The dynamics of maps close to z2 + c

Bruce Peckham

University of Minnesota DuluthPhysics Seminar

October 31, 2013

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Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Overview

General family to study: z 7→ zn + c + β/zd

Dynamical Systems Philosophy

Background: z2 + c

Motivation from earlier studies

Escape sets for z 7→ zn + c + β/zd : holes in filled Julia sets

Special case: zn + β/zn; radial symmetry, proofs

Polar coordinate representationModulus mapAngular mapFull dynamicsβ parameter spaceComparison with zn + λ/zn

A parameter space picture for z2 + β/z

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Dynamical Systems Introduction

A discrete dynamical system is defined by

xn+1 = fa(xn)

A recursion function fa and an initial condition x0 generate asequence: x0, x1 = fa(x0), x2 = fa(x1), ...

Goal: describe the (long term) behavior of the sequence

How does it depend on x0? (Basins of attraction)How does it depend on fa (ie, on a)? (Bifurcations)Phase space or dynamic space (x ∈ X ) vs parameter space(a ∈ A).

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Graphical Iteration

Example: Qc(x) = x2 + c where x is a real (dynamic) variable andc is a real parameter.Graphical iteration for x2 + 0. Bounded orbits??x0 = −3/4, x1 = (−3/4)2 = 9/16, x2 = (9/16)2, ...

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

z2 + c . Both z and c are now complex!

A brief summary of the dynamics of z2 + c :

Kc (filled Julia set) = the set of bounded orbits: Connectedvs totally disconnectedJc (Julia set) = boundary of Kc , chaos

c = 0 c = 0.27

z0 inside, outside, boundary z0 in Cantor set, outside

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Julia Sets for z2 + c

Escape algorithms (rely on Escape Theorem)Definitions of Jc : boundary of Kc , closure of repelling periodicpoints, chaotic dynamicsSoftware demo for different c valueshttp://lycophron.com/Math/Devaney.htmlThe Mandelbrot set

The connectedness locus: Dichotomy for Julia setsThe critical orbit bounded setBulbs

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Motivation

Motivation: combine studies

Holomorphic singular perturbations (Devaney, Blanchard,Josic, Uminsky, ... ):

z 7→ zn + c + λ/zd

Nonholomorphic nonsingular perturbations (BP earlier workpubl. in 1998 and 2000, UMD grad student Jon Drexler1996). C vs. <2:

z2 + c + Az

General Family (escape pictures)

z 7→ zn + c + β/zd

Special case (results - coauthor UMD grad student Brett Bozyk2012, paper in press)

z 7→ zn + β/zn

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Escape pictures - Symmetry

(a) z2 + 0.05/z2 (b) z2 + 0.25− .004/z2 (c) z2 − 1− 0.001/z2

(e)z3 + (0.49 + 0.049i)(d) z3 − 0.125/z3 −0.001/z3 (f)z3 + (0.1 + 0.1i)/z1

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Escape pictures: compare with z2 + c

z 7→ z2 + c + β/z2

Punching holes in the filled Julia sets for z2 + c :

(a) z2 + 0.05/z2 (b) z2 + 0.25− .004/z2 (c) z2 − 1− 0.001/z2

(a’) z2 (b’) z2 + 0.25 (c’) z2 − 1

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Special case: n = d AND c = 0

Restrict toz 7→ zn + β/zn

Observations:

1 In polar coordinates, the modulus component decouples:circles map to circles

2 Escape results are completely determined by the modulus map

3 The full planar dynamics - attractors and basins – isdominated by 1D unimodal dynamics of the modulus map:bifurcation sequences will resemble those for x2 + c (x , c real)

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

The Modulus and Angular Maps

Restrict toz 7→ zn + β/zn

Substituting z = re iθ,

Fn,β

(re iθ)

=(re iθ)n

+β

(re−iθ)n =

(rn +

β

rn

)e inθ

=

(rn +

β1rn

+ iβ2rn

)e inθ

Pn,β

(rθ

)≡(Mn,β (r)An,β (r , θ)

)=

√r2n + 2β1 +

β21+β

22

r2n

nθ + Arg(rn + β

rn

) .

r decouples!

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

(Conjugate) Modulus Map dynamics

a) b) c)

d) e) f)

Six graphs and critical orbits for M̃3,β for β values decreasingalong the ray φ = arg(β) = π/3. Not symmetric about the criticalpoint.

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Full dynamics example

T I B0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

Full dynamics: Red attractor Modulus map

z3 + 0.04/z3

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Parameter plane trichotomy

The Escape Set Trichotomy for z3 + β/z3:

i all orbits go off to infinity,

ii only an annulus of points stays bounded (shaded)

iii only a Cantor set of circles stays bounded.

In cases (ii) and (iii), there is a transitive invariant set; this set isan attractor in case (ii).

iii

a

b

c

de

f

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i

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Full dynamics

Results: Radial map M̃ from unimodal (1D) map theory. Resultsfor F (2D) proved.

M̃ F

K bounded K × S bounded

Critical point c Critical circle {c} × SA attractor A× S attractor

A transitive A× S transitive

A periodic pts dense A× S periodic pts dense

A chaotic A× S chaotic

1D results imply three cases for A when β is in the “strip”:

Periodic orbit

Cantor set (Feigenbaum points)

Union of intervals (homoclinic points)

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Other values of n

n > 3 similar, but width of strip shrinks

n = 2 interesting near β = 0: “full family unless path goesthrough origin, then miss only last parameter point

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Parameter plane β2 = 0 cut b) β2 = 0.1 cut

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Holomorphic Comparison - Dynamic Space

Agreement on the real line!!

z3 − 0.125z3

z3 − 0.125z3

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Holomorphic Comparison - Parameter Space

Agreement on the POSITIVE real line!!z3 + β/z3 z3 + λ/z3

β plane λ plane

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

A Generalization

Numerical experiments for more general cases: z2 + β/z1. Follow3 points on critical circle.

β plane Phase planes in parameter array

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

Summary

Take-home messages and themes

Complex analytic maps are a very special case of maps of theplane.

Perturbation/Continuation

Singularities

Reduction/Generalization

Experimentation/Analysis

Mathematical vs applications approach

Levels of appreciation

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

References

Bozyk B D and Peckham B B, “Nonholomorphic SingularContinuation: a case with radial symmetry,” to appear inIJBC, 2013.

Email bpeckham@d.umn.edu for preprint with full list of references.

See especially:

Guckenheimer J [1979], “Sensitive dependence to initialconditions for one dimensional maps,” Commun. Math. Phys.70, 133-160.

De Melo W and van Strein S [1993], One-dimensionalDynamics, A Series of Modern Surveys in Mathematics, 3(25)(Springer-Verlag, Berlin, New York).

Bozyk B., Non-Analytic Singular Continuations of ComplexAnalytic Dynamical Systems [2012], Master’s thesis,University of Minnesota Duluth, 2012.

Kraft, R, Some One-dimensional Dynamics, preprint, 2012.

Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar

The End

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Bruce Peckham UMD Math/Stat Maps close to z2 + c - UMD Physics Seminar