How to create a large response from chaotic systems: Optimal forcing functions complement the natural dynamics of a system

Abstract

O ne of the biggest problems in controlling complex systems is that they are not very responsive to external signals unless an overwhelming control force is applied. Usually the response is small and irregular. The response of harmonic oscillators to random input is typically very small as well, unless the driving force has a certain frequency (the resonance or resonant frequency). Then the response can be extremely pronounced (resonance). When a dynamical system is driven at its resonance frequency, its energy absorption peaks, and the resulting pattern of absorption (and frequently its subsequent emission) form the "fingerprints" used in spectroscopic identification. X-ray spectroscopy, nuclear magnetic resonance (NMR), and other spectroscopic techniques employ resonances to identify cancer tissue, crystals, molecules, and the parts of the nucleus of atoms with high resolution.If there is no forcing, an oscillator will swing with its natural frequency. One of Galileo's great discoveries was that the resonance frequency of a harmonic oscillator is equal to the natural frequency of the system. This phenomenon is exemplified in the tidal resonances in the Bay of Fundy, home of the world's highest tides. High tides create large waves, which travel from the end of the bay to the mouth of the bay, reflect, and then travel back. The traveling time in the Bay of Fundy coincidentally matches the time from one high tide to the next, and because of this frequency match, the wave is amplified by the tides.The dynamics of the driving force does not necessarily have to match the driven system exactly; a series of impulses at the natural frequency (as opposed to a smooth sinusoidal forcing) of a system does a wonderful job of increasing the amplitude of a child on a swing set, but the energy transfer is most efficient when the driving dynamics is exactly matched to the natural dynamics.If the motion of the oscillator is recorded and then this recording is used as the input of an actuator that drives the same oscillator, then the resulting response is very large. Thus the natural frequency of a water glass (the sound produced when the glass is lightly struck) is the frequency to which the glass will respond most strongly, so playing this frequency back to the glass (with some amplification) can excite the oscillation of the glass with often spectacular results and eventually break it. We could excite the glass as well with a square-wave or saw-tooth signal that matched the resonance frequency, but the One of Galileo's great discoveries was that the resonance frequency of a harmonic oscillator is equal to the natural frequency of the system.

Harvesting wave energy with resonant observers

“…]. The targeted flow of energy is due to the synchronization of the motion of the sender and the motion of the observer . In this simple model, the dynamics of the waves is modeled by a single linear oscillator [see Eqs.…”

Nonlinear response of chemical reaction dynamics

“…The waveform and frequency of the impressed AC current could match these time scales to maximize the level of protection. Nonlinear optimal control methods could be used to optimize the time dependence of impressed currents systematically.…”

Information engines: Converting information into energy

“…Therefore, finding new sources of energy and efficient energy conversion has been an important research topic for a long time. In recent years, energy efficient forcing of nonlinear systems was studied systematically [1][2][3][4]. It was found that nonlinear dynamical systems react most sensitively to perturbations that complement the natural dynamics of the system [1].…”

“…In recent years, energy efficient forcing of nonlinear systems was studied systematically [1][2][3][4]. It was found that nonlinear dynamical systems react most sensitively to perturbations that complement the natural dynamics of the system [1]. In this case there is a perfect impedance match [2] and the energy transfer is most effective.…”