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Reasoning about the physical world requires models that are endowed with the right inductive biases to learn the underlying dynamics. Recent works improve generalization for predicting trajectories by learning the Hamiltonian or Lagrangian…
Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational…
Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be…
A fundamental issue in nonlinear dynamics and statistical physics is how to distinguish chaotic from stochastic fluctuations in short experimental recordings. This dilemma underlies many complex systems models from stochastic gene…
In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of $\R^N$, whose most familiar example is provided by the $N-$dimensional torus $\T ^N$. It is proved that…
Lagrangian coherent structures (LCSs) are material surfaces that shape finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from…
We propose a dynamic domain semi-Lagrangian method for stochastic Vlasov equations driven by transport noises, which arise in plasma physics and astrophysics. This method combines the volume-preserving property of stochastic characteristics…
In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta et al. (2000), Mean Exponential Growth factor of Nearby…
In the application of the Expectation Maximization algorithm to identification of dynamical systems, internal states are typically chosen as latent variables, for simplicity. In this work, we propose a different choice of latent variables,…
We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally,…
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the…
The study of the phase space of multidimensional systems is one of the central open problems in dynamical systems. Being able to distinguish chaoticity from regularity in nonlinear dynamical systems, as well as to determine the subspace of…
We introduce a new dynamical indicator of stability based on the Extreme Value statistics showing that it provides an insight on the local stability properties of dynamical systems. The indicator perform faster than other based on the…
Simple dynamical systems -- with a small number of degrees of freedom -- can behave in a complex manner due to the presence of chaos. Such systems are most often (idealized) limiting cases of more realistic situations. Isolating a small…
In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We…
While classical chaos is defined via a system's sensitive dependence on its initial conditions (SDIC), this notion does not directly extend to quantum systems. Instead, recent works have established defining both quantum and classical chaos…
A 3D dynamical model with a quasi-homogeneous core and a disk component is used for the chaos control in the central parts of elliptical galaxy. Numerical experiments in the 2D system show a very complicated phase plane with a large chaotic…
This paper investigates manipulability of interactive Lagrangian systems with parametric uncertainty and communication/sensing constraints. Two standard examples are teleoperation with a master-slave system and teaching operation of robots.…
We study coupled dynamics on networks using symbolic dynamics. The symbolic dynamics is defined by dividing the state space into a small number of regions (typically 2), and considering the relative frequencies of the transitions between…
In this paper, we establish the sufficient conditions guaranteeing global uniform exponential stability, or at least global asymptotic stability, of all solutions for nonlinear dynamical systems, also known as global incremental stability…