Geometry of Resonance Tongues

Transcription

1 Geometry of Resonance Tongues Henk Broer with Martin Golubitsky, Sijbo Holtman, Mark Levi, Carles Simó & Gert Vegter Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen Resonance p.1/36

2 Resonance Resonance: type of periodic dynamics Ratio of frequencies of two (or more) oscillatory parts of a dynamical system is rational Dynamical systems usually depend on parameters Resonance tongues: "Domains of the parameter space where resonance occurs: Part of the bifurcation diagram Dissipative and conservative settings Resonance p.2/36

3 The Problem (dissipative) Consider dynamical system: Ẋ = F µ (X, t) with periodic orbit γ for µ = 0 Problem: Find periodic orbits near γ of period q ( subharmonics of order q) Use Poincaré map P µ : V V on transversal section V : P 0 (0) = 0 γ q-periodic orbit of P: P q µ(x) = x Resonance p.3/36

4 HNS-bifurcation NS-bifurcation of periodic orbit Ẋ = f µ (X) + ɛg µ (X, t) HNS-bifurcation of fixed point of (Poincaré) map P µ Conditions: - D 0 P 0 has two conjugate eigenvalues on unit circle - Eigenvalues cross unit circle with positive speed - Higher order stability condition Resonance if eigenvalues of D 0 P 0 (Floquet multipliers) are e ±2πip/q Resonance p.4/36

5 Subharmonic orbits q-subharmonic orbit q-periodic orbit of P µ : V V solve P q µ(x) = x, given that P 0 (0) = 0 Alternative formulation: P µ (x 1 ) = x 2, P µ (x 2 ) = x 3,..., P µ (x q ) = x 1 ˆP µ : V q V q (x 1,...,x q ) (P(x 1 ) x 2,...,P(x q ) x 1 ) Resonance p.5/36

6 Alternative formulation ctd. Take ˆP µ : V q V q and solve Symmetry solve ˆP µ (y) = 0 with ˆP 0 (0) = 0 σ : V q V q (x 1,...,x q ) (x 2,...,x q,x 1 ) σ ˆP µ = ˆP µ σ σ generates Z q ˆP µ is Z q -equivariant Resonance p.6/36

7 Lyapunov-Schmidt reduction I Notation: Assume D 0 ˆP 0 is semi-simple: V q = ker(d 0 ˆP0 ) im(d 0 ˆP0 ) D 0 ˆP0 has full rank here Projections: E : V q im(d 0 ˆP 0 ) y w (I E) : V q ker(d 0 ˆP0 ) y v Resonance p.7/36

8 Lyapunov-Schmidt reduction II The problem splits up: ˆP µ (y) = 0 E ˆP µ (v, w) = 0 & (I E) ˆP µ (v, w) = 0 Implicit Function Theorem: If E ˆP 0 (0) = 0 and w E ˆP µ (v, w) 0 then w = w(v) s.t. E ˆP(v, w(v)) = 0 Problem reduces as follows: then solve g µ (v) = 0 If g µ (v) := (I E) ˆP µ (v, w(v)), Resonance p.8/36

9 Lyapunov-Schmidt reduction III D 0 P 0 has only two complex conjugate eigenvalues ω, ω = e ±2πip/q dim(ker(d 0 ˆP0 )) = 2 Need to study g µ (z) = 0 With properties: g 0 (0) = 0 D 0 g 0 = 0 Symmetry expressed by ker(d 0 ˆP0 ) R 2 C g µ (ωz) = ωg µ (z) Resonance p.9/36

10 Singularity theory I Consider germ g µ at (0, 0) Z q -Symmetry g µ leads to format: g µ (z) = K(u, v)z + L(u, v) z q 1 u = z z, v = z q + z q Wanted: classification of possible forms of g µ under Z q contact equivalence (keeps track of zero set) h µ g µ : h µ (z) = S(z)g µ (Z(z)) S(0) 1, (D 0 Z) 1 exist Z(ωz) = ωz(z), S(ωz) = S(z) Resonance p.10/36

11 Singularity theory II Equivalence classes indicated by simplest element: Normal form always in format g µ (z) = K(u, v)z + L(u, v) z q 1 Non-degenerate normal form for q = 3 Conditions: K(0, 0) = 0, L(0, 0) 0 Normal form: h 0 (z) = z 2 Universal unfolding: h σ (z) = σz + z 2 Non-degenerate normal form for q 5 Conditions: K(0, 0) = 0, K u (0, 0) 0, L(0, 0) 0 Normal form: h 0 (z) = z 2 z + z q 1 Universal unfolding: h σ (z) = (σ + z 2 )z + z q 1 Resonance p.11/36

12 Singularity theory III Degenerate normal form for q 7. Conditions: K(0, 0) = K u (0, 0) = 0, K uu (0, 0) 0 L(0, 0) 0 Normal form: h 0,0 (z) = z 4 z + z q 1 Universal unfolding: h σ,τ (z) = (σ + τ z 2 + z 4 )z + z q 1 Resonance p.12/36

13 Resonance tongue I Tongue boundary where # of zeroes of H µ changes h µ = 0, det(dh µ ) = 0 q 5, h σ (z) = (σ + z 2 )z + z q 1 σ = α + iβ β 2 ( α) q 2 This is a q 2/2-cusp, thus Arnold tongues recovered h σ,τ (z) = (σ + τ z 2 + z 4 )z + z q 1. σ = α + iβ, τ = µ + iν 4D parameter space with universal geometry Characterization by Swallowtail and Whitney Umbrella Resonance p.13/36

14 Resonance tongue II ν β µ α Resonance p.14/36

15 Resonance tongue III β ν µ α = 0.01 α = 0.01 Resonance p.15/36

16 Resonance tongue IV α = 0.05 α = 0.05 Resonance p.16/36

17 Resonance tongue V α = 0.1 α = 0.1 Resonance p.17/36

18 Resonance tongue VI α = 0.15 α = 0.15 Resonance p.18/36

19 Conclusion I Case study of Forced Duffing - Van der Pol oscillator: ÿ + (a + cy 2 )ẏ + by + dy 3 = ɛf(y, ε, a, b, c, d, ω 1 t,...,ω n t) Procedure: Ẋ = F µ (X, t) Poincare map (Poincaré Normal Form Theory / Averaging) Complex Z q -equivariant germ (Lyapunov-Schmidt reduction) Bifurcation diagrams (Z q -equivariant Singularity Theory) Questions: How does universal geometry pull back to µ-space? More on the dynamics ((quasi-) periodicity, chaos, stability, bifurcation, etc.)? Resonance p.19/36

20 References F. Takens: Forced oscillations and bifurcations. Applications of Global Analysis I, Comm. Math. Inst. University of Utrecht 3 (1974) 1-59 Reprinted in H.W. Broer, B. Krauskopf, G. Vegter (Eds.), Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday, pp Bristol and Philadelphia IOP, 2001 V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag 1983 Resonance p.20/36

21 References J. Knobloch, A. Vanderbauwhede: A general method for periodic solutions in conservative and reversible systems. J. Dynamics Diff. Eqns. 8 (1996) HWB, M. Golubitsky, G. Vegter: The geometry of resonance tongues: a Singularity Theory approach, Nonlinearity 16 (2003) HWB, M. Golubitsky, G. Vegter, Geometry of resonance tongues. To appear Luminy Proceedings on Singularity Theory 2006 Resonance p.21/36

22 The Problem (conservative) I Hill s equation ẍ + (a + b p(t))x = 0, with p(t) p(t + 2π), (a,b) plane of parameters, assume p C System format ẋ = y ẏ = (a + bp(t))x ṫ = 1 Resonance p.22/36

23 The Problem (conservative) II Resonance: Trivial solution x = 0 = y unstable k-th unstability domain ( k = 1, 2,...): tongue emanating from (a,b) = (( k 2 )2, 0), subharmonics as before Geometry and density of stability diagram? Resonance p.23/36

24 Poincaré map F : R 2 R 2 Poincaré map of section t = 0 mod 2πZ F = F a,b Sp(1, R) Stability domain {(a,b) R 2 Tr F a,b < 2} Circle map f : S 1 S 1, f(θ) = arg F(e iθ ) f has Z 2 -symmetry Rotation number rot(f) = 1 2π lim j f j (θ) Instability when F hyperbolic f has fixed points and rotf Z TrF = Z TrF = 2 Resonance p.24/36

25 Related... First variation equation of (quasi-) periodic solutions in Hamiltonian systems Models parametrically forced oscillators Schrödinger operator (potential V = bp) (H V x)(t) = ẍ(t) + V (t)x(t), essentially self-adjoint operator on L 2 (R) Hill s equation has eigenvalue format H V x = ax results relevant for spectrum of H V Resonance p.25/36

26 Instability pockets (reversible) Resonance p.26/36

27 Geometry I Geometry tongue tips (b 0) in reversible case (where p is even) Contains A 2k 1 hierarchy of singularities Stability channels in general case ( ) exponentially narrow around families of parabolæ Fold between pictures 1 and 2 A 3 Resonance p.27/36

28 Geometry II, b 0 Scheme A 5 in deformed Mathieu for 3rd tongue... Resonance p.28/36

29 Geometry III, b In deformed Mathieu p(t) = cost + c 2 cos 2t + c 3 cos 3t Resonance p.29/36

30 Geometry IV, b Parabolæ centering channel TrF a,b = 0 Action angle variables (x,y,t) (I,ϕ,t), averaging TrF a,b = 0 ϕ = Φ(a,b), with Φ(a,b) = const. + πz 0 Family of parabolæ: quadratic approximation a + p(t) =: q(t) h κt 2 remainders, Airy,... Φ(a, b) = π 2 t + t q(t)dt a + bp(0) bp (0) h κt2 dt = πh 2κ Resonance p.30/36

31 Tongues at large I b p(t) = cos(t) cos(2t) two negativity intervals a Resonance p.31/36

32 Tongues at large II p(t) = cos(t) + cos(2t) + cos(4t) cos(6t) four negativity intervals Resonance p.32/36

33 Tongues at large III Resonance p.33/36

34 Density of stability Ray-wise density ϱ of stability domain ( ) Define sector I (a,b)-plane in between outmost tangencies of q with zero level When q(t) > 0 for all t [0, 2π], then ϱ = 1 (averaging, adiabatic invariants ellipticity) When q(t) < 0 for all t [0, 2π], then ϱ = 0 (hyperbolicity) Within sector I in between also ϱ = 0 (mixture, hyperbolicity dominant) Interesting things occur when p is less regular Resonance p.34/36

35 References HWB, G. Vegter: Bifurcational aspects of parametric resonance, Dynamics Reported, New Series 1 (1992) 1-51 HWB, M. Levi: Geometrical aspects of stability theory for Hill s equations, Archive Rat. Mech. An. 131 (1995) HWB, C. Simó, Hill s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bol. Soc. Bras. Mat. 29 (1998) Resonance p.35/36

36 References HWB, C. Simó: Resonance tongues in Hill s equations: a geometric approach, Journ. Diff. Eqns. 166 (2000) HWB, C. Simó, J. Puig: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation, Commun. Math. Phys. 241 (2003) Resonance p.36/36