scholarly journals A non-linear mathematical model for the X-ray variability classes of the microquasar GRS 1915+105 – II. Transition and swaying classes

2020 ◽  
Vol 496 (2) ◽  
pp. 1697-1705 ◽  
Author(s):  
E Massaro ◽  
F Capitanio ◽  
M Feroci ◽  
T Mineo ◽  
A Ardito ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Input Function ◽  
Unstable Equilibrium ◽  
Light Curves ◽  
Extended Model ◽  
X Ray ◽  
Non Linear ◽  
Non Linear System

ABSTRACT The complex time evolution in the X-ray light curves of the peculiar black hole binary GRS 1915+105 can be obtained as solutions of a non-linear system of ordinary differential equations derived from the Hindmarsh–Rose model and modified introducing an input function depending on time. In the first paper, assuming a constant input with a superposed white noise, we reproduced light curves of the classes ρ, χ, and δ. We use this mathematical model to reproduce light curves, including some interesting details, of other eight GRS 1915+105 variability classes either considering a variable input function or with small changes of the equation parameters. On the basis of this extended model and its equilibrium states, we can arrange most of the classes in three main types: (i) stable equilibrium patterns (classes ϕ, χ, α″, θ, ξ, and ω) whose light curve modulation follows the same time-scale of the input function, because changes occur around stable equilibrium points; (ii) unstable equilibrium patterns characterized by series of spikes (class ρ) originated by a limit cycle around an unstable equilibrium point; and (iii) transition pattern (classes δ, γ, λ, κ, and α′), in which random changes of the input function induce transitions from stable to unstable regions originating either slow changes or spiking, and the occurrence of dips and red noise. We present a possible physical interpretation of the model based on the similarity between an equilibrium curve and literature results obtained by numerical integrations of slim disc equations.

scholarly journals A non-linear mathematical model for the X-ray variability classes of the microquasar GRS 1915+105 – I. Quiescent, spiking states, and quasi-periodic oscillations

2020 ◽  
Vol 495 (1) ◽  
pp. 1110-1121 ◽  
Author(s):  
E Massaro ◽  
F Capitanio ◽  
M Feroci ◽  
T Mineo ◽  
A Ardito ◽  
...  
Keyword(s):  
Differential Equations ◽  
Equilibrium Points ◽  
Low Frequency ◽  
Input Function ◽  
Light Curves ◽  
Stable Region ◽  
Periodic Oscillations ◽  
X Ray ◽  
Unstable Solutions ◽  
Unified View

ABSTRACT The microquasar GRS 1915+105 is known to exhibit a very variable X-ray emission on different time-scales and patterns. We propose a system of two ordinary differential equations, adapted from the Hindmarsh–Rose model, with two dynamical variables x(t), y(t), and an input constant parameter J0, to which we added a random white noise, whose solutions for the x(t) variable reproduce consistently the X-ray light curves of several variability classes as well as the development of low-frequency quasi-periodic oscillations (QPO). We show that changing only the value of J0, the system moves from stable to unstable solutions and the resulting light curves reproduce those of the quiescent classes like ϕ and χ, the δ class and the spiking ρ class. Moreover, we found that increasing the values of J0 the system induces high-frequency oscillations that evolve into QPO when it moves into another stable region. This system of differential equations gives then a unified view of the variability of GRS 1915+105 in term of transitions between stable and unstable states driven by a single input function J0. We also present the results of a stability analysis of the equilibrium points and some considerations on the existence of periodic solutions.

scholarly journals A non-linear mathematical model for the X-ray variability of the microquasar GRS 1915+105 – III. Low-frequency quasi-periodic oscillations

2020 ◽  
Vol 497 (1) ◽  
pp. 405-415
Author(s):  
E Massaro ◽  
F Capitanio ◽  
M Feroci ◽  
T Mineo
Keyword(s):  
Mathematical Model ◽  
Equilibrium Points ◽  
Low Frequency ◽  
Input Function ◽  
Power Density Spectrum ◽  
Periodic Oscillations ◽  
Density Spectrum ◽  
X Ray ◽  
Almost All ◽  
The Mathematical Model

ABSTRACT The X-ray emission from the microquasar GRS 1915+105 shows, together with a very complex variability on different time-scales, the presence of low-frequency quasi-periodic oscillations (LFQPOs) at frequencies lower than ∼30 Hz. In this paper, we demonstrate that these oscillations can be consistently and naturally obtained as solutions of a system of two ordinary differential equations, which is able to reproduce almost all variability classes of GRS 1915+105. We modified the Hindmarsh–Rose model and obtained a system with two dynamical variables x(t), y(t), where the first one represents the X-ray flux from the source, and an input function J(t), whose mean level J0 and its time evolution is responsible of the variability class. We found that for values of J0 around the boundary between the unstable and the stable interval, where the equilibrium points are of spiral type, one obtains an oscillating behaviour in the model light curve similar to the observed ones with a broad Lorentzian feature in the power density spectrum and, occasionally, with one or two harmonics. Rapid fluctuations of J(t), as those originating from turbulence, stabilize the LFQPOs, resulting in a slowly amplitude modulated pattern. To validate the model, we compared the results with real RXTE data, which resulted remarkably similar to those obtained from the mathematical model. Our results allow us to favour an intrinsic hypothesis on the origin of LFQPOs in accretion discs ultimately related to the same mechanism responsible for the spiking limit cycle.

scholarly journals Non-Linear Resonances in Accretion Disks and QPOs

2004 ◽  
Vol 194 ◽  
pp. 128-129
Author(s):  
Włodek Kluźniak
Keyword(s):  
Black Holes ◽  
Neutron Stars ◽  
Accretion Disk ◽  
Accretion Disks ◽  
High Frequency ◽  
Light Curves ◽  
X Ray ◽  
Non Linear ◽  
Low Mass ◽  
X Ray Binaries

AbstractNon-linear oscillations in the accretion disk are favored as an explanation of high-frequency QPOs observed in the light curves of low-mass X-ray binaries containing neutron stars, black holes, or white dwarfs.

scholarly journals Simulations of LQR Control on Linear Spacecraft Dynamics about Asteroid Equilibrium Points

2021 ◽  
Author(s):  
Abhijeet Aryal
Keyword(s):  
Equilibrium Point ◽  
Equilibrium Points ◽  
Primary Objective ◽  
Spacecraft Dynamics ◽  
Position Change ◽  
Non Linear ◽  
Lqr Controller ◽  
Dynamics Simulations ◽  
Non Linear System ◽  
Application Requirements

Future exploratory missions to asteroids may require a spacecraft to perform attitude and position change maneuvers within small perturbations of the equilibrium point to conduct measurements and make observations based on mission requirements. The non-linear dynamics of the spacecraft can be approximated to be linear given that the system operates about an equilibrium point and the signals are small. Based on this, the linearized system is equivalent to the non-linear system within a limited operating range. This project follows this precedent and applies a closed loop LQR controller to perturbations of 1%, 2% and 5% from asteroid 101955 Bennu’s equilibrium points. The LQR controller methodology requires that weighting matrices Q and R which penalize the states and the controls respectively to be iterated for – depending on the application requirements. The iteration procedure is the primary objective of this project and is conducted for six different spacecraft orientations about eight different equilibrium points. The procedure examines the settling times and response plots to critique the performance of the controller. This paper presents the underlying control theory, the modelling scenario, the simulation procedure, results and some patterns discovered in the results. This project was conducted in conjunction with [3] and presents the results of simulations based on linearized spacecraft dynamics. The results of the non-linear spacecraft dynamics simulations can be found in [3].

scholarly journals Dynamics and Control of a Magnetic Transducer Array Using Multi-Physics Models and Artificial Neural Networks

Sensors ◽  
2021 ◽  
Vol 21 (20) ◽  
pp. 6788
Author(s):  
Georgios Tsakyridis ◽  
Nikolaos I. Xiros
Keyword(s):  
Equilibrium Points ◽  
Current Source ◽  
Pole Placement ◽  
Design Parameters ◽  
System Response ◽  
State Variables ◽  
Fast Tracking ◽  
Non Linear ◽  
Artificial Neural ◽  
Non Linear System

A linear mechanical oscillator is non-linearly coupled with an electromagnet and its driving circuit through a magnetic field. The resulting non-linear dynamics are investigated using magnetic circuit approximations without major loss of accuracy and in the interest of brevity. Different computational approaches to simulate the setup in terms of dynamical system response and design parameters optimization are pursued. A current source operating in baseband without modulation directly feeds the electromagnet, which consists commonly of a solenoid and a horseshoe-shaped core. The electromagnet is then magnetically coupled to a mass made of soft magnetic material and attached to a spring with damping. The non-linear system is described by a linearized steady-space representation while is examined for controllability and observability. A controller using a pole placement approach is built to stabilize the element. Drawing upon the fact that coupling works both ways, enabling estimation of the mass position and velocity (state variables) by processing the induced voltage across the electromagnet, a state observer is constructed. Accurate and fast tracking of the state variables, along with the possibility of driving more than one module from the same source using modulation, proves the applicability of the electro-magneto-mechanical transducer for sensor applications. Next, a three-layer feed-forward artificial neural network (ANN) system equivalent was trained using the non-linear plant-linear controller-linear observer configuration. Simulations to investigate the robustness of the system with respect to different equilibrium points and input currents were carried out. The ANN proved robust with respect to position accuracy.

scholarly journals Analysis of the Dynamics of SI-SI-SEIR Avian Influenza A(H7N9) Epidemic Model with Re-infection

2020 ◽  
pp. 43-73
Author(s):  
Oluwafemi I. Bada ◽  
Abayomi S. Oke ◽  
Winfred N. Mutuku ◽  
Patrick O. Aye
Keyword(s):  
Mathematical Model ◽  
Avian Influenza ◽  
Global Stability ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Influenza A ◽  
Equilibrium Points ◽  
Local And Global Stability ◽  
Continuous Contact ◽  
Non Linear

The spread of Avian influenza in Asia, Europe and Africa ever since its emergence in 2003, has been endemic in many countries. In this study, a non-linear SI-SI-SEIR Mathematical model with re-infection as a result of continuous contact with both infected poultry from farm and market is proposed. Local and global stability of the three equilibrium points are established and numerical simulations are used to validate the results.

scholarly journals Local Stability Analysis of a Mathematical Model of the Interaction of Two Populations of Differential Equations (Host-Parasitoid)

2016 ◽  
Vol 5 (1) ◽  
pp. 9
Author(s):  
Dewi Anggreini
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Stable Equilibrium ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Real Life ◽  
Host Population ◽  
Matrix Analysis ◽  
The Stability ◽  
Two Populations

<p>Mathematical model has many benefits in life, especially the development of science and application to other fields. The mathematical model seeks to represent real-life problems formulated mathematically to get the right solution. This research is the application of mathematical models in the field of biology that examines the interaction of the two populations that host populations and parasitoid populations. This study differs from previous studies that examine the interaction of two more species that prey and predators where predators kill prey quickly. In this study the parasitoid population slowly killing the host population by living aboard and take food from the host population it occupies. In this study of differential equations are used to construct a mathematical model was particularly focused on the stability of the local mathematical model of interaction of two differential equations that host and parasitoid populations. Stability discussed in this study are stable equilibrium points are obtained from the characteristic equation systems of differential equations host and parasitoid interactions. Type the stability of the equilibrium point is determined on the eigenvalues of the Jacobian matrix. Analysis of stability is obtained by determining the eigenvalues of the Jacobian matrix around equilibrium points. Having obtained the stable equilibrium points are then given in the form of charts and portraits simulation phase to determine the behavior of the system in the future.</p>

Neural Network Modelling Applied for Model-Based Fault Detection

Volume 3 ◽  
2004 ◽  
Author(s):  
Zhanqun Shi ◽  
Yibo Fan ◽  
Fengshou Gu ◽  
Abdul-Hannan Ali ◽  
Andrew Ball
Keyword(s):  
Neural Network ◽  
Mathematical Model ◽  
Fault Detection ◽  
Linear Part ◽  
Model Parameters ◽  
Network Modelling ◽  
Model Based ◽  
Neural Network Modelling ◽  
Non Linear ◽  
Non Linear System

This paper aims to combine neural network modelling with model-based fault detection. An accurate and robust model is critical in model-based fault detection. However, the development of such a model is the most difficult task especially when a non-linear system is involved. The problem comes not only from the lack of concerned information about model parameters, but also from the inevitable linearization. In order to solve this problem, neural networks are introduced in this paper. Instead of using conventional neural network modelling, the neural network is only used to approximate the non-linear part of the system, leaving the linear part to be represented by a mathematical model. This new scheme of integration between neural network and mathematical model (NNMM) allows the compensation of the error from conventional modelling methods. Simultaneously, it keeps the residual signatures physically interpretable.

scholarly journals Simulations of LQR Control on Linear Spacecraft Dynamics about Asteroid Equilibrium Points

2021 ◽  
Author(s):  
Abhijeet Aryal
Keyword(s):  
Equilibrium Point ◽  
Equilibrium Points ◽  
Primary Objective ◽  
Spacecraft Dynamics ◽  
Position Change ◽  
Non Linear ◽  
Lqr Controller ◽  
Dynamics Simulations ◽  
Non Linear System ◽  
Application Requirements

Future exploratory missions to asteroids may require a spacecraft to perform attitude and position change maneuvers within small perturbations of the equilibrium point to conduct measurements and make observations based on mission requirements. The non-linear dynamics of the spacecraft can be approximated to be linear given that the system operates about an equilibrium point and the signals are small. Based on this, the linearized system is equivalent to the non-linear system within a limited operating range. This project follows this precedent and applies a closed loop LQR controller to perturbations of 1%, 2% and 5% from asteroid 101955 Bennu’s equilibrium points. The LQR controller methodology requires that weighting matrices Q and R which penalize the states and the controls respectively to be iterated for – depending on the application requirements. The iteration procedure is the primary objective of this project and is conducted for six different spacecraft orientations about eight different equilibrium points. The procedure examines the settling times and response plots to critique the performance of the controller. This paper presents the underlying control theory, the modelling scenario, the simulation procedure, results and some patterns discovered in the results. This project was conducted in conjunction with [3] and presents the results of simulations based on linearized spacecraft dynamics. The results of the non-linear spacecraft dynamics simulations can be found in [3].

Imprecision of Laboratory Determinations and Diagnostic Accuracy: Theoretical Considerations

1974 ◽  
Vol 13 (03) ◽  
pp. 151-158 ◽  
Author(s):  
D. A. B. Lindbebo ◽  
Fr. R. Watson
Keyword(s):  
Mathematical Model ◽  
Diagnostic Accuracy ◽  
Diagnostic Process ◽  
Clinical Laboratories ◽  
Non Linear ◽  
Theoretical Considerations ◽  
Made In

Recent studies suggest the determinations of clinical laboratories must be made more precise than at present. This paper presents a means of examining benefits of improvement in precision. To do this we use a mathematical model of the effect upon the diagnostic process of imprecision in measurements and the influence upon these two of Importance of Diagnosis and Prevalence of Disease. The interaction of these effects is grossly non-linear. There is therefore no proper intuitive answer to questions involving these matters. The effects can always, however, be calculated.Including a great many assumptions the modeling suggests that improvements in precision of any determination ought probably to be made in hospital rather than screening laboratories, unless Importance of Diagnosis is extremely high.

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