Energy-based theory of autoresonance phenomena: Application to Duffing-like systems

A general energy-based theory of autoresonance (self-sustained resonance) in low-dimensional nonautonomous systems is presented. The equations that together govern the autoresonance solutions and excitations are derived with the aid of a variational principle concerning the power functional. These equations provide a feedback autoresonance-controlling mechanism. The theory is applied to Duffing-like systems to obtain exact analytical expressions for autoresonance excitations and solutions which explain all the phenomenological and approximate results arising from a previous (adiabatic) approach to autoresonance phenomena in such systems. The theory is also applied to obtain new, general, and exact properties concerning autoresonance phenomena in a broad class of dissipative and Hamiltonian systems, including (as a particular case) Duffing-like systems. PACS number: 05.45.-a Nonlinear dynamics and nonlinear dynamical systems It has been well known for about half a century that autoresonance (AR) phenomena occur when a system continuously adjusts its amplitude so that its instantaneous nonlinear period matches the driving period, the effect being a growth of the system’s energy. Autoresonant effects were first observed in particle accelerators [1,2], and have since been noted in nonlinear waves [3,4], fluid dynamics [5,6], atomic and molecular physics [7,8], plasmas [9-11], nonlinear oscillators [12,13], and planetary dynamics [14-17]. Apparently, the first mention of the notion of resonance (“risonanza”) was by Galileo [18]. Remarkably, this linear-system-based concept has survived up to now: resonance (nonlinear resonance) is identified with how well the driving period fits (a rational fraction of) a natural period of the underlying conservative system [19]. However, the genuine effect of the frequency (Galilean) resonance (FR) (i.e., the secular growth of the oscillation amplitude) can no longer be observed in a periodically driven nonlinear system. As is well known, the reason is simple: a