Speaker: Prof. Arnaud Lazarus

Abstract

In this talk, we investigate particles in periodically varying potentials, believing those periodically time-varying systems (PTVS) could eventually give new interesting physical insights in wave-particles interactions. When the local evolution function of a PTVS is periodically varied in synchrony with one of its natural time scales, it is known parametric instabilities can occur, according notably to Floquet theory. But unlike classical PTVS encountered in mechanics where the modulation of local evolution function is usually small, we consider here finite and possibly large modulations so that parametric instabilities are extremely enhanced and new dynamical phenomena can emerge. Based on those concepts, we will show i) how to trigger and efficiently sustain the natural vibrations of an oscillator, ii) a new particle- wave “duality” that can emerge in fundamental PTVS and iii) an analogy between parametric instabilities in Initial and Boundary Value Problems with an application in the buckling of elastic structures in periodic compressive state.

Biography

Arnaud Lazarus received a PhD in mechanical engineering from École polytechnique in France in 2008. After 2 postdoctoral years in Paris, he joined the Massachusetts Institute of Technology as an associate researcher from 2010 to 2013. Since 2013, he is an assistant professor at Sorbonne Université (Paris, France), doing his research at Institut Jean le Rond d'Alembert. His current interests are in the stability of dynamical systems and the mechanical behavior of slender elastic structures.

Notes

• Oscillating systems with periodic (temporal or spatial) variations can lead to unstable responses upon excitation. The instability domains are captured in Matthieu's instability diagrams, dividing the stable and the unstable regions (called tongues) with respect to the parameters of the problem.


• A review of the existing literature suggests that most periodically varying systems are close to the tip of the tongues, meaning that the rest of the diagrams are not investigated.


• First studied problem: a magnetic pendulum where the potential is a periodic function of time. Its Mathieu’s instability diagram features a series of small instability zones in the parameter space called "instability pockets", which correspond to the situation where the variation of the potential is synchronized with the movement of the pendulum to pump energy into the system.


• Second studied problem : 1D axially compressed beam on a space varying periodic elastic foundation. Using Floquet series, the system is expected to be unstable when the Floquet exponent is purely imaginary.

Suggested readings

• Alvaro A Grandi, Suzie Protière, Arnaud Lazarus
  Enhancing and controlling parametric instabilities in mechanical systems
  Extreme Mech. Lett. 42, 101195, 2021.


• Mas Arabi, C. and Kudlinski, A. and Mussot, A. and Conforti, M
  Geometric parametric instability in periodically modulated graded-index multimode fibers
  Phys. Rev. A 97, 023803, 2018.


• Gang, Hu and Daffertshofer, A. and Haken, H.
  Diffusion of Periodically Forced Brownian Particles Moving in Space-Periodic Potentials
  Phys. Rev. Lett. 76, 4874--4877, 1996.