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Linear finite dynamical systems play an important role, for example, in coding theory and simulations. Methods for analyzing such systems are often restricted to cases in which the system is defined over a field %and usually strive to…
Outliers contaminating data sets are a challenge to statistical estimators. Even a small fraction of outlying observations can heavily influence most classical statistical methods. In this paper we propose generalized spherical principal…
The dynamics of coherent structures present in real-world environmental data is analyzed. The method developed in this Paper combines the power of the Proper Orthogonal Decomposition (POD) technique to identify these coherent structures in…
This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear…
Smooth autonomous dynamical systems modeled by ordinary differential equations (ODEs) cannot robustly and globally stabilize a point in compact, boundaryless manifolds. This obstruction, which is topological in nature, implies that…
Hybrid dynamical systems have proven to be a powerful modeling abstraction, yet fundamental questions regarding the dynamical properties of these systems remain. In this paper, we develop a novel class of relaxations which we use to recover…
This paper is devoted to constructing and studying exactly solvable dynamical systems in discrete time obtained from some algebraic operations on matrices, to reductions of such systems leading to classical field theory models in…
We present a new class of stochastic, geometrically-driven optimization algorithms on the orthogonal group $O(d)$ and naturally reductive homogeneous manifolds obtained from the action of the rotation group $SO(d)$. We theoretically and…
We introduce a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. Motivated by models of regulatory networks, we construct a state transition graph from a piecewise affine…
We analyse the dynamics of a hard-sphere lattice gas on generalised Bethe lattices using a projective approximation scheme (PAS). The latter consists in mapping the system's dynamics to a finite set of global observables, closure of the…
This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each…
We provide a self-contained geometric description of the geodesic flow in the three-dimensional Lie group $\mathrm{Sol}$, one of Thurston's eight model geometries. The geometry of geodesics is governed by a single invariant $k\in[0,1]$, its…
When developing control laws for robotic systems, the principle factor when examining their performance is choosing inputs that allow smooth tracking to a reference input. In the context of robotic manipulation, this involves translating an…
Many applications of geometry modeling and computer graphics necessite accurate curvature estimations of curves on the plane or on manifolds. In this paper, we define the notion of the discrete geodesic curvature of a geodesic polygon on a…
In this work, we utilize discrete geometric mechanics to derive a 2nd-order variational integrator so as to simulate rigid body dynamics. The developed integrator is to simulate the motion of a free rigid body and a quad-rotor. We…
In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and…
The long-term dynamics of the geostationary Earth orbits (GEO) is revisited through the application of canonical perturbation theory. We consider a Hamiltonian model accounting for all major perturbations: geopotential at order and degree…
Algorithms that use derivatives of governing equations have accelerated rigid robot simulations and improved their accuracy, enabling the modeling of complex, real-world capabilities. However, extending these methods to soft and hybrid…
Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid…
Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented.…