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

One of the biggest problems with controlling complex systems is that they are not very responsive to external signals unless an overwhelming control force is applied; otherwise the response is usually small and irregular. The response of harmonic oscillators to random input is typically very small as well, but if 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.