DRIVEN and COUPLED OSCILLATORS. I Parametric forcing The pendulum in 1:2 resonance Santiago de Compostela


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1 DRIVEN and COUPLED OSCILLATORS I Parametric forcing The pendulum in 1:2 resonance Santiago de Compostela II Coupled oscillators Resonance tongues Huygens s synchronisation III Coupled cell system with symmetry synchronicity and related things IV Outlook quasiperiodic forcing bifurcations Cantor spectrum Henk Broer Dept of Mathematics, RuG URL: 1
2 Parametric resonance (I) ẍ = (a + bp(t))sin x vs. ẍ = (a + bp(t))x Trivial periodic solution: x 0 ẋ ( lower equilibrium ) Mathieu case p(t) = cos t RESONANCE lower equilibrium unstable tongues in (a, b)plane tip in (a, b) = (( k 2 )2,0), k = 1,2,... Consider tongue emanating from (a, b) = ( 1 4,0), here lower equilibrium x 0 ẋ unstable period doubling (Santiago de Compostela) Write ẋ = y ẏ = (a + bp(t))sin x ṫ = 1 consider Poincaré (or period / strobocopic) map P = P(x, y) 2
3
4
5 y x y x 5
6 y x W s(0) y W u(0) x 6
7 7
8 8
9 W u(p) p W s(p) 9
10 10
11 References (I) H.W. Broer: De chaotische schommel, Pythagoras 35(5), (1997), H.W. Broer, G. Vegter: Bifurcational aspects of parametric resonance, Dynamics Reported, New Series 1 (1992) 151 H.W. Broer, M. Levi: Geometrical aspects of stability theory for Hill s equations, Archive Rat. Mech. An. 131 (1995) H.W. Broer, C. Simó: Resonance tongues in Hill s equations: a geometric approach, Journ. Diff. Eqns. 166 (2000)
12 Coupled oscillators (II) Van der Pol type oscillator ẍ + cẋ + ax + f(x, ẋ) = 0, e.g., f(x, ẋ) = bx 2 ẋ, with a, b, c constants Periodic forcing or coupling ẍ + cẋ + ax + f(x, ẋ) = εg(x, ẋ, t) ẍ 1 + cẋ 1 + ax + f 1 (x 1, ẋ 1 ) = εg 1 (x 1, ẋ 1, x 2, ẋ 2 ) ẍ 2 + cẋ 2 + ax + f 2 (x 2, ẋ 2 ) = εg 2 (x 1, ẋ 1, x 2, ẋ 2 ) invariant 2torus T and Poincaré circle mapping P : S 1 S 1 RESONANCE phase lock periodic solution Metaphoric: Arnold family of circle maps P α,β : ϕ ϕ + 2πα + β sin ϕ Tongues emanate from (α, β) = ( p q,0) saddlenodes at tongue boundaries Huygens s synchronisation in 1 : 1tongue 12
13 β α 13
14 References (II) V.I. Arnold: Geometrical methods in the theory of ordinary differential equations, Springer Verlag 1983 H.W. Broer, M. Golubitsky, G. Vegter: The geometry of resonance tongues: a Singularity Theory approach, Nonlinearity 16 (2003) H.W. Broer, C. Simó and J.C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity 11(3) (1998) F. Takens: Forced oscillations and bifurcations. Applications of Global Analysis I, Comm. Math. Inst. University of Utrecht 3 (1974) 159 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,
15 Symmetric coupled cell system (III) Three identical oscillators with Z 3 symmetry ẋ 1 = f(x 1, x 2, x 3 ) ẋ 2 = f(x 2, x 1, x 3 ) ẋ 3 = f(x 3, x 1, x 2 ) f(a, b, c) = f(a, c, b) undergoes Hopf bifurcation from synchronous equilibrium x 1 = x 2 = x 3. Generically three kinds of periodic solutions can bifurcate. Let T be the period. 1. Discrete rotating wave x 2 (t) = x 1 (t + T/3) and x 3 (t) = x 2 (t + T/3) 2. Discrete standing wave : x 1 = x 2 (synchronicity) and x 3 oscillates differently (with period T) 3. Discrete standing wave : x 1 (t) = x 2 (t + T/2) and x 3 oscillates at TWICE the frequency 15
16 References (III) H.W. Broer, M. Golubitsky, G. Vegter, Geometry of resonance tongues, To appear Luminy Proceedings on Singularity Theory 2005 M. Golubitsky, M. Nicol and I. Stewart, Some curious phenomena in coupled cell networks, J. Nonlinear Sci. 14(2) (2004)
17 Outlook (IV) Quasiperiodicity (forcing, attractors) Bifurcations (HopfLandauLifschitzRuelleTakens: from periodic and quasiperiodic attractors to strange attractors and chaos) Cantor spectrum The general theory of dynamical systems 17
18 References (IV) V.I. Arnold and F. Takens as under II H.W. Broer, KAM theory: the legacy of Kolmogorov s 1954 paper, Bull. AMS (New Series), 41(4), (2004), H.W. Broer and M.B. Sevryuk, KAM theory: quasiperiodicity in dynamical systems. In H.W. Broer, B. Hasselblatt and F. Takens (Eds.): Handbook of Dynamical Systems, Volume 3. To be published by NorthHolland, H.W. Broer, C. Simó, Hill s equation with quasiperiodic forcing: resonance tongues, instability pockets and global phenomena, Bol. Soc. Bras. Mat. 29 (1998) H.W. Broer, C. Simó, J. Puig: Resonance tongues and instability pockets in the quasiperiodic HillSchrödinger equation, Commun. Math. Phys. 241 (2003)
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