracting FP
• For which values of c is p- attracting
  ??Condition Qc(p-) lt 1, solving this yields

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• This phenomenon is called a bifurcation
• Saddle-node (or tangent bifurcation)

• Summarized in Proposition on page 54
• For the family Qc(x)x2c
• All orbits tend to infinity if c gt ¼.
• When c ¼, Qc has a single neutral FP at pp-
  ½.
• For c lt ¼, Qc has 2 FP at p and p-. FP p is
  always repelling,a. If -¾ lt c lt ¼, p- is
  attracting,b. If c -¾, p- is neutral,c. If c
  lt -¾ , p- is repelling.

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Second part 6.1

• For c ¼
• interesting dynamics occur in p x p
• Orbits for x lt -p and x gt p tend to infinity
• Qc(-p) p eventually FP !

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• For ¾ lt c lt ¼ orbits in (-p,p) tend to
  attracting FP p-.

• Case c gt ¼, c ¼ and -¾ lt c lt ¼ known, but what
  about c lt -¾ ??

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• When c lt -¾1) p- becomes repelling2) and a
  cycle of period 2 occurs !!To see this solve
  Qc2(x) x
• (x2c)2 c x ? x4 2cx2 c2 c x
  04th order difficult ! But we know 2 solutions
  !(x-p) and (x-p-)

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Solving x2 x (c 1) gives

• FP q and q- of Qc2(x) are FP of
• Qc(x) of period 2 !

Note that (-4c-3) should be 0 c -¾. As
c decreases below ¾, p- changes from attracting
to repelling and a 2-cycle occurs at q and q-.
If c -¾, q q- p- -½.
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• Is this 2-cylce attracting/neutral/repelling
  ?Condition if (Qc2)(q) lt 1, then
  attracting2 ways to solve this (Qc2)
  4x34xc, substitute q 44c lt 1
  (Qc2)(q) lt 1 ? Qc(q)Qc(q-) lt 1 ?
  
  44c lt 1Both cases if -1¼ ltclt -¾, attracting
  2-cycle !And if clt-1¼ repelling 2 cycle.

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• This phenomenon is also called a bifurcation
• Period doubling bifurcation.

• Summarized in Proposition on page 56
• For the family Qc(x)x2c
• For ¾ lt c lt ¼, Qc has attracting FP at p- and
  no 2-cycles.
• For c -¾, Qc has neutral FP at p-q and no
  2-cycles.
• For -1¼ lt c lt -¾, Qc has repelling FP at p and
  an attracting 2 cycle at q.

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Section 6.2

• Look at more general family of functions F? to
  study bifurcations.
• For each ? F?(x) is function of x
• Examples ?x(1-x), ? sin(x), ex ?
• Bifurcations occur in F? when there is change in
  fixed or period point structure as ? passes
  through some particular value ?0.

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Saddle-Node bifurcation

• DefinitionA one parameter family of functions
  F? undergoes a saddle-node (or tangent)
  bifurcation at the parameter value ?0, if there
  is open interval I and e gt 0 st1) For ?0-e lt ?
  lt ?0, F? has no FP in I2) For ??0, F? has one
  neutral FP in I.3) For ?0 lt ? lt ?0e, F? has 2
  FP in I, one attracting, one repelling.

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Remarks

• Reversed direction also bifurcation possible
• Also saddle-node bifurcation of periodic points.
  
• Saddle-node bifurcation typically occurs
  whenF?0(x0) 1 and F?0(x0) ? 0 (concave and
  one FP x0).

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Typical saddle-node bifurcation
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Bifurcation diagram

• 2 dimensional plot
• X-axis value of ?
• Y-axis fixed points xExample Q?(x) x2 ?,
  saddle-node bifur. at ?¼ ? gt ¼ 0
  FP ? ¼ 1 FP ? lt ¼ 2 FP

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• Another example
• family E?(x)ex? saddle-node bifurcation when
  ?-1E-1 (0)0E-1 (0)1E-1(0)1

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6.3 Period-Doubling bifurcation

• Definition A one-parameter family of functions
  F? undergoes a period-doubling bifurcation at
  ??0 if there is open interval I and e gt 0
  st1. For each ? in ?0- e, ?0 e, there is
  unique FP p? for F? in I.2. For ?0- e lt ?
  ?0, F? has no 2-cycles in I and p is
  attracting (resp. repelling).3. For ?0 lt ? lt
  ?0e, there is unique 2-cycle q1?, q2? in I
  with F?(q1?) q2?.This 2-cycle is attracting
  (resp. repelling). Meanwhile the FP p? is
  repelling (resp. attracting).4. As ? ? ?0,
  we have qi? ? p?0

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Remarks

• 2 cases for period-doubling when ? changesFP
  changes attracting?repelling attracting 2-cycle
  orFP changes repelling? attracting repelling
  2-cycle
• Direction of ? may be reversed.
• Cycle can period-double too n-cycle ? 2n-cycle
• Period-doubling occurs when F? - diagonal yx
  F?(x) -1

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• Example Qc(x) x2 c, period-doubling at
  c -¾. Look at p-

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saddle-node
period-doubling
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Last example page 65
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• F?(x) ?x-x3
• FP F?(0)0 for all ?
• -1 lt ? lt 1 FP attracting ? -1 FP neutral ?
  lt -1 FP repellingWhy ? F?(x) ? 3x2 and
  F?(0) ? Obvious !
• Odd function F?(-x) -F?(x) for all x.F(x0)
  -x0 after 1 applyF(-x0) -F(x0) F2(x0) x0,
  so solve F?(x0) -x
• ?x-x3x0, x0 or x1,2 Sqrt(?1)
• Repelling or attracting ?(F2?)(x1) (F?)(x1)
  (F?)(x2) (?-3x2)(?-3x2) 4?2 12? 9For ?
  -1 value 1 neutralFor ? -1/2 value 4
  repelling