Related papers: McMillan map and nonlinear Twiss parameters
If one wants to explore the properties of a dynamical system systematically one has to be able to track equilibria and periodic orbits regardless of their stability. If the dynamical system is a controllable experiment then one approach is…
In this article, we present an extension of the formulation recently developed by the authors (A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural dynamics…
In the framework of the variational principle the canonical variables describing ideal magnetohydrodynamic (MHD) flows of general type (i.e., with spatially varying entropy and nonzero values of all topological invariants) are introduced.…
This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of…
The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical…
Willems et al. showed that all input-output trajectories of a discrete-time linear time-invariant system can be obtained using linear combinations of time shifts of a single, persistently exciting, input-output trajectory of that system. In…
We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in…
Periodic orbits are among the simplest non-equilibrium solutions to dynamical systems, and they play a significant role in our modern understanding of the rich structures observed in many systems. For example, it is known that embedded…
Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the…
Recently patterns were found in the least significant digits in the logistic map orbits in the chaotic regime. However, the dynamic of these digits was not explored in deep. We propose a new interpretation of the patterns found in the least…
Beam instabilities and resonances affect the transverse dynamics in particle accelerators and, when encountered, can trigger emittance growth and beam loss. Resonance lines originate from non linear elements and effects in the lattice,…
Some results of numerical study of the canonical map with a sawtooth force are given and discovered new unexpected dynamical effects are described. In particular, it is shown that if the values of the system parameter K belong to the…
We propose a robust nonlinear model predictive control (MPC) scheme for trajectory-tracking control of autonomous vehicles at the limits of handling on non-planar road surfaces. We derive the dynamics from first principles and selectively…
We propose a method for developing the flows of stochastic dynamical systems, posed as Ito's stochastic differential equations, on a Riemannian manifold identified through a suitably constructed metric. The framework used for the stochastic…
Topological entropy is a measure of complex dynamics. In this regard, multimodal maps play an important role when it comes to study low-dimensional chaotic dynamics or explain some features of higher dimensional complex dynamics with…
The Eulerian system of dynamic equations for the ideal fluid is closed but incomplete. The complete system of dynamic equations arises after appending Lin constraints which describe motion of fluid particles in a given velocity field. The…
We establish stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota-Yorke maps under a variety of perturbations. Our family of random maps need not be close to a fixed map; thus, our results can…
Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which…
We modified the way in which the Universal Map is obtained in the regular dynamics to derive the Universal $\alpha$-Family of Maps depending on a single parameter $\alpha > 0$ which is the order of the fractional derivative in the nonlinear…
We establish almost sure invariance principles, a strong form of approximation by Brownian motion, for non-stationary time-series arising as observations on dynamical systems. Our examples include observations on sequential expanding maps,…