Secular and oscillatory motions in dynamical systems Henk Broer Johann Bernoulli Instituut voor Wiskunde en Informatica Rijksuniversiteit Groningen
Contents 1. Toroidal symmetry 2. Secular (slow) versus oscillatory (rapid) 3. Structures in slow dynamics 4. Examples & Conclusions V.I. Arnold, Mathematical Methods of Classical Mechanics. GTM 60, Springer 1989 V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren de mathematische Wissenschafte 250, Springer 1983 HWB, Normal forms in perturbation theory. In: R. Meyers (ed.), Encyclopædia of Complexity & System Science Springer 2009 HWB and G. Vegter, Generic Hopf-Neĭmark-Sacker bifurcations in feed forward systems, Nonlinearity 21 (2008) 1547-1578 HWB, H. Hanßmann, Á. Jorba, J. Villanueva and F.O.O. Wagener, Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity 16 (2003) 1751-1791
Toroidal symmetry Action-angle variables given H : R 2 R q = H q, Define A := A(H), then T = da dh, I := 1 2π A, ϕ := T 2π t I = 0, ω(i) = dh di (I) General format of perturbation in n dof: ṗ = H p ϕ = ω(i), ϕ = ω(i) + εf(i, ϕ), I = εg(i, ϕ) where H = H 0 + εh 1, ω(i) = dh 0 di, f = H 1 I, and g = H 1 ϕ Perturbation problems as usual
Scholium I Integrability for ε = 0 Examples: 1 dof systems, central force field, Lagrange top, geodesics on ellipsoid, combinations of the above,... Near integrability occurs persistently: think of Solar system as a perturbation of a number of uncoupled central force systems (Kepler, Newton) Also in small neighbourhood of elliptic equilibrium points (see below...) Slow-fast dynamics: I slow (secular), ϕ fast (oscillatory)
Toroidal symmetry II Local theory at 0 R 2n = {p, q} Birkhoff normal form at equilibrium H(p, q) = 1 2 ω, (p2 + q 2 ) + F (p 2 + q 2 ) + G(p, q) with σ = dp dq, F = O(4) and G = O(N + 1) (N depends on non-resonance properties of ω) Define q = 2I sin ϕ, p = 2I cos ϕ, then p 2 + q 2 = 2I, σ = di dϕ and H(I, ϕ) = ω, I + F (I) + G(I, ϕ) system ϕ = ω + F I + G I, where G is small whenever I is Locally same perturbation problem I G = ϕ
Scholium II Proofs: by action of adjoint operator ω ( p q q ) p on homogeneous parts of Taylor series Simplification leaves only terms in kernel Resonant examples later Similar theory near periodic solutions, action by, e.g., α ( t + ω p q q ) p and analogously near quasi-periodic tori
A simple Averaging Theorem Given a (general) 2π periodic system ϕ = ω(i) + εf(ϕ, I), I = εg(ϕ, I), (ϕ, I) T 1 R n Suitable near-identity transformation (ϕ, I) (ϕ, J) (truncating at order O(ε 2 )) J = εḡ(j) where ḡ(j) = 1 2π 2π 0 g(ϕ, J)dϕ Theorem: If ω(j) > 0 is bounded away from 0, then, for a constant c > 0 I(t) J(t) < c ε for 0 t 1 ε.
Proof Any transformation J = I + εk(ϕ, I) I = J + εh(j, ϕ, ε) J = I + ε k I I + ε k ϕ ϕ ( = ε g(ϕ, I) + k ) ϕ ω(i) + O(ε 2 ) Define 2π k(ϕ, I) = 1 (g(ϕ, I) ḡ(i)) dϕ, then ω(i) 0 J = εḡ(i) + O(ε 2 ) = εḡ(j) + O(ε 2 )
Scholium III Extensions to many classes of systems, for instance to Hamiltonian systems Generalization to the immediate vicinity of a quasi-periodic torus Further normalization give estimates that are polynomial or exponential in ε (in real analytic case) Passage through resonance: then condition on ω not valid. J.A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems. Springer 2008 N.N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Russ. Math. Surv. Usp. Mat. Nauk 32(6) (1977) 1-65
Adiabatic invariants Application of Averaging Theorem Hamiltonian time dependent, slowly varying system ϕ = ω(i, λ) + εf(i, ϕ, λ), where (ϕ, I, λ) T 1 R R Compare with averaged system I = εg(i, ϕ, λ), λ = ε Hamiltonian character gives g 0, hence J = εg(j), Λ = ε I(t) I(0) < c ε for 0 t 1 ε I is adiabatic invariant
Scholium IV Planck s: In the above take H(p, q, l) = p2 2l 2 + lgq2 2 with l = ε Energy level H(p, q, l) = E is ellipse: axes a = l 2E and b = 2E/lg Then action variable I = 1 2π A = 1 2π πab = E g = E ν l Laplace (more degrees of freedom) semi-major axes of Keplerian ellipses of the planets have no secular variations KAM theory and adiabatic invariants: 2 dof versus more; Arnold diffusion...
Parametric resonance I Consider time periodic Hamiltonian (swing) H α,β (p, q, t) = 1 2 p2 + (α 2 + β cos t)(1 cos q) where (p, q) (0, 0) and (α, β) ( 1 2, 0) (near 1 : 2 resonance; (p, q, t) R T 1 T 1 ) Two reversibilities: R(p, q, t) = ( p, q, t) and S(p, q, t) = (p, q, t) Complex notation z = p + 1 2iq ż = 1iz (δ + β cos t) sin q 1 2 4 (sin q q) ṫ = 1 where δ = α 2 1 4 detunes the 1 : 2 resonance Normalization/averaging (resonant) as described before H.W.B. and G. Vegter, Bifurcational aspects of parametric resonance, Dynamics Reported, New Series 1 (1992) 1-51.
Parametric resonance II Double cover given by map Π : C R/(4πZ) C R/(2πZ) Deck transformation (ζ, t) (ζe πi, t 2π) (ζ, t) (ζe 1 2 it, t(mod) 2π)) Symmetry induced by reversions R and S Z 2 Z 2 -symmetry (all compatible) Takens normal form Poincaré map P α,δ (ζ) = ( Id) N 2π α,δ(ζ) + h.o.t.(ζ, α, δ) Singularity theory describes planar Hamilton functions of N α,δ in generic setting Similarly for the other k : 2 resonances
Parametric resonance III Bifurcation diagram of N α,δ : showing the slow or secular dynamics near 1 : 2 resonance
Parametric resonance IV Poincaré map P α,β for α 1 2 and β.4 toy model for botafumeiro (Santiago de Compostela) co-existence of periodicity, quasi-periodicity and chaos
Quasi-periodic forcing Swing forced with fixed ω 1 and ω 2 : H α,β,ω1,ω 2 (p, q, t) = 1 2 p2 + (α 2 + β(cos(ω 1 t) + cos(ω 2 t)))(1 cos q) gives on T 2 R 2 system q = p, ϕ 1 = ω 1, ϕ 2 = ω 2 ṗ = (α 2 + β (cos ϕ 1 + cos ϕ 2 )) sin q Resonances given by k 1 ω 1 + k 2 ω 2 + lα = 0, for k 1, k 2, l Z If ω 1, ω 2 are rationally independent, then dense set of α-values
A dense set of resonances Arnold resonance tongues as these occur in many contexts
Normalization as before Diophantine conditions: For given τ > 1 and γ > 0 k 1 ω 1 + k 2 ω 2 + lα γ( k 1 + k 2 ) τ, for all (k 1, k 2 ) Z \ {(0, 0)} and for all l Z with l N Nowhere dense, large positive measure (for small γ) Normalized system (need Diophantine conditions) has same planar vector field truncation N α,β as before; same secular motion as before HWB, G.B. Huitema, F. Takens and B.L.J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori, Mem. AMS 83(421), 1990 HWB, H. Hanßmann, À. Jorba, J. Villanueva, and F.O.O. Wagener, Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach. Nonlinearity 16 (2003) 1751-1791 H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems Results and Examples, LNM 1893 Springer 2007
Cantorized fold catastrophe Model for quasi-periodic center-saddle bifurcation H λ (ϕ, I, p, q) = ω, I + 1 2 p2 + (λq q 3 ) + h.o.t.(p, q, λ) Cantorized fold: for q < 0 hyperbolic 2-tori occur and for q > 0 elliptic 2-tori
Scholium V Universal model, in particular robust Many further models in H. Hanßmann, The Quasi-Periodic Centre-Saddle Bifurcation, J. Differential Equations 142 (1998) 305-370 Co-existing dynamics with periodicity and chaos in gaps; often infinite regress inside gaps Oxtoby paradox nowhere dense versus positive measure meagre versus full measure Principle: mix of Singularity Theory and Kolmogorov Arnold Moser Theory H.W.B., H. Hanßmann and F.O.O. Wagener. Quasi-periodic bifurcation theory. in preparation
Dissipative analogues Same idea of oscillatory versus secular dynamics, normalizing/averaging the former away Periodic attractors and Cantorized families of quasi-periodic attractors of positive measure in parameter space Hopf-Neĭmark-Sacker bifurcation between them Quasi-periodic Hopf bifurcation from n- to (n + 1)-tori Onset of turbulence: Hopf-Landau-Lifschitz-Ruelle-Takens in generic families coexistence of periodicity, quasi-periodicity (of positive measure in parameter space) and chaos (often idem dito) Cantorized fold diagram also for bifurcations of diffeomorphisms from invariant circles to 2-tori Oxtoby paradox as before
Scholium VI Analogies of Averaging Theorem valid in some cases (e.g., in cases with an attractor) Singular perturbation theory is challenging HWB, T.J. Kaper and M. Krupa, Geometric desingularization of a cusp singularity in slow fast systems with applications to Zeeman s examples. JDDE 25(4) (2013) 925-958