Geometry of Resonance Tongues

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. 1-61. Bristol and Philadelphia IOP, 2001 V.I. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag 1983 Resonance p.20/36

References J. Knobloch, A. Vanderbauwhede: A general method for periodic solutions in conservative and reversible systems. J. Dynamics Diff. Eqns. 8 (1996) 71-102 HWB, M. Golubitsky, G. Vegter: The geometry of resonance tongues: a Singularity Theory approach, Nonlinearity 16 (2003) 1511-1538 HWB, M. Golubitsky, G. Vegter, Geometry of resonance tongues. To appear Luminy Proceedings on Singularity Theory 2006 Resonance p.21/36

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

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

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 = 2 1 2 + Z TrF = 2 Resonance p.24/36

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

Instability pockets (reversible) 0 1 2 3 4 5 6-2 -1 0 1 2 3 4 5 6 0 1 2 3 4 5 6-2 -1 0 1 2 3 4 5 6 0 1 2 3 4 5 6-2 -1 0 1 2 3 4 5 6 Resonance p.26/36

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

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

Geometry III, b 0 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0-0.1-0.1-0.2 2.25 2.255 2.26 2.265 2.27-0.2 2.25 2.255 2.26 2.265 2.27 2.275 0.4 0.3 0.2 0.1 0-0.1-0.2-0.3-0.4 2.25 2.252 2.254 2.256 2.258 2.26 In deformed Mathieu p(t) = cost + c 2 cos 2t + c 3 cos 3t Resonance p.29/36

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

Tongues at large I b p(t) = cos(t) + 3 4 cos(2t) two negativity intervals a Resonance p.31/36

Tongues at large II 5 4 3 2 1 0-1 -2-3 0 1 2 3 4 5 6 p(t) = cos(t) + cos(2t) + cos(4t) + 3 2 cos(6t) four negativity intervals Resonance p.32/36

Tongues at large III 500 450 400 350 300 250 200 150 100 50 0-500 -400-300 -200-100 0 100 200 300 400 500 Resonance p.33/36

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

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) 225-240 HWB, C. Simó, Hill s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bol. Soc. Bras. Mat. 29 (1998) 253-293 Resonance p.35/36

References HWB, C. Simó: Resonance tongues in Hill s equations: a geometric approach, Journ. Diff. Eqns. 166 (2000) 290-327 HWB, C. Simó, J. Puig: Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation, Commun. Math. Phys. 241 (2003) 467-503 Resonance p.36/36