Author:
Cheverry Christophe,Farhat Shahnaz
Abstract
The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.
Reference29 articles.
1. L. Ambrosio, H. Frid; Multiscale Young measures in almost periodic homogenization and applications. Arch. Ration. Mech. Anal., 192 (2009) (1): 37-85. https://doi.org/10.1007/s00205-008-0127-3
2. G. Benettin, P. Sempio; Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field. Nonlinearity, 7 (1984) (1): 281-303. https://doi.org/10.1088/0951-7715/7/1/014
3. S. H. Benton, Jr.; The Hamilton-Jacobi equation. Academic Press [Harcourt Brace Jo- vanovich, Publishers], New York-London, 1977. A global approach, Mathematics in Science and Engineering, Vol. 131.
4. M. Bostan; The Vlasov-Maxwell system with strong initial magnetic field: guiding-center approximation. Multiscale Model. Simul., 6(3) (2007), 1026-1058. https://doi.org/10.1137/070689383
5. F. Camilli, O. Ley, P. Loreti; Homogenization of monotone systems of Hamilton-Jacobi equa- tions. ESAIM Control Optim. Calc. Var., 16(1) (2010), 58-76. https://doi.org/10.1051/cocv:2008061