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The decomposition of non-stationary signals is an important and challenging task in the field of signal time-frequency analysis. In the recent two decades, many signal decomposition methods led by the empirical mode decomposition, which was…

A neural networks (NN) compensator is designed for systems with multi-segment piecewise-linear nonlinearities. The compensator uses the back stepping technique with NN for inverting the multi-segment piecewise-linear nonlinearities in the…

This paper is concerned with the estimation of time-varying networks for high-dimensional nonstationary time series. Two types of dynamic behaviors are considered: structural breaks (i.e., abrupt change points) and smooth changes. To…

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a…

Cortical networks are strongly recurrent, and neurons have intrinsic temporal dynamics. This sets them apart from deep feed-forward networks. Despite the tremendous progress in the application of feed-forward networks and their theoretical…

We consider the relay version of the generalized Hutchinson's equation as a phenomenological model of an isolated neuron. After an exponential substitution, the equation takes the form of a differential-difference equation with a…

We propose a parameterization of a nonlinear dynamic controller based on the recurrent equilibrium network, a generalization of the recurrent neural network. We derive constraints on the parameterization under which the controller…

We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…

We provide novel sufficient conditions for stability of nonlinear and time-varying impulsive systems. These conditions generalize, extend, and strengthen many existing results. Different types of input-to-state stability (ISS), as well as…

This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of…

It is widely accepted that the complex dynamics characteristic of recurrent neural circuits contributes in a fundamental manner to brain function. Progress has been slow in understanding and exploiting the computational power of recurrent…

Dynamical balance of excitation and inhibition is usually invoked to explain the irregular low firing activity observed in the cortex. We propose a robust nonlinear balancing mechanism for a random network of spiking neurons, which works…

This paper studies deterministic consensus networks with discrete-time dynamics under persistent flows and non-reciprocal agent interactions. An arc describing the interaction strength between two agents is said to be persistent if its…

Although artificial neural networks have shown great promise in applications including computer vision and speech recognition, there remains considerable practical and theoretical difficulty in optimizing their parameters. The seemingly…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

Classical numerical methods solve partial differential equations (PDEs) efficiently on regular meshes, but many of them become unstable on irregular domains. In practice, multiphysics interactions such as diffusion, damage, and healing…

We consider a nonlinear non-autonomous system with time-varying delays $$ \dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which has a large number of applications in the theory of artificial neural networks. Via…

We study the expressive power of positive neural networks. The model uses positive connection weights and multiple input neurons. Different behaviors can be expressed by varying the connection weights. We show that in discrete time, and in…

This paper deals with traveling wavefronts for temporally delayed, spatially discrete reaction-diffusion equations. Using a combination of the weighted energy method and the Green function technique, we prove that all noncritical wavefronts…

Point process data are becoming ubiquitous in modern applications, such as social networks, health care, and finance. Despite the powerful expressiveness of the popular recurrent neural network (RNN) models for point process data, they may…