Related papers: The Extended Alpha Group Dynamic Mapping
We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations (ODEs), which may be generated from partial differential equations in plasma physics. We map a…
This paper studies the contraction property of time-varying differential-algebraic equation (DAE) systems by embedding them to higher-dimension ordinary differential equation (ODE) systems. The first result pertains to the equivalence of…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…
The influence of nonequilibrium initial values of the order parameter on its evolution at a critical point is described using a renormalization group approach of the field theory. The dynamic critical exponent $\theta'$ of the short time…
We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters $a,b,c,d$) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a $\ell$-Volterra quadratic…
The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions…
In this paper we study dynamical systems generated by an evolution operator of a dioecious population. This evolution operator is a six-parametric, non-linear operator mapping $[0,1]^2$ to itself. We find all fixed points and under some…
Quantum double models, such as the toric code, can be constructed from transfer matrices of lattice gauge theories with discrete gauge groups and parametrized by the center of the gauge group algebra and its dual. For general choices of…
Chaos in classical systems has been studied in plenty over many years. Although the search for chaos in quantum systems has been an area of prominent research over the last few decades, the detailed analysis of many inherently chaotic…
The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not…
We investigate the geometric phases and the Bargmann invariants associated with a multi-level quantum systems. In particular, we show that a full set of `gauge-invariant' objects for an $n$-level system consists of $n$ geometric phases and…
We introduce simple models of genetic regulatory networks and we proceed to the mathematical analysis of their dynamics. The models are discrete time dynamical systems generated by piecewise affine contracting mappings whose variables…
We consider the problem of the determination of the isotropy classes of the orbit spaces of all the real linear groups, with three independent basic invariants satisfying only one independent relation. The results are obtained in the $\hat…
We analyze the behavior of a non-Hermitian opened one-dimensional quantum system with $\mathcal{PT}$ symmetry. This system is built by a dimer, with balanced gains and losses described by a parameter $\gamma$. By varying $\gamma$ the system…
It is shown, under weak conditions, that the dynamical evolution of an important class of large systems of globally coupled, heterogeneous frequency, phase oscillators is, in an appropriate physical sense, time-asymptotically attracted…
Artificial neural networks, widely recognised for their role in machine learning, are now transforming the study of ordinary differential equations (ODEs), bridging data-driven modelling with classical dynamical systems and enabling the…
In this paper, we propose a framework to investigate the collective dynamics in ensembles of globally coupled phase oscillators when higher-order modes dominate the coupling. The spatiotemporal properties of the attractors in various…
In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when…
Humans are often incapable of precisely identifying and implementing the desired control strategy in controlling unstable dynamical systems. That is, the operator of a dynamical system treats the current control effort as acceptable even if…