The third lecture in honour of Eduard Cech
Jean Mawhin
Resonance and Nonlinearity
March 28, 2006
Resonance is one of the most versatile concepts of
science. It is present under various aspects in astronomy, physics,
technology, musics, and, of course, mathematics. At resonance, the
response of a vibrating system to a periodic excitation becomes
unbounded; resonance can destroy the stability of a bridge or of the
solar system.
The mathematical study of resonance is closely linked to the concept of
spectrum, a name coined in optics by Newton in 1672 and part of the
mathematical language since the end of the XIXth century only.
The presence of nonlinear terms in an equation modifies the resonance
phenomenon in essential ways, and the concept of spectrum can be
extended to some classes of nonlinear operators, leading to new
resonance conditions. The lecture describes some recent results
relating spectra and nonlinearities in differential equations.
