Targeted activation in deterministic and stochastic systems

Abstract

Metastable escape is ubiquitous in many physical systems and is becoming a concern in engineering design as these designs (e.g., swarms of vehicles, coupled building energetics, nanoengineering, etc.) become more inspired by dynamics of biological, molecular and other natural systems. In light of this, we study a chain of coupled bistable oscillators which has two global conformations and we investigate how specialized or targeted disturbance is funneled in an inverse energy cascade and ultimately influences the transition process between the conformations. We derive a multiphase averaged approximation to these dynamics which illustrates the influence of actions in modal coordinates on the coarse behavior of this process. An activation condition that predicts how the disturbance influences the rate of transition is then derived. The prediction tools are derived for deterministic dynamics and we also present analogous behavior in the stochastic setting and show a divergence from Kramers activation behavior under targeted activation conditions.

Control of a model of DNA division via parametric resonance

“…Note that, as explained in Ref. 12, the study provided in Refs. 10-12 was limited to the simpler Duffing potential because, "The exponential form of the Morse potential makes analytical progress difficult, ….…”

“…As pointed out in Ref. 12, the Morse potential is indeed a difficult function to work with. Since the geometry of the Morse potential requires a polynomial of degree at least 26 for an accurate approximation, we decide to use such a high degree expansion for our computation of the averaged reduced Hamiltonian.…”

“…By applying the method of partial averaging 1,12,14,18,26,28,35 for nearly 0:1 resonance, we obtained the averaged equations for the reduced models of this chain of a) Author to whom correspondence should be addressed. Electronic mail: koon@cds.caltech.edu.…”

“…By using the Fourier modal coordinates, 8,12,23 this model can be seen as a small nonlinear perturbation of n harmonic oscillators with frequencies x a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð1 À cos 2pa=nÞ p ; a ¼ 0; …; n À 1. The coarse variable of the (approximate) 0th mode, which is the average angle of the pendula, corresponds to the angle of the frame defined by the radius of gyration in our molecular studies.…”

Coherent Swing Instability of Power Grids

“…The escape criterion developed in Eisenhower and Mezić (2010) for static perturbations can now be based on the Hamiltonian of the steady part of the above system (26); it is…”