Research

1. Countably Infinite Utilitarian Aggregation (joint with Shaowei Ke and Tangren Feng)

Journal of Mathematical Economics, Vol. 98, January 2022, 102576

Abstract: We extend Harsanyi's (1955) Utilitarianism theorem to an infinite-horizon multi-generation setting: Under some additional assumptions, the Pareto condition is equivalent to utilitarian aggregation and the utilitarian weights are unique. Our results facilitate analysis of the properties of utilitarian weights, such as the limiting behavior of utilitarian weights for distant future generations, and the comparative statics of utilitarian weights as the social discount factor or the social risk attitudes. Among other findings, we show that a higher social discount rate is associated with a more unequal assignment of utilitarian weights across generations.

2. Optimal Time Averages in Non-Autonomous Nonlinear Dynamical Systems (joint with Charles Doering)

Journal of Pure and Applied Functional Analysis, Vol. 7 (1), February 2022, 231-251

Abstract: The auxiliary function method allows computation of extremal long-time averages of functions of dynamical variables in autonomous nonlinear ordinary differential equations via convex optimization. For dynamical systems defined by autonomous polynomial vector fields, it is operationally realized as a semidefinite program utilizing sum of squares technology. In this contribution we review the method and extend it for application to periodically driven non-autonomous and non-linear vector fields involving trigonometric functions of the dynamical variables.The damped driven Duffing oscillator and periodically driven pendulum are presented as examples to illustrate the auxiliary function method's utility.

3. Stability of a parametrically driven, coupled oscillator system: An Auxiliary Function Method Approach (joint with Yin Lu Young)

Journal of Applied Physics, Vol. 131, April 2022, 154901

Abstract: Coupled, parametric oscillators are often studied in applied biology, physics, fluids, and many other disciplines. In this paper, we study a parametrically driven, coupled oscillator system where the individual oscillators are subjected to varying frequency and phase with a focus on the influence of the damping and coupling parameters away from parametric resonance frequencies. In particular, we study the long-term statistics of the oscillator system's trajectories and stability. We present a novel, robust and computationally efficient method come to be known as an auxiliary function method for long-time averages, and we pair this method with classical, perturbative-asymptotic analysis to corroborate the results of this auxiliary function method. These paired methods are then used to compute the regions of stability for a coupled oscillator system. The objective is to explore the influence of higher order, coupling effects on the stability region across a broad range of modulation frequencies, including frequencies away from parametric resonances. We show that both simplified and more general asymptotic methods can be dangerously un-conservative in predicting the true regions of stability due to high order effects caused by coupling parameters. The differences between the true stability region and the approximate stability region can occur at physically relevant parameter values in regions away from parametric resonance. As an alternative to asymptotic methods, we show that the auxiliary function method for long-time averages is an efficient and robust means of computing true regions of stability across all possible initial conditions.

Accompanying code