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We propose a minimal model for the emergence of a directed flow in autonomous Hamiltonian systems. It is shown that internal breaking of the spatio-temporal symmetries, via localised initial conditions, that are unbiased with respect to the…
We explore the use of Physics Informed Neural Networks to analyse nonlinear Hamiltonian Dynamical Systems with a first integral of motion. In this work, we propose an architecture which combines existing Hamiltonian Neural Network…
We detail how incorporating physics into neural network design can significantly improve the learning and forecasting of dynamical systems, even nonlinear systems of many dimensions. A map building perspective elucidates the superiority of…
We review in detail the Hamiltonian dynamics for constrained systems. Emphasis is put on the total Hamiltonian system rather than on the extended Hamiltonian system. We provide a systematic analysis of (global and local) symmetries in total…
We describe the Hamiltonian structures, including the Poisson brackets and Hamiltonians, for free boundary problems for incompressible fluid flows with vorticity. The Hamiltonian structure is used to obtain variational principles for…
Hamiltonian systems with multiple timescales arise in molecular dynamics, classical mechanics, and theoretical physics. Long-time numerical integration of such systems requires resolving fast dynamics with very small time steps, which…
We consider the description of open quantum systems with probability sinks (or sources) in terms of general non-Hermitian Hamiltonians.~Within such a framework, we study novel possible definitions of the quantum linear entropy as an…
The {\it curvature} and the {\it reduced curvature} are basic differential invariants of the pair: (Hamiltonian system, Lagrange distribution) on the symplectic manifold. We show that negativity of the curvature implies that any bounded…
The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set or a $W$-set in a manifold. These are certain classes of compact invariant sets in whose vicinity the…
Dynamical systems can be quantised only if they are Hamiltonian. This prompts the question from which our talk gets its title. We show how the simple predator-prey equation and the damped harmonic oscillator can be considered to be…
We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous…
The static and dynamical properties of a one-dimensional quantum system described by a non-Hermitian Hamiltonian of the so-called Hatano-Nelson type; a tight-binding model with asymmetric (or non-reciprocal) hopping, are studied. The static…
This paper studies conditions for invariance of dynamical systems on stratified do- mains as originally introduced by Bressan and Hong. We establish Hamiltonian conditions for both weak and strong invariance of trajectories on systems with…
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…
This paper considers systems subject to nonholonomic constraints which are not uniform on the whole configuration manifold. When the constraints change, the system undergoes a transition in order to comply with the new imposed conditions.…
Newtonian dynamical systems accepting the normal shift on an arbitrary Riemannian manifold are considered. Partial differential equations forming the weak and additional normality conditions for them are reported.
A reduced dynamical model is derived which describes the interaction of weak inertia-gravity waves with nonlinear vortical motion in the context of rotating shallow-water flow. The formal scaling assumptions are (i) that there is a…
Hamiltonian systems are a classical example in the ergodic theory of flows with an invariant measure. In this matter, we present a brief introduction to measure theory and prove the Poincare recurrence theorem to present the conditions for…
Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered. For them the determinating equations making the weak normality condition are derived. The expansion for the algebra of tensor…
Based on two dissipative models, universal asymptotic behavior of flow equations for Hamiltonians is found and discussed. Universal asymptotic behavior only depends on fundamental bath properties but not on initial system parameters, and…