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We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear…
Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two…
Self-organization is ubiquitous in nature and mind. However, machine learning and theories of cognition still barely touch the subject. The hurdle is that general patterns are difficult to define in terms of dynamical equations and…
We consider the dynamics of a droplet on a vibrating fluid bath. This hydrodynamic quantum analog system is shown to elicit the canonical behavior of damped-driven systems, including a period doubling route to chaos. By approximating the…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
This paper is concerned with the online estimation of a nonlinear dynamic system from a series of noisy measurements. The focus is on cases wherein outliers are present in-between normal noises. We assume that the outliers follow an unknown…
Modern learning systems increasingly interact with data that evolve over time and depend on hidden internal state. We ask a basic question: when is such a dynamical system learnable from observations alone? This paper proposes a research…
Empirically observed time series in physics, biology, or medicine, are commonly generated by some underlying dynamical system (DS) which is the target of scientific interest. There is an increasing interest to harvest machine learning…
We introduce a simple method to estimate the system parameters in continuous dynamical systems from the time series. In this method, we construct a modified system by introducing some constants (controlling constants) into the given…
In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces…
The parameter space of dynamical systems arising in applications is often found to be high-dimensional and difficult to explore. We construct a fast algorithm to numerically analyze "quantitative features" of dynamical systems depending on…
Since its foundations, more than one hundred years ago, the field of structural biology has strived to understand and analyze the properties of molecules and their interactions by studying the structure that they take in 3D space. However,…
Dynamical systems, that are used to model power grids, the brain, and other physical systems, can exhibit coexisting stable states known as attractors. A powerful tool to understand such systems, as well as to better predict when they may…
The dynamical properties of finite dynamical systems (FDSs) have been investigated in the context of coding theoretic problems, such as network coding and index coding, and in the context of hat games, such as the guessing game and…
Dynamical systems (DSs) provide a framework for high flexibility, robustness, and control reliability and are widely used in motion planning and physical human-robot interaction. The properties of the DS directly determine the robot's…
Complex systems may often be characterized by their hierarchical dynamics. In this paper do we present a method and an operational algorithm that automatically infer this property in a broad range of systems; discrete stochastic processes.…
Engineered infrastructure systems pose inverse problems in which hidden states, unknown parameters, and subsystem couplings must be inferred from sparse and noisy measurements. These problems are difficult because physical subsystems are…
Dimension reduction is a common strategy to study non-linear dynamical systems composed by a large number of variables. The goal is to find a smaller version of the system whose time evolution is easier to predict while preserving some of…
We present a data-driven modeling strategy to overcome improperly modeled dynamics for systems exhibiting complex spatio-temporal behaviors. We propose a Deep Learning framework to resolve the differences between the true dynamics of the…
Dynamical systems are typically governed by a set of linear/nonlinear differential equations. Distilling the analytical form of these equations from very limited data remains intractable in many disciplines such as physics, biology, climate…