Related papers: Poincar\'e-Bendixson Theorem for Hybrid Systems
A novel theory of hybrid quantum-classical systems is developed, utilizing the mathematical framework of constrained dynamical systems on the quantum-classical phase space. Both, the quantum and the classical descriptions of the respective…
We present a consistent formalism to describe the dynamics of hybrid systems with mixed classical and quantum degrees of freedom. The probability function of the system, which, in general, will be a combination of the classical distribution…
Hamiltonian theory of hybrid quantum-classical systems is used to study dynamics of the classical subsystem coupled to different types of quantum systems. It is shown that the qualitative properties of orbits of the classical subsystem…
We consider how the reduced dynamics of an open quantum system coupled to an environment admits the Poincar\'e symmetry. The reduced dynamics is described by a dynamical map, which is given by tracing out the environment from the total…
In this article we present a Lagrangian representation for evolutionary systems with a Hamiltonian structure determined by a differential-geometric Poisson bracket of the first order associated with metrics of constant curvature.…
This paper focuses on the equilibria and their regions of attraction for continuous-time nonlinear dynamical systems. The classical Poincar\'e--Hopf Theorem from differential topology is used to derive a result on a sufficient condition on…
In this paper we consider a family of system with 2 predators feeding on one prey. We show how to construct a positively invariant set in which it is possible to define a Poincar\'e map for examining the behaviour of the system, mainly in…
Nonlinear dynamics of a bouncing ball moving vertically in a gravitational field and colliding with a moving limiter is considered and the Poincar\'e map, describing evolution from an impact to the next impact, is described. Displacement of…
Constrained Hamiltonian description of the classical limit is utilized in order to derive consistent dynamical equations for hybrid quantum-classical systems. Starting with a compound quantum system in the Hamiltonian formulation conditions…
We present AI Poincar\'e, a machine learning algorithm for auto-discovering conserved quantities using trajectory data from unknown dynamical systems. We test it on five Hamiltonian systems, including the gravitational 3-body problem, and…
Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. As typical in dynamical analysis, an essential goal is to study the long-term behavior of these systems. In this work, we present two different…
Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the…
In this article we provide a proof of the so called absolute continuity theorem for random dynamical systems on $R^d$ which have an invariant probability measure. First we present the construction of local stable manifolds in this case.…
We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is…
We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies…
The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical…
Statistics of Poincar\' e recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
The presence of two compatible Hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of Hamiltonian structures, there are associated conservation laws (first integrals).…
We present how a probabilistic model can describe the asymptotic behaviour of the iterations, especially for ODE with an approach of the Poincar\'e-Bendixon's problem in $\mathbb{R}^d$. On pr\'esente un mod\`ele probabiliste pour d\'ecrire…