scholarly journals On periodic solutions associated with the nonlinear feedback loop in the non-dissipative Lorenz model

2016 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Homoclinic Orbit ◽  
Feedback Loop ◽  
The Other ◽  
Nonlinear Feedback ◽  
Restoring Force ◽  
Oscillatory Solutions ◽  
Energy Cycle ◽  
Large Cycle

Abstract. In this study, we discuss the role of the linear heating term and nonlinear terms associated with a nonlinear feedback loop in the energy cycle of the three-dimensional (X–Y–Z) non-dissipative Lorenz model (3D-NLM), where (X, Y, Z) represent the solutions in the phase space. Using trigonometric functions, we first present the closed-form solution of the nonlinear equation d2X/dτ2 + (X2/2)X = 0 without the heating term (i.e., rX), (where τ is a non-dimensional time and r is the normalized Rayleigh number), a solution that has not been previously documented. Since the solution of the simplified 3D-NLM is oscillatory (wave-like) and since the nonlinear term (X3) is associated with the nonlinear feedback loop, here, we suggest that the nonlinear feedback loop may act as a restoring force. When the heating term is considered, the system yields three critical points. A linear analysis suggests that the origin (i.e., the trivial critical point) is a saddle point and that the other two non-trivial critical points are stable. Here, we provide an analysis for three types of solutions that are associated with these critical points. Two of the solutions represent closed curves that travel around one non-trivial critical point or all three critical points. The third type of solution, appearing to connect the stable and unstable manifolds of the saddle point, is called the homoclinic orbit. Using the solution that contains one non-trivial critical point, here, we show that the competing impact of the nonlinear restoring force and the linear (heating) force determines the partitions of the averaged available potential energy from the Y and Z modes. Based on the energy analysis, an energy cycle with four different regimes is identified. The cycle is only half of a "large" cycle, displaying the wing pattern of a glasswinged butterfly. The other half cycle is antisymmetric with respect to the origin. The two types of oscillatory solutions with either a small cycle or a large cycle are orbitally stable. As compared to the oscillatory solutions, the homoclinic orbit is not periodic because it "takes forever" to reach the origin. Two trajectories with starting points near the homoclinic orbit may be diverged because one moves with a small cycle and the other moves with a large cycle. Therefore, the homoclinic orbit is not orbitally stable. In a future study, dissipation and/or additional nonlinear terms will be included in order to determine how their interactions with the original nonlinear feedback loop and the heating term may change the periodic orbits (as well as homoclinic orbits) to become quasi-periodic orbits and chaotic solutions.

Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop

2018 ◽  
Vol 28 (06) ◽  
pp. 1850072 ◽  
Author(s):  
Sara Faghih-Naini ◽  
Bo-Wen Shen
Keyword(s):  
Periodic Solution ◽  
Feedback Loop ◽  
Three Dimensional ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Oscillatory Solutions ◽  
Nonlinear Temperature

A recent study suggested that the nonlinear feedback loop (NFL) of the three-dimensional nondissipative Lorenz model (3D-NLM) serves as a nonlinear restoring force by producing nonlinear oscillatory solutions as well as linear periodic solutions near a nontrivial critical point. This study discusses the role of the extension of the NFL in producing quasi-periodic trajectories using a five-dimensional nondissipative Lorenz model (5D-NLM). An analytical solution to the locally linear 5D-NLM is first obtained to illustrate the association of the extended NFL and two incommensurate frequencies whose ratio is irrational, yielding a quasi-periodic solution. The quasi-periodic solution trajectory moves endlessly on a torus but never intersects itself. While the NFL of the 3D-NLM consists of a pair of downscaling and upscaling processes, the extended NFL within the 5D-NLM additionally introduces two new pairs of downscaling and upscaling processes that are enabled by two high wavenumber modes. One pair of downscaling and upscaling processes provides a two-way interaction between the original (primary) Fourier modes of the 3D-NLM and the newly-added (secondary) Fourier modes of the 5D-NLM. The other pair of downscaling and upscaling processes involves interactions amongst the secondary modes. By comparing the numerical simulations using one- and two-way interactions, we illustrate that the two-way interaction is crucial for producing the quasi-periodic solution. A follow-up study using a 7D nondissipative LM shows that a further extension of NFL, which may appear throughout the spatial mode-mode interactions rooted in the nonlinear temperature advection, is capable of producing one more incommensurate frequency.

scholarly journals On the nonlinear feedback loop and energy cycle of the non-dissipative Lorenz model

2014 ◽  
Vol 1 (1) ◽  
pp. 519-541 ◽  
Author(s):  
B.-W. Shen
Keyword(s):  
Saddle Point ◽  
Feedback Loop ◽  
Initial Perturbation ◽  
Half Cycle ◽  
Nonlinear Feedback ◽  
Lorenz Model ◽  
Restoring Force ◽  
Linear Heating ◽  
Nonlinear Restoring Force ◽  
Energy Cycle

Abstract. In this study, we discuss the role of the nonlinear terms and linear (heating) term in the energy cycle of the three-dimensional (X–Y–Z) non-dissipative Lorenz model (3D-NLM). (X, Y, Z) represent the solutions in the phase space. We first present the closed-form solution to the nonlinear equation d2 X/dτ2+ (X2/2)X = 0, τ is a non-dimensional time, which was never documented in the literature. As the solution is oscillatory (wave-like) and the nonlinear term (X2) is associated with the nonlinear feedback loop, it is suggested that the nonlinear feedback loop may act as a restoring force. We then show that the competing impact of nonlinear restoring force and linear (heating) force determines the partitions of the averaged available potential energy from Y and Z modes, respectively, denoted as APEY and APEZ. Based on the energy analysis, an energy cycle with four different regimes is identified with the following four points: A(X, Y) = (0,0), B = (Xt, Yt), C = (Xm, Ym), and D = (Xt, -Yt). Point A is a saddle point. The initial perturbation (X, Y, Z) = (0, 1, 0) gives (Xt, Yt) = ( √ 2σr , r) and (Xm, Ym) = (2√  σr , 0). σ is the Prandtl number, and r is the normalized Rayleigh number. The energy cycle starts at (near) point A, A+ = (0, 0+) to be specific, goes through B, C, and D, and returns back to A, i.e., A- = (0,0-). From point A to point B, denoted as Leg A–B, where the linear (heating) force dominates, the solution X grows gradually with { KE↑, APEY↓, APEZ↓}. KE is the averaged kinetic energy. We use the upper arrow (↑) and down arrow (↓) to indicate an increase and decrease, respectively. In Leg B–C (or C–D) where nonlinear restoring force becomes dominant, the solution X increases (or decreases) rapidly with {KE↑, APEY↑, APEZ↓} (or {KE↓, APEY↓, APEZ↑}). In Leg D–A, the solution X decreases slowly with {KE↓, APEY↑, APEZ↑ }. As point A is a saddle point, the aforementioned cycle may be only half of a "big" cycle, displaying the wing pattern of a glasswinged butterfly, and the other half cycle is antisymmetric with respect to the origin, namely B = (-Xt, -Yt), C = (-Xm, 0), and D = (-Xt, Yt).

scholarly journals Multi-critical point principle as the origin of classical conformality and its generalizations

2021 ◽  
Author(s):  
Hikaru Kawai ◽  
Kiyoharu Kawana
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Effective Potential ◽  
Vacuum Expectation ◽  
High Energy ◽  
Low Energy ◽  
Coupling Constants ◽  
The Other ◽  
Fine Tuning ◽  
General Scalar

Abstract Multi-critical point principle (MPP) is one of the interesting theoretical possibilities that can explain the fine-tuning problems of the Universe. It simply claims that “the coupling constants of a theory are tuned to one of the multi-critical points, where some of the extrema of the effective potential are degenerate.” One of the simplest examples is the vanishing of the second derivative of the effective potential around a minimum. This corresponds to the so-called classical conformality, because it implies that the renormalized mass m2 vanishes. More generally, the form of the effective potential of a model depends on several coupling constants, and we should sweep them to find all the multi-critical points. In this paper, we study the multi-critical points of a general scalar field φ at one-loop level under the circumstance that the vacuum expectation values of the other fields are all zero. For simplicity, we also assume that the other fields are either massless or so heavy that they do not contribute to the low energy effective potential of φ. This assumption makes our discussion very simple because the resultant one-loop effective potential is parametrized by only four effective couplings. Although our analysis is not completely general because of the assumption, it still can be widely applicable to many models of the Coleman-Weinberg mechanism and its generalizations. After classifying the multi-critical points at low-energy scales, we will briefly mention the possibility of criticalities at high-energy scales and their implications for cosmology.

The Propaganda Feedback Loop

2018 ◽  
Author(s):  
Yochai Benkler ◽  
Robert Faris ◽  
Hal Roberts
Keyword(s):  
Case Studies ◽  
Feedback Loop ◽  
The Other ◽  
Hillary Clinton ◽  
Donald Trump ◽  
The Public ◽  
False Reporting ◽  
Media Ecosystem ◽  
Crowd Out ◽  
Selection Of

This chapter presents a model of the interaction of media outlets, politicians, and the public with an emphasis on the tension between truth-seeking and narratives that confirm partisan identities. This model is used to describe the emergence and mechanics of an insular media ecosystem and how two fundamentally different media ecosystems can coexist. In one, false narratives that reinforce partisan identity not only flourish, but crowd-out true narratives even when these are presented by leading insiders. In the other, false narratives are tested, confronted, and contained by diverse outlets and actors operating in a truth-oriented norms dynamic. Two case studies are analyzed: the first focuses on false reporting on a selection of television networks; the second looks at parallel but politically divergent false rumors—an allegation that Donald Trump raped a 13-yearold and allegations tying Hillary Clinton to pedophilia—and tracks the amplification and resistance these stories faced.

Phase space correspondence between Jordan and Einstein frames

2021 ◽  
pp. 2150087
Author(s):  
Amin Salehi
Keyword(s):  
Phase Space ◽  
Critical Point ◽  
Critical Points ◽  
Dynamical Behavior ◽  
Cosmological Parameters ◽  
Jordan Frame ◽  
Field Equations ◽  
Theories Of Gravity ◽  
The Stability ◽  
Jordan And Einstein Frames

Scalar–tensor theories of gravity can be formulated in the Einstein frame or in the Jordan frame (JF) which are related with each other by conformal transformations. Although the two frames describe the same physics and are equivalent, the stability of the field equations in the two frames is not the same. Here, we implement dynamical system and phase space approach as a robustness tool to investigate this issue. We concentrate on the Brans–Dicke theory in a Friedmann–Lemaitre–Robertson–Walker universe, but the results can easily be generalized. Our analysis shows that while there is a one-to-one correspondence between critical points in two frames and each critical point in one frame is mapped to its corresponds in another frame, however, stability of a critical point in one frame does not guarantee the stability in another frame. Hence, an unstable point in one frame may be mapped to a stable point in another frame. All trajectories between two critical points in phase space in one frame are different from their corresponding in other ones. This indicates that the dynamical behavior of variables and cosmological parameters is different in two frames. Hence, for those features of the study, which focus on observational measurements, we must use the JF where experimental data have their usual interpretation.

scholarly journals Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Juliana Vargas Sánchez
Keyword(s):  
Linear System ◽  
Critical Point ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Critical Points ◽  
Global Dynamics ◽  
Parametric Surfaces ◽  
Critical Points At Infinity ◽  
Global Phase Portrait ◽  
The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

DESIGN AND DEVELOPMENT OF REGENERATIVE DAMPERS

2022 ◽  
Vol 2 (1) ◽  
pp. 406-410
Author(s):  
Aditya Singh ◽  
Manish Thakur ◽  
Akshat shah ◽  
Neerake Bajaj ◽  
Hardik Taneja ◽  
...  
Keyword(s):  
Magnetic Flux ◽  
Electric Vehicle ◽  
Feedback Loop ◽  
Mechanical Energy ◽  
Cost Effective ◽  
The Other ◽  
Minimal Cost ◽  
Cost Increase ◽  
Suspension Systems ◽  
On Demand

Each and every automobile in service or being developed in the industry is benchmarked on the basis of its efficiency in running real conditions. So in our project here we have tried to develop a complete new damper and spring setup which can be used in all sorts of suspension systems and in turn provides a feedback loop of voltage which can then be used charge the batteries and upscale the efficiency of bikes by (5-6)% & and for HUV or Sedans by (2-4)% (can even go higher) depending on the terrain. In this setup we harness the mechanical energy into electrical where earlier it was left as heat and vibrational losses. This setup is as cost effective as the earlier dampers where as providing an efficient output in minimal cost increase due to its novelty. The other Features include Electronic height adjustment & on demand suspension softness or stiffness. Keywords: Dampers, Automobile, Electromagnet, EV (Electric vehicle), Voltage, Magnetic flux, Suspension

scholarly journals A method to estimate statistical errors of properties derived from charge-density modelling

2018 ◽  
Vol 74 (3) ◽  
pp. 170-183 ◽  
Author(s):  
Bertrand Fournier ◽  
Benoît Guillot ◽  
Claude Lecomte ◽  
Eduardo C. Escudero-Adán ◽  
Christian Jelsch
Keyword(s):  
Charge Density ◽  
Least Squares ◽  
Critical Point ◽  
Critical Points ◽  
Property Values ◽  
Density Model ◽  
Interaction Energies ◽  
Topological Properties ◽  
A Charge ◽  
Density Modelling

Estimating uncertainties of property values derived from a charge-density model is not straightforward. A methodology, based on calculation of sample standard deviations (SSD) of properties using randomly deviating charge-density models, is proposed with theMoProsoftware. The parameter shifts applied in the deviating models are generated in order to respect the variance–covariance matrix issued from the least-squares refinement. This `SSD methodology' procedure can be applied to estimate uncertainties ofanyproperty related to a charge-density model obtained by least-squares fitting. This includes topological properties such as critical point coordinates, electron density, Laplacian and ellipticity at critical points and charges integrated over atomic basins. Errors on electrostatic potentials and interaction energies are also available now through this procedure. The method is exemplified with the charge density of compound (E)-5-phenylpent-1-enylboronic acid, refined at 0.45 Å resolution. The procedure is implemented in the freely availableMoProprogram dedicated to charge-density refinement and modelling.

scholarly journals Slow oscillations in blood pressure via a nonlinear feedback model

2001 ◽  
Vol 280 (4) ◽  
pp. R1105-R1115 ◽  
Author(s):  
John V. Ringwood ◽  
Simon C. Malpas
Keyword(s):  
Blood Pressure ◽  
Feedback Loop ◽  
Sympathetic Nerve Activity ◽  
Linear Feedback ◽  
Scaling Effects ◽  
Nonlinear Feedback ◽  
Nerve Activity ◽  
Feedback Model ◽  
The Central Nervous System ◽  
Proportional Derivative Controller

Blood pressure is well established to contain a potential oscillation between 0.1 and 0.4 Hz, which is proposed to reflect resonant feedback in the baroreflex loop. A linear feedback model, comprising delay and lag terms for the vasculature, and a linear proportional derivative controller have been proposed to account for the 0.4-Hz oscillation in blood pressure in rats. However, although this model can produce oscillations at the required frequency, some strict relationships between the controller and vasculature parameters must be true for the oscillations to be stable. We developed a nonlinear model, containing an amplitude-limiting nonlinearity that allows for similar oscillations under a very mild set of assumptions. Models constructed from arterial pressure and sympathetic nerve activity recordings obtained from conscious rabbits under resting conditions suggest that the nonlinearity in the feedback loop is not contained within the vasculature, but rather is confined to the central nervous system. The advantage of the model is that it provides for sustained stable oscillations under a wide variety of situations even where gain at various points along the feedback loop may be altered, a situation that is not possible with a linear feedback model. Our model shows how variations in some of the nonlinearity characteristics can account for growth or decay in the oscillations and situations where the oscillations can disappear altogether. Such variations are shown to accord well with observed experimental data. Additionally, using a nonlinear feedback model, it is straightforward to show that the variation in frequency of the oscillations in blood pressure in rats (0.4 Hz), rabbits (0.3 Hz), and humans (0.1 Hz) is primarily due to scaling effects of conduction times between species.

scholarly journals II. On the critical state of gases

1880 ◽  
Vol 30 (200-205) ◽  
pp. 323-329 ◽  
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Critical State ◽  
Atmospheric Pressure ◽  
Boiling Point ◽  
Chemical Society ◽  
Boiling Points ◽  
Intermediate Condition ◽  
Specific Volumes

In a paper read before the Chemical Society, in May, 1879, I gave an account of a method of determining what is termed by Kopp the “specific volumes” of liquids; that was shown to be the volume of liquid at its boiling-point, at ordinary atmospheric pressure, obtainable from 22,326 volumes of its gas, supposed to exist at 0°. Being desirous of extending these researches, with the view of ascertaining such relations at higher temperatures, since April, 1879, I have made numerous experiments, the results of, and deductions from which I hope to publish before long. The temperatures observed vary from the boiling-points of the liquids examined, to about 50° above their critical points; and in course of these experiments I have noticed some curious facts, which may not be unworthy of the attention of the Society. It is well known that at temperatures above that which produces what is termed by Dr. Andrews the “critical point” of a liquid, the substance is supposed to exist in a peculiar condition, and Dr Andrews purposely abstained from speculating on the nature of the matter, whether it be liquid or gaseous, or in an intermediate condition, to which no name has been given. As my observations bear directly on this point, it may be advisable first to describe the experiments I have made, and then to draw the deductions which appear to follow from them.

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