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Subharmonic response is a well known phenomena in, e.g., deterministic nonlinear dynamical systems. We investigate the conditions under which such subharmonic oscillations can persist for a long time in open systems with stochastic dynamics…

Statistical properties of environments experienced by biological signaling systems in the real world change, which necessitate adaptive responses to achieve high fidelity information transmission. One form of such adaptive response is gain…

This SSFG Lp stability theorem can predict permissible forcing amplitudes below which a finite nonlinear input-output gain can be maintained. Our analysis employs Linear Matrix Inequalities (LMI) and Sum-of-Squares (SOS) as the primary…

The stochastic gradient descent (SGD) algorithm is the algorithm we use to train neural networks. However, it remains poorly understood how the SGD navigates the highly nonlinear and degenerate loss landscape of a neural network. In this…

For most stochastic dynamical systems, variables which are tightly regulated tend to respond slowly to external changes. This idea is often discussed for applicable systems, within a linear response regime, through the Fluctuation…

We study synchronous phenomena of a coarse-grained power grid model, the swing equation, on small-world networks. We show that its steady state, which stands for the normal operation of the power systems, can be realized even if the phases…

Loss functions with non-isolated minima have emerged in several machine learning problems, creating a gap between theory and practice. In this paper, we formulate a new type of local convexity condition that is suitable to describe the…

This paper considers small gain theorems for the global asymptotic and exponential input-to-state stability for discrete time time-delay systems using Razumikhin-type Lyapunov function. Among other things, unlike the existing literature, it…

The input/output stability of an interconnected system composed of an ordinary differential equation and a damped string equation is studied. Issued from the literature on time-delay systems, an exact stability result is firstly derived…

We report on the possibilities of using the method of normal fundamental systems for solving some problems of oscillation theory. Large elastic dynamical systems with continuous and discrete parameters are considered, which have many…

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm…

This paper is devoted to proving the small noise asymptotic behaviour, particularly large deviation principle, for multi-scale stochastic dynamical systems with fully local monotone coefficients driven by multiplicative noise. The main…

The strong growth condition (SGC) is known to be a sufficient condition for linear convergence of the stochastic gradient method using a constant step-size $\gamma$ (SGM-CS). In this paper, we provide a necessary condition, for the linear…

In a power grid with growing penetrations of renewable energy sources, inverters play a larger role in the dynamic interactions among network components. However, much is yet to be studied regarding inverter-dominated grid stability. This…

Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyze the future asymptotics of models which exhibit oscillatory behavior, we provide a theorem concerning the large-time…

In this paper, we consider asymptotic behaviors of multiscale multivalued stochastic systems with small noises. First of all, for general, fully coupled systems for multivalued stochastic differential equations of slow and fast motions with…

Behavioral homogeneity is often critical for the functioning of network systems of interacting entities. In power grids, whose stable operation requires generator frequencies to be synchronized--and thus homogeneous--across the network,…

Discrete-time models of non-uniformly sampled nonlinear systems under zero-order hold relate the next state sample to the current state sample, (constant) input value, and sampling interval. The exact discrete-time model, that is, the…

Stochastic gradient (SG) methods are fundamental to system identification and machine learning, enabling online parameter estimation in large-scale and streaming-data settings. As a classical identification method, the SG algorithm has been…

In this article, two types of methods from different perspectives based on spectral normalization are described for ensuring the stability of the system controlled by a neural network. The first one is that the L2 gain of the feedback…