0.1 0.05 0.05 0.1 0.15 0.2 0.3 0.2 0.1 0.1 The dynamics of maps close to z 2 + c Bruce Peckham University of Minnesota Duluth Physics Seminar October 31, 2013 Β
Overview General family to study: z z n + c + β/z d Dynamical Systems Philosophy Background: z 2 + c Motivation from earlier studies Escape sets for z z n + c + β/z d : holes in filled Julia sets Special case: z n + β/z n ; radial symmetry, proofs Polar coordinate representation Modulus map Angular map Full dynamics β parameter space Comparison with z n + λ/z n A parameter space picture for z 2 + β/z
Dynamical Systems Introduction A discrete dynamical system is defined by x n+1 = f a (x n ) A recursion function f a and an initial condition x 0 generate a sequence: x 0, x 1 = f a (x 0 ), x 2 = f a (x 1 ),... Goal: describe the (long term) behavior of the sequence How does it depend on x 0? (Basins of attraction) How does it depend on f a (ie, on a)? (Bifurcations) Phase space or dynamic space (x X ) vs parameter space (a A).
Graphical Iteration Example: Q c (x) = x 2 + c where x is a real (dynamic) variable and c is a real parameter. Graphical iteration for x 2 + 0. Bounded orbits?? x 0 = 3/4, x 1 = ( 3/4) 2 = 9/16, x 2 = (9/16) 2,...
z 2 + c. Both z and c are now complex! A brief summary of the dynamics of z 2 + c: K c (filled Julia set) = the set of bounded orbits: Connected vs totally disconnected J c (Julia set) = boundary of K c, chaos c = 0 c = 0.27 z 0 inside, outside, boundary z 0 in Cantor set, outside
Julia Sets for z 2 + c Escape algorithms (rely on Escape Theorem) Definitions of J c : boundary of K c, closure of repelling periodic points, chaotic dynamics Software demo for different c values http://lycophron.com/math/devaney.html The Mandelbrot set The connectedness locus: Dichotomy for Julia sets The critical orbit bounded set Bulbs
Motivation Motivation: combine studies Holomorphic singular perturbations (Devaney, Blanchard, Josic, Uminsky,... ): z z n + c + λ/z d Nonholomorphic nonsingular perturbations (BP earlier work publ. in 1998 and 2000, UMD grad student Jon Drexler 1996). C vs. R 2 : z 2 + c + Az General Family (escape pictures) z z n + c + β/z d Special case (results - coauthor UMD grad student Brett Bozyk 2012, paper in press) z z n + β/z n
Escape pictures - Symmetry (a) z 2 + 0.05/z 2 (d) z3 0.125/z 3 (b) z 2 + 0.25.004/z 2 (c) z 2 1 0.001/z 2 (e)z 3 + (0.49 + 0.049i) 0.001/z 3 (f)z 3 + (0.1 + 0.1i)/z 1
Escape pictures: compare with z 2 + c z z 2 + c + β/z 2 Punching holes in the filled Julia sets for z 2 + c: (a) z 2 + 0.05/z 2 (b) z 2 + 0.25.004/z 2 (c) z 2 1 0.001/z 2 (a ) z 2 (b ) z 2 + 0.25 (c ) z 2 1
Special case: n = d AND c = 0 Restrict to Observations: z z n + β/z n 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 is dominated by 1D unimodal dynamics of the modulus map: bifurcation sequences will resemble those for x 2 + c (x, c real)
The Modulus and Angular Maps Restrict to Substituting z = re iθ, F n,β (re iθ) = z z n + β/z n ( re iθ) n + = ( r n + β 1 r n + i β 2 r n β (re iθ ) n = (r n + βr ) n e inθ ) e inθ P n,β ( r θ ) ( Mn,β (r) A n,β (r, θ) ) = r 2n + 2β 1 + β2 1 +β2 2 r 2n ) nθ + Arg (r n + β r n. r decouples!
(Conjugate) Modulus Map dynamics a) b) c) d) e) f) Six graphs and critical orbits for M 3,β for β values decreasing along the ray φ = arg(β) = π/3. Not symmetric about the critical point.
Full dynamics example 1 0.8 0.6 0.4 0.2 Full dynamics: Red attractor T 0.2 0.4 0.6 0.8 1 Modulus map B z 3 + 0.04/z 3
Parameter plane trichotomy The Escape Set Trichotomy for z 3 + β/z 3 : 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 is an attractor in case (ii). Β 0.3 0.2 c b a 0.1 d e f 0.1 0.05 0.05 0.1 0.15 0.2 iii 0.1 i
Full dynamics Results: Radial map M from unimodal (1D) map theory. Results for F (2D) proved. M F K bounded K S bounded Critical point c Critical circle {c} S A 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)
Other values of n n > 3 similar, but width of strip shrinks n = 2 interesting near β = 0: full family unless path goes through origin, then miss only last parameter point Β 0.3 0.2 0.1 0.1 0.05 0.05 0.1 0.15 0.2 0.1 Parameter plane β 2 = 0 cut b) β 2 = 0.1 cut
Holomorphic Comparison - Dynamic Space Agreement on the real line!! z 3 0.125 z 3 0.125 z 3 z 3
Holomorphic Comparison - Parameter Space Agreement on the POSITIVE real line!! z 3 + β/z 3 z 3 + λ/z 3 β plane λ plane
A Generalization Numerical experiments for more general cases: z 2 + β/z 1. Follow 3 points on critical circle. β plane Phase planes in parameter array
Summary Take-home messages and themes Complex analytic maps are a very special case of maps of the plane. Perturbation/Continuation Singularities Reduction/Generalization Experimentation/Analysis Mathematical vs applications approach Levels of appreciation
References Bozyk B D and Peckham B B, Nonholomorphic Singular Continuation: a case with radial symmetry, to appear in IJBC, 2013. Email bpeckham@d.umn.edu for preprint with full list of references. See especially: Guckenheimer J [1979], Sensitive dependence to initial conditions for one dimensional maps, Commun. Math. Phys. 70, 133-160. De Melo W and van Strein S [1993], One-dimensional Dynamics, A Series of Modern Surveys in Mathematics, 3(25) (Springer-Verlag, Berlin, New York). Bozyk B., Non-Analytic Singular Continuations of Complex Analytic Dynamical Systems [2012], Master s thesis, University of Minnesota Duluth, 2012. Kraft, R, Some One-dimensional Dynamics, preprint, 2012.
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