One-dimensional Autonomous System Research Articles

A Quantum Formalism for Abstract Dynamical Systems

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract” because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case.

A study of bouncing cosmology in framework of f(T,𝒯 ) gravity with probing of cosmographic parameters

This paper is devoted to explore bouncing cosmology in [Formula: see text] modified gravity. In the background of [Formula: see text] modified theory with [Formula: see text] being Torsion scalar and [Formula: see text] being the trace of energy–momentum tensor, the Hubble parameter is considered for the isotropic, flat, and homogenous universe. The standard bouncing scale factor has employed the exponential term to unify bounce with late-time acceleration. Phase portrait analysis reveals us that Minkowskian origin transfers to de Sitter origin in [Formula: see text]-plane and as past infinite time [Formula: see text][Formula: see text][Formula: see text] and [Formula: see text] constant which corresponds to some physical cosmological scenarios. Since field equations of [Formula: see text] gravity are of second order, a one-dimensional autonomous system has been extracted. Finally, outcomes have been plotted graphically, and any type of singular behavior has not been experienced. The study of the equation of state parameter against cosmic time reveals that phantom phase is quite significant for both models. The inertial force in terms of Hubble parameter and cosmic time gives rise to pseudo Rip. The study of cosmographic parameters tells us that the present model corresponds to [Formula: see text]CDM at a large value of cosmic time.

Phase portraits of general f(T) cosmology

We use dynamical system methods to explore the general behaviour of f(T) cosmology. In contrast to the standard applications of dynamical analysis, we present a way to transform the equations into a one-dimensional autonomous system, taking advantage of the crucial property that the torsion scalar in flat FRW geometry is just a function of the Hubble function, thus the field equations include only up to first derivatives of it, and therefore in a general f(T) cosmological scenario every quantity is expressed only in terms of the Hubble function. The great advantage is that for one-dimensional systems it is easy to construct the phase space portraits, and thus extract information and explore in detail the features and possible behaviours of f(T) cosmology. We utilize the phase space portraits and we show that f(T) cosmology can describe the universe evolution in agreement with observations, namely starting from a Big Bang singularity, evolving into the subsequent thermal history and the matter domination, entering into a late-time accelerated expansion, and resulting to the de Sitter phase in the far future. Nevertheless, f(T) cosmology can present a rich class of more exotic behaviours, such as the cosmological bounce and turnaround, the phantom-divide crossing, the Big Brake and the Big Crunch, and it may exhibit various singularities, including the non-harmful ones of type II and type IV. We study the phase space of three specific viable f(T) models offering a complete picture. Moreover, we present a new model of f(T) gravity that can lead to a universe in agreement with observations, free of perturbative instabilities, and applying the Om(z) diagnostic test we confirm that it is in agreement with the combination of SNIa, BAO and CMB data at 1σ confidence level.

Generic phase portrait analysis of finite-time singularities and generalized Teleparallel gravity**Supported by the Egyptian Ministry of Scientific Research (24-2-12)

We analyze the four common types of finite-time singularity using a generic framework of the phase portrait geometric approach. This technique requires the Friedmann system to be written as a one-dimensional autonomous system. We employ a scale factor that has been used widely in the literature to realize the four finite-time singularity types, then we give a detailed discussion for each case showing possible novel models. Moreover, we show how different singularity types can play essential roles in different cosmological scenarios. Among several modified gravity theories, we show that the f(T) cosmology is compatible with the phase portrait analysis, since the field equations include Hubble derivatives only up to first order. Therefore, we reconstruct the f(T) theory which generates these phase portraits. We also perform a complementary analysis using the effective equation of state. Furthermore, we investigate the role of the torsion fluid in realizing the cosmic singularities.

Lagrangians and Hamiltonians for One-Dimensional Autonomous Systems

An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system that has certain quasi-relativistic properties. A new method based on a Taylor series expansion is used to obtain the associated Hamiltonian for this system. These results have the usual expression for a conservative system when the dissipation parameter goes to zero. An example of this approach is given.

One-Dimensional Autonomous Systems and Dissipative Systems

The Lagrangian and the Generalized Linear Momentum are given in terms of a constant of motion for a one-dimensional autonomous system. The possibility of having an explicit Hamiltonian expression is also analyzed. The approach is applied to some dissipative systems.