The Cusp-cusp Bifurcation for noninvertible maps of the plane

Transcription

1 The Cusp-cusp Bifurcation for noninvertible maps of the plane Bruce Peckham Dept of Mathematics and Statistics Universit of Minnesota, Duluth Duluth, Minnesota Coauthors: Bernd Krauskopf and Hinke Osinga Universit of Bristol C 0 W J 0 S Ĵ 0 f(w ) C 1 Midwest Dnamical Sstems Conference, October 21, 2007, The Universit of Michigan, Ann Arbor

2 Talk Outline 1 Background on noninvertible maps of the plane: some features and bifurcations 2 The codimension-two cusp-cusp bifurcation (a) The two cusps (b) The normal form of the model (c) Definition of equivalence (d) Analsis of the model: five codimension-one bifurcations, eight generic cases 3 An adaptive control application 4 Summar

3 Some research groups in noninvertible maps of the plane Gumowski and Mira 1980 book Recurrences and Discrete Dnamic Sstems (, ) (a +, b + 2 ) Christian Mira group : Laura Gardini, Milleriou, Barugola, Cathala, Carcasses, Ralph Abraham Several books and lots of articles Yannis Kevrekidis group : Christos Frouakis, Ra Adomaitis, Rafael de Llave, Rico-Martine, BP Dick McGehee group: Eveln Sander, Josh Nien, Rick Wicklin, Loren (1989 paper): (, ) ((1 aτ) τ, (1 τ) + τ 2 ) Bristol Group: Bernd Krauskopf, Hinke Osinge, James England, BP, V Maistrenko and Y Maistrenko A good introductor reference to Noninvertible maps of the plane: On some properties of invariant sets of two-dimensional noninvertible maps, Frouakis, Gardini, Kevrekidis, Milleriou, and Mira, IJBC, Vol 7, No 6, (1997),

4 Features and bifurcations unique to NONINVERTIBLE maps of the plane

5 Some features unique to noninvertible maps of the plane Critical curves: J 0 ( LC 1 ), ( LC) J 0 = { : det(df ()) = 0}; = F (J 0 ) Folding of phase space: Z 0 Z 2, Z 1 < Z 3, Figs from FGKMM (1997)

6 Some features unique to noninvertible maps of the plane (cont) Interaction of critical curves with fied/periodic points (eigenvalue ero) Unstable manifolds (outsets): self-intersections, loops, cusps Stable manifolds (insets): disconnected, allowing disconnected and multipl connected basins of attraction Invariant circles Chaotic attractors

7 Eample: Loren (1989) (, ) 7 ((1 aτ ) τ, (1 τ ) + τ 2) (a) O 03 O 02 Γ Γ (c) (d) (b) 06 O O Frouakis, Kevrekidis, P, (b) J0 J1 L(C) 02 (c) C 00 L(C) J

8 Some bifurcations unique to noninvertible maps of the plane Codimension-one Interactions of J 0 with 1 fied/periodic points - transition from orientation preserving to orientation reversing, 2 creating self-intersections of W u via tangenc (Mira: contact bifurcation) or loop formation 3 creating disconnected or multipl connected basin boundaries 4 breakup of an invariant circle (loops on what used to be the smooth invariant curve - Loren) 5 creating intersections of basin boundaries with J 0 Codimension-two 1 (0,1) eigenvalues (Josh Nien thesis with Dick McGehee ) (Unfoldings not finitel determined) 2 Sander (2000) A transverse homoclinic point with no tangle 3 The Cusp-cusp bifurcation

9 Global bifurcation diagrams E Mira 1991 Note to Kevrekidis

10 The CUSP-CUSP bifurcation

11 Motivation from FGKMM (1997): Interactions in the neighborhood of a cusp point

12 The first cusp - along the phase space fold The Cusp point C 0 and its image C 1 Persists under perturbation The green line field denotes the direction of the ero eigenspaces at points of J 0 (a) and (b) General position; (c) and (d) normal form position (a) C 0 Ĵ 0 (b) C 1 Z 1 J 0 E U Z 3 V (c) (d) Z 1 0 C 0 Ĵ 0 0 C 1 E J 0 0 Z 3 0

13 The second cusp - on the image of W Cusps on images of curves which cross J 0 tangent to the line field Codimensionone occurrence W W W 0 E 0 E 0 E J 0 (a) J 0 (b) J 0 (c) Z 0 Z 0 Z 0 f( 0 ) f(w ) f( 0 ) f( 0 ) Z 2 f(w ) Z 2 f(w ) Z 2 Assume W is described locall b α(t) with α(t 0 ) = 0 and tangent vector α (t) At f( 0 ), W is described locall b f(α(t)), and has tangent vector Df( 0 )α (t), which implies W is still smooth at f( 0 ) unlees α (t) is an eigenvector for eigenvalue ero at 0

14 The normal forms of the cusp map and W (cont) ( ) ( ) ( ) F : = 3 3 (1) ( ) [ ] 3 DF = is singular along the particularl simple critical curve J 0 := { = 2 } The image of the parabola J 0 is the standard cusp := { = ±2( ) 3 0} (3) A straightforward calculation shows that it has the second pre-image Ĵ 0 := { = } The ero-eigenvector line field is horiontal (independent of J 0 ) (2) (4) Normal form (truncated) of W : W := { = γ( a) 2 + b} (5)

15 The projection of the graph of the normal forms The cusp maps is fied; W = {γ( a) 2 + b} varies The parameters a and b are the pimar parameters while γ is a secondar parameter We construct bifurcation diagrams in the (a, b) plane for fied γ There turn out to be eight generic cases, depending on γ C 0 W J 0 S Ĵ 0 f(w ) C 1

16 The organiing center for different values of γ Focus on case (i): γ = 05 (a) i (b) (f) (c) ii iii iv (g) vi (d) (h) (e) v (i) vii viii

17 All possible normal form configurations

18 The cusp-cusp unfolding for Case VII (γ = 05) b vii a

19 The codimension-one bifurcations: (a) Cusp transition (pass thru the cusp point C) (b) Loop creation (cusp on W at critical parameter value) (c) Intersecion at tangenc (topological change in intersections in range) C C (a) Ĵ 0 J 0 L L (b) Ĵ 0 J 0 I I (c) Ĵ 0 J 0

20 The codimension-one bifurcations (continued) (a) tangenc-creation (W tangent to J 0 ) (b) Enter-eit (W tangent to Ĵ 0 ) T T (a) Ĵ 0 J 0 E E (b) Ĵ 0 J 0 Definition of equivalence: Toplogical equivalence of the two curves in the image

21 Eplicit computation of bifurcation curves: all parabola in (a,b) 1 The cusp transition, denoted b C, where the curve W passes through J 0 at the pre-cusp point C 0, which means that F (W ) passes eactl through the cusp point C 1 on ; The locus of this bifurcation in the (a, b)-plane is the parabola b = c C (γ) a 2 = γ a 2 (6) 2 The loop-creation bifurcation, denoted b L, where W crosses J 0 tangent to the (horiontal) line field E; b = c L (γ) a 2 = a 2 3 The intersection-at-tangenc bifurcation, denoted b I, where F (W ) selfintersects at a tangenc point with ; b = c I (γ) a 2 = (9γ + 4) a 2 (8) 4 The tangenc-creation bifurcation, denoted b T, where W is tangent to J 0 ; b = c T (γ) a 2 = γ 1 + γ a2 (9) 5 The enter-eit bifurcation, denoted b E, where W is tangent to Ĵ 0 ; b = c E (γ) a 2 = γ a 2 (10) (7)

22 Classification in the γ-coefficient space The eight cases, depending on γ, the quadratic coefficient in the normal form of the curve W := { = γ( a) 2 + b}: 6 c (γ) c C (γ) c E (γ) c L (γ) c T (γ) i ii iii iv v vi vii viii c I (γ) γ

23 Adaptive Control Application ( ) ( ) + η g : β + p( +η 1), (11) c+ 2 Frouakis, Adomaitis, Kevrekidis, Golden and Ydstie 1992 (β = 10) Searched for cusp-cusp point numericall in (p, η) plane for c = 12, β = 10??? In KOP: Fied p = 081, searched in (η, β) plane!!! 101 β (a) SN 03 (b) p W u (p 0 ) Ĵ 0 SN NS 02 C 0 J γ (c) W u (p 1 ) 050 (d) 054 p 1 W u (p 1 ) 050 C1 049

24 Adaptive Control Application (continued) Case VII! (a) p 0 W u (p 0 ) C 0 J 0 Ĵ β c b vii 28 (b) p W u (p 1 ) C γ c a Application Normal form

25 Future plans Find or create a better eample - with period less than 30 Implement algorithms for all codimension-one points, as well as codimensiontwo points Identif other codimension-two points Connect behavior inside Arnold tongues to behavior outside (irrational rotation arcs for invertible maps)

26 Summar: Sstematic analsis of a codimension-two point using singularit theor Definition of equivalence class allowed identification of five codimension-one bifurcations Ke transitions can be studied with onl the first iterate of the map under investigation Normal form allowed eplicit computations of bifucation curves in (a, b) parameter space, and classification diagram for cases I - VIII in (γ, coefficient) space Allows classification of tpe I - VIII in applications Might provide a framework for analing other noninvertible bifurcations