Dynamics of nonlinear resonances in Hamiltonian systems

Abstract

It is well known that the dynamics of a Hamiltonian system depends crucially on whether or not it possesses nonlinear resonances. In the generic case, the set of nonlinear resonances consists of independent clusters of resonantly interacting modes, described by a few low-dimensional dynamical systems. We show that 1) most frequently met clusters are described by integrable dynamical systems, and 2) construction of clusters can be used as the base for the Clipping method, substantially more effective for these systems than the Galerkin method. The results can be used directly for system with cubic Hamiltonian. PACS numbers: 47.10.Df, 47.10.Fg, 02.70.Dh 1. Introduction. A notion of resonance runs through all our life. Without resonance we wouldn't have radio, television, music, etc. The general properties of linear resonances are quite well-known; their nonlinear counterpart is substantially less studied though interest in understanding nonlinear resonances is enormous. Famous experiments of Tesla show how disastrous resonances can be: he studied experimentally vibrations of an iron column which ran downward into the foundation of the building, and caused sort of a small earthquake in Manhattan, with smashed windows and swayed buildings [1]. Another example is Tacoma Narrows Bridge which tore itself apart and collapsed (in 1940) under a wind of only 42 mph, though designed for winds of 120 mph. Nonlinear resonances are ubiquitous in physics. Euler equations, regarded with various boundary conditions and specific values of some parameters, describe an enormous number of nonlinear dispersive wave systems (capillary waves, surface water waves, atmospheric planetary waves, drift waves in plasma, etc.) all possessing nonlinear resonances [2]. Nonlinear resonances appear in a great amount of typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate [3].The so-called "nonlinear resonance jump", important for the analysis of a turbine governor positioning system of hydroelectric power plants, can cause severe damage to the mechanical, hydraulic and electrical systems [4]. It was recently established that nonlinear resonance is the dominant mechanism behind outer ionization and energy absorption in near infrared laser-driven rare-gas or metal clusters [5]. The characteristic resonant frequencies observed in accretion disks allow astronomers to determine whether the object is a black hole, a neutron star, or a quark star [6]. Thermally induced variations of the helium dielectric permittivity in superconductors are due to microwave nonlinear resonances [7]. Temporal processing in the central auditory nervous system analyzes sounds using networks of nonlinear neural resonators [8]. The non-linear resonant response of biological tissue to the action of an electromagnetic field is used to investigate cases of suspected disease or cancer [9].

Resonance Clustering in Wave Turbulent Regimes: Integrable Dynamics

“…In order to demonstrate it we will use the notion of dynamical invariant first introduced in [2] which is given in the next section and illustrated by the example of harmonic oscillator.…”

“…In this case the system can be reduced to N = 4 (see [2] for more details), the Theorem on (N − 2)-integrability can be applied and we obtain the following CL:…”

“…A dynamical invariant for this system was originally presented in [2], in terms of the three real roots R 1 < R 2 < R 3 of the cubic polynomial , but these roots' dependence on the coordinates or the CLs was not made explicit. Moreover, the explicit solution for the amplitudes C j and phases θ j in the amplitude-phase representation B j = C j exp (iθ j ) was not provided.…”

“…If H 3 = 0, fourwave resonances have to be studied, and so on. To confirm that H 3 = 0 and three-wave resonances are dominant, one has to find solutions of (2) and check that V 3 12 = 0 at least at some resonant triads. Afterwards the corresponding dynamical system has to be studied.…”

“…In [17], construction of a set of reduced dynamical systems corresponding to the solutions of (2) and the systems themselves are given explicitly (as an example, resonances of oceanic planetary waves were considered in the spectral domain 0 ≤ m, n ≤ 50). The integrability of some resonance clusters has been studied in [2].…”