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Lunch with Hamilton: Parabolic resonances and other non-separable structures September 05, 2018 (12:00 PM PDT - 01:00 PM PDT)
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Location: MSRI: Baker Board Room
Speaker(s) Vered Rom-Kedar (The Weizmann Institute)
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The parabolic resonance instability emerges when an integrable system that has an invariant torus which is normally parabolic and is resonant is perturbed [1-7]. It persistently appears in near integrable n d.o.f. Hamiltonian families depending on p parameters provided n+p≥3 and the underlying integrable system is non-separable, namely, it is a ubiquitous instability [1]. Indeed, parabolic resonances appear in many models with nearly-preserved rotational symmetry, such as models in which the angular momentum is nearly preserved [2,3] and in perturbed integrable Hamiltonian PDEs such as the forced periodic 1D NLS equation and the driven surface waves equation. For the forced 1D NLS this instability can lead to spatial decoherence of small amplitude nearly flat solutions [4,5,6].

To analyze these systems, geometrical tools such as the Energy-Momentum bifurcation diagrams, Fomenko graphs and the hierarchy of bifurcations framework are utilized. These provide insight on the global structure of near integrable systems with PR, and, more generally, on the structure of near integrable non-separable Hamiltonian systems with n≥2 d.o.f. [1-7].

Finally, the analysis of the parabolic resonance instability via local transformation to a slow-fast system shows that when properly scaled, the PR orbits exhibit adiabatic chaos – they follow level sets of an adiabatic invariant, with chaotic motion arising as a result of separatrix crossings. The structure of the adiabatic level sets are quite far from the base resonances, leading to chaotic dynamics which is distinct from the elliptic and hyperbolic resonance structures [7].  

[1] A. Litvak-Hinenzon and V. Rom-Kedar; On energy surfaces and the resonance web; SIAM J. Appl. Dyn. Syst. 3(4), 525, 2004.

[2] V. Rom-Kedar; Parabolic resonances and instabilities, Chaos, 7(1):148-158, 1997.

[3]  V. Rom-Kedar and N. Paldor; From the tropic to the poles in forty days. Bull. Amer. Mete. Soc., 78(12):2779-2784, 1997.

[4]  E. Shlizermann and V. Rom-Kedar;  Parabolic Resonance: A Route to Hamiltonian Spatio-Temporal Chaos, Phys Rev Letters, 2009.

[5] E. Shlizermann and V Rom-Kedar; Three types of chaos in the forced nonlinear Schrodinger equation.   Physical Review Letters, 96, 024104, 2006.

[6] E. Shlizermann and V. Rom-Kedar; Classification of solutions of the forced periodic nonlinear Schroedinger equation,  Nonlinearity 23 (9), 2183-2218, 2010.

[7] V. Rom-Kedar and D. Turaev; The Parabolic resonance instability, Nonlinearity 23 1325-1351, 2010.

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