Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Persistent nonlinear phase-locking and nonmonotonic energy dissipation in micromechanical resonators



Mingkang Wang, Diego J. Perez, Omar Lopez, Vladimir Aksyuk


Many nonlinear systems can be described by eigenmodes with amplitude dependent frequencies, interacting strongly whenever the frequencies become commensurate. Fast energy exchange via internal resonances holds the key to rich dynamical behavior, such as time-varying relaxation rates and signatures of nonergodicity in thermal equilibrium, revealed in the recent experimental and theoretical studies of micro and nanomechanical resonators. However, a universal physical description for these diverse and sometimes apparently contradictory experimental observations remains elusive. Here we reveal persistent nonlinear phase-locked states occurring at internal resonances and demonstrate that they are essential to understand the dynamics of nonlinear systems with coupled eigenmodes. The experimentally measured dynamics of a fully observable micromechanical resonator system is quantitatively described by the lower frequency mode entering, maintaining and exiting a period tripling state generated by the nonlinear driving force exerted by the higher frequency mode. This model describes the observed phase-locked coherence times, the direction and magnitude of the energy exchange resulting in a non-monotonic mode energy evolution. Depending on the initial relative phase, the system selects distinct relaxation pathways, either entering or bypassing the persistent locked state. The described persistent nonlinear phase- locking is not limited to particular frequency fractions or types of nonlinearities and may advance nonlinear resonator systems engineering across physical domains, including photonics as well as nanomechanics.
Physical Review X


micromechanical resonators, internal resonance, period-tripling states, ringdown


Wang, M. , Perez, D. , Lopez, O. and Aksyuk, V. (2022), Persistent nonlinear phase-locking and nonmonotonic energy dissipation in micromechanical resonators, Physical Review X, [online], (Accessed May 30, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created December 5, 2022, Updated December 12, 2022