Resonance and fractal geometry Henk Broer Johann Bernoulli Institute for Mathematics and Computer Science Rijksuniversiteit Groningen
Summary i. resonance in parametrized systems ii. two oscillators: torus- en circle dynamics iii. driven oscillators, examples: - Hopf-Neĭmark-Sacker bifurcation - parametric resonance - Hopf saddle-node bifurcation for maps iv. universal properties for parameter space - resonance open & dense or residual - quasi-periodicity nowhere dense or meagre a fractal set of positive measure - and chaos emerging
What is resonance? Resonance: interaction of oscillating subsystems, where rational ratio of frequencies and with compatible motion examples: 1 : 1 resonance: Tuning radio frequency, Huygens s clocks, Moon and Earth, Charon and Pluto 1 : 2 resonance: Botafumeiro (Santiago de Compostela) 2 : 3 resonance: Mercury
J.J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience 1950; Wiley 1992 1 : 1 resonance R R α = ω α = ω Amplitude response diagram driven oscillators resonance peaks ẍ = α 2 x cẋ+εsin(ωt) (harmonic) ẍ = α 2 x cẋ δx 3 +εsin(ωt) (Duffing)
Christiaan Huygens (1629-1695) Christiaan Huygens and title page Horologium Oscillatorium 1673 cycloids and synchronisation
Huygens s clocks synchronize Chr. Huygens, Œuvres Complètes de Christiaan Huygens, publiées par la Société Hollandaise des Sciences 16, Martinus Nijhoff, The Hague 1929, Vol. 5, 241-262; Vol. 17, 156-189
Tidal resonance Moon caught by Earth in 1 : 1 resonance Pluto and Charon caught each other: als the ultimate fate of the Earth-Moon system... Mercury caught in3 : 2 resonance A. Correia and J. Laskar, Mercury s capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics. Nature 429 (2004) 848-850
Botafumeiro Santiago de Compostela incense container brought into 1:2 resonance by pully: period exactly equals twice that of the forcing
Mathematical programme modelling in terms of dynamical systems depending on parameters emergence of several kinds of dynamics: periodic, quasi-periodic and chaotic bifurcations (fase transitions) in between applications from climate change to (biological) cell systems
Torus en circle dynamics simplest model: torus dynamics P(ϕ) ϕ Poincaré map: ϕ ϕ+2πα+εf(ϕ) dynamics of the circle (by iteration) resonant periodic
Example: Arnol d family ϕ ϕ+2πα+εsinϕ 3 2.5 ε 2 1.5 1 0.5 0-1/2-4/9-3/7-2/5-3/8-1/3-2/7-1/4-2/9-1/5-1/6-1/7-1/8 α 0 1/8 1/7 1/6 1/5 2/9 1/4 2/7 1/3 3/8 2/5 3/7 4/9 1/2 resonance tongues in (α,ε)-vlak catalogue of circle and torus dynamics
Explanation torus dynamics within tongues: periodicity resonance phase-locking synchronisation within main tongue : 1 : 1 resonance entrainment outside tongues: quasi-periodicity each orbit densely fills torus / circle H.W. Broer and F. Takens, Dynamical Systems and Chaos. Epsilon-Uitgaven 64 (2009); Appl. Math. Sc. 172 Springer (2011) D.G.M. Beersma, H.W. Broer, K. Efstathiou, K.A. Gargar and I. Hoveijn, Pacer cell response to periodic Zeitgebers. Physica D 19 (2011) 1516-1527 H.W. Broer, K. Efstathiou and E. Subramanian, Robustness of unstable attractors in arbitrarily sized pulse-coupled systems with delay. Nonlinearity 21(1) (2008) 13-49 H.W. Broer, K. Efstathiou and E. Subramanian, Heteroclinic cycles between unstable attractors.
Devil s staircase 0.6 0.4 0.2 0-0.2-0.4-0.6-0.4-0.2 0 0.2 0.4 α rotation number mean rotation as a function ofα, for small ε > 0 fixed (α) continuous, non-decreasing and constant on plateaux for rational values of resonance rational H.W. Broer, C. Simó and J.C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms. Nonlinearity 11 (1998) 667-770
Geometry in parameter space non-resonance fractal geometry nowhere dense set, topologically small positive Lebesgue measure fractal: Hausdorff dimension > topological dimension (= 0) quasi-periodic motion with irrational ; strongly irrational differentiable conjugation with rigid rotation V.I. Arnol d, Geometrical Methods in the Theory of Ordinary Differential Equations. Springer (1983) B.B. Mandelbrot, The Fractal Geometry of Nature. Freeman (1977) J. Oxtoby, Measure and Category, Springer (1971)
Conclusions Huygens s clocks torus / circle model weakly coupled oscillators as before almost identical oscillators 0 modulo 1 parameters(α,ε) in main tongue NB: mind the universal nature of this explanation. For more details on the phases of Huygens s clocks see the below references. Here the dynamics of the connecting beam has to play a role. M. Bennett, M.F. Schatz, H. Rockwood and K. Wiesenfeld, Huygens s clocks. Proc. R. Soc. Lond. A 458 (2002), 563-579 A. Pogromsky, D. Rijlaarsdam and H. Nijmeijer, Experimental Huygens synchronization of oscillators. In: M. Thiel, J. Kurths, M.C. Romano, A. Moura and G. Károlyi, Nonlinear Dynamics and Chaos: Advances and Perspectives. Springer Complexity 2010, 195-210
Hopf-Neĭmark-Sacker I more general: mapp : R 2 R 2 fixed point P(0) = 0 eigenvalues derivative e 2π(α±iβ) with α 0 and β p/q example: P is Poincaré map of driven oscillator ẍ+ax+cẋ = εf(x,ẋ,t) with f(x,ẋ,t+2π) f(x,ẋ,t) locally: geometry with universal singularities (saddle node / fold and other bifurcations) globally: quasi-periodicity and fractal geometry
Hopf-Neĭmark-Sacker II β α non-degenerate case q 5 F. Takens, Forced oscillations and bifurcations. In: Applications of Global Analysis I, Comm. of the Math. Inst. Rijksuniversiteit Utrecht (1974). In: H.W. Broer, B. Krauskopf and G. Vegter (eds.), Global Analysis of Dynamical Systems. IoP Publishing (2001) 1-62
Hopf-Neĭmark-Sacker III mildly-degenerate case q 7
Bifurcation set Periodic points of periodq via - Lyapunov Schmidt reduction and - equivariant contact equivalence singularity theory p(u;σ,τ;ε) = σ +τu+u 2 2 ε q 6 u q 2 with u,ε R and σ,τ C Discriminant set: p(u;σ,τ;ε) = 0 & D u p(u;σ,τ;ε) = 0 eliminatingu 3D set in 4-space = {σ 1,σ 2,τ 1,τ 2 } tomographyτ 1 = constant:
Tomogram σ 2 σ1 τ 2
Continued σ 2 σ 1 τ 2
Conclusions Arnol d tongues universal: fold, cusp, swallowtail and Whitney umbrella H.W. Broer, M. Golubitsky and G. Vegter, The geometry of resonance tongues: A Singularity Theory approach. Nonlinearity 16 (2003) 1511-1538 H.W. Broer, S.J. Holtman and G. Vegter, Recognition of the bifurcation type of resonance in mildly degenerate Hopf-Neĭmark-Sacker families. Nonlinearity 21 (2008) 2463-2482 H.W. Broer, S.J. Holtman, G. Vegter and R. Vitolo, Geometry and dynamics of mildly degenerate Hopf-Neĭmarck-Sacker families near resonance. Nonlinearity 22 (2009) 2161-2200 H.W. Broer, S.J. Holtman and G. Vegter, Recognition of resonance type in periodically forced oscillators. Physica-D 239(17) (2010) 1627-1636 H.W. Broer, S.J. Holtman, G. Vegter, and R. Vitolo, Dynamics and Geometry Near Resonant Bifurcations. Regular and Chaotic Dynamics 16(1-2) (2011) 39-50
Parametric resonance driven oscillator ẍ+(a+εf(t))sinx = 0 (swing) with f(t+2π) f(t) for example: f ε (t) = cost+εcos(2t) f(t) = signum (cost) loss of stability x 0 ẋ in discrete tongues emanating from(a,ε) = ( 1 4 k2,0), k = 0,1,2,... subharmonic bifurcations covering spaces H.W. Broer and G. Vegter, Bifurcational aspects of parametric resonance. Dynamics Reported, New Series 1 (1992) 1-51 H.W. Broer and G. Vegter, Generic Hopf-Neĭmark-Sacker bifurcations in feed forward systems. Nonlinearity 21 (2008) 1547-1578
Resonance tongues swing 6 6 5 5 4 4 3 3 2 2 1 1 0-2 -1 0 1 2 3 4 5 6 0-2 -1 0 1 2 3 4 5 6 6 5 4 3 2 1 0-2 -1 0 1 2 3 4 5 6 H.W. Broer and M. Levi, Geometrical aspects of stability theory for Hill s equations. Archive Rat. Mech. An. 131 (1995) 225-240 H.W. Broer and C. Simó, Resonance tongues in Hill s equations: a geometric approach. Journal of Differential Equations 166 (2000) 290-327
Botafumeiro revisited Poincaré map swing in 1 : 2 resonance period doubling quasi-periodicity chaos...
From gaps to tongues universal geometry from gaps to tongues with ε extra parameter collapse theory of gaps with Singularity Theory A 2k 1 quasi-periodic analogue ẍ+(a+εf(t))x = 0 with f(t) = F(ω 1 t,ω 2 t,...,ω n t) where F : T n R geometry per tongue as before globally fractal geometry with infinite regress H.W. Broer, H. Hanßmann, Á. Jorba, J. Villanueva and F.O.O. Wagener, Normal-internal resonances in quasi-periodically forces oscillators: a conservative approach. Nonlinearity 16 (2003) 1751-1791
Quasi-periodic Schrödinger tongues gaps spectrum Schrödinger operator (H εq x)(t) = ẍ(t) εf(t)x(t) with potentialεf, forx = x(t) L 2 (R) Cantor spectrum and... J. Moser and J. Pöschel, An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici 59 (1984) 39-85 L.H. Eliasson, Floquet solutions for the one-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146 (1992) 447-482 H.W. Broer, J. Puig and C. Simó, Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. Commun. Math. Phys. 241 (2003) 467-503
... devil s staircase revisited 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52 rotation number (as before) as a function of a in example with n = 2, ω 1 = 1 and ω 2 = 1 2 ( 5 1)
Hopf saddle-node I map P : R 3 R 3, fixed point P(0) = 0 eigenvalues derivative 1 en e 2π(α±iβ) with α 0 and β p/q math more experimental inspired by climate models... H.W. Broer, C. Simó and R. Vitolo, Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity 15(4) (2002) 1205-1267 A.E. Sterk, R. Vitolo, H.W. Broer, C. Simó and H.A. Dijkstra, New nonlinear mechanisms of midlatitude atmospheric low-frequency variability. Physica D: Nonlinear Phenomena 239 (2010) 701-718 H.W. Broer, H.A. Dijkstra, C. Simó, A.E. Sterk and R. itolo, The dynamics of a low-order model for the Atlantic Multidecadal Oscillation, DCDS-B 16(1) (2011) 73-102
Hopf saddle-node II box H 0.1 d1 0.05 0-0.05-0.1 0.2 O 0.4 0.6 I 0.8 d2 1 1.2 1.4 H H.W. Broer, C. Simó and R. Vitolo, The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeomorphisms, analysis of a resonance bubble. Physica D 237 (2008) 1773-1799 H.W. Broer, C. Simó and R. Vitolo, The Hopf-Saddle-Node bifurcation for fixed points of 3D-diffeomorphisms, the Arnol d resonance web. Bull. Belgian Math. Soc. Simon Stevin 15 (2008) 769-787 H.W. Broer, C. Simó and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus. DCDS-B 14(3) (2010) 871-905
Hopf saddle-node III 1 n=0.46 C C1 1 0 0 y -1 z x x -1-1 0 1-1 0 1 1 n=0.4555 D D1 1 0 0 y -1 z x x -1-1 0 1-1 0 1 corresponding dynamics: quasi-periodicity and chaos...
Conclusions I co-existence periodicity (including resonance), quasi-periodicity and chaos in product state- and parameter space bifurcations (phase transitions): singularities non-resonances: Kolmogorov-Arnol d-moser theory H.W. Broer, KAM theory: the legacy of Kolmogorov s 1954 paper. Bull. AMS (New Series) 41(4) (2004) 507-521 H.W. Broer, H. Hanßmann and F.O.O. Wagener, Quasi-Periodic Bifurcation Theory, the geometry of KAM. (Monograph in preparation)
Conclusions II fractal geometry with infinite regress nowhere dense meagre modelling at larger scale D. Ruelle and F. Takens, On the nature of turbulence. Comm. Math. Phys. 20 (1971) 167-192; 23 (1971) 343-344 H.W. Broer, B. Hasselblatt and F. Takens (eds.): Handbook of Dynamical Systems. Volume 3 North-Holland (2010) H.W. Broer, Resonance and Fractal Geometry. Acta Applicandæ Mathematicæ 120(1) (2012) 61-86