ARTICLE | doi:10.20944/preprints201801.0277.v1
Subject: Computer Science And Mathematics, Mathematics Keywords: Lorenz system; implicit time scheme; counterexamples
Online: 30 January 2018 (04:38:31 CET)
In nonlinear self-consistent system, Lorenz system (Lorenz equations) is a classic case with chaos solutions which are sensitively dependent on the initial conditions. As it is difficult to get the analytical solution, the numerical methods and qualitative analytical methods are widely used in many studies. In these papers, Runge-Kutta method is the one most often used to solve these differential equations. However, this method is still a method based on explicit time scheme, which would be the main reason for the chaotic solutions to Lorenz system. In this work, numerical experiments based on implicit time scheme and explicit scheme are setup for comparison, the results show that: in implicit time scheme, the numerical solutions (counterexamples) are without chaos; for an original volume, the volume shrinks exponentially fast to 0 in common.
ARTICLE | doi:10.20944/preprints202304.0127.v1
Subject: Physical Sciences, Theoretical Physics Keywords: Master-slave synchronization; data assimilation; parameter estimation; Lorenz system
Online: 7 April 2023 (13:02:59 CEST)
We explore several aspects of replica synchronization with the goas of retrieving the value of parameters for the Lorenz system. The idea is that of having a computer replica (slave) of a natural system (master, simulated in this paper), and exploit the fact that slave synchronizes with the master only if they evolve with the same parameters. As a byproduct, in the synchronized phase the state variables of the slave and that of the master are the same, thus allowing to perform measurements impossible on the real system. We review some aspects of master-slave synchronization using a subset of variables, with intermittent coupling. We show how synchronization can be achieved when some of the state variables are available for direct measurement using a simulated annealing approach, and also when they are accessible only through a scalar function, using a pruned-enriching ensemble approach, similar to genetic algorithms without cross-over.
ARTICLE | doi:10.20944/preprints202206.0341.v1
Subject: Social Sciences, Education Keywords: Inequality; Lorenz Curve; Education; Equity; demand and supply; Ethiopia
Online: 24 June 2022 (16:03:48 CEST)
Globally, inequalities in educational provision are prevalent between genders, various geographical regions, and among different socio-economic backgrounds. Consequently, this study set-out to assess the level of disparity among the Federal Regional States of Ethiopia using Gini-coefficient and Lorenz curve from the statistical data of MoE. Moreover, data were collected from 656 respondents found in the sample regions. The result of the Gini-coefficient indicated that disparity in educational provision has been reduced over the past couple of decades both at primary (0.145 to 0.032) and secondary levels (0.277 to 0.126). Emerging regions are by far lagging behind the central and established regions. The sources of variation were mainly the demand-side variables, especially the economic and contextual related issues like drought and conflicts. Therefore, educational policies designed at the central level are advised to consider the strategies to bridge the existing inequalities through equitable provision of the education system to its citizen.
ARTICLE | doi:10.20944/preprints202304.0816.v1
Subject: Computer Science And Mathematics, Applied Mathematics Keywords: discrete-time dynamic systems; positive invariance; ellipsoid; Lorenz cone; optimization
Online: 24 April 2023 (04:10:49 CEST)
Positive invariant set is an important concept of dynamic systems. The purpose of this paper is to study the sufficient and necessary conditions that the set of ellipsoids and the Lorenz cone are positive invariant sets of discrete time systems. By means of nonlinear programming and induced norm, the problem of positive invariance is formulated as an optimization problem, and the equivalent dual form optimization is also presented using the dual optimization method. Our results provide more alternative methods for determining the positive invariance of quadratic form convex sets from the point of view of optimization and dual optimization. The effectiveness of this method is demonstrated by numerical examples.
ARTICLE | doi:10.20944/preprints202102.0419.v1
Subject: Computer Science And Mathematics, Algebra And Number Theory Keywords: Lorenz equation; Hopf bifurcation; Unstable limit cycle; Calculation process; Continuation method
Online: 18 February 2021 (13:47:19 CET)
This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.
ARTICLE | doi:10.20944/preprints202301.0077.v1
Subject: Engineering, Control And Systems Engineering Keywords: chaos; linear feedback control; Lorenz system; nonlinear feedforward control; oscillator; sinusoidal forcing
Online: 4 January 2023 (10:35:26 CET)
Chaotic systems are systems whose results are sensitive to the initial conditions. Chaotic systems generally have nonlinearities that can be difficult to model. One example of a chaotic system with nonlinearities is the Lorenz system. The Lorenz system can be used to model weather, wind disturbances, and electronic circuit design, among other applications. In this manuscript, the Lorenz system is modeled and various control methods are applied in an effort to dictate the system’s state and rate trajectories. The combination of the linear feedback and nonlinear feedforward controllers can show over an 80% improvement in state trajectory errors when compared to the baseline run. However, the improved state trajectory performance comes at a cost of as much as 1800% error in the rate trajectories.
ARTICLE | doi:10.20944/preprints201807.0576.v1
Subject: Engineering, Mechanical Engineering Keywords: Ball bearing; Fractional Lorenz chaos system; Extension theory; Chua’s Circuit Fault diagnosis
Online: 30 July 2018 (10:00:57 CEST)
In this study we used a non-autonomous Chua’s Circuit, and the fractional Lorenz chaos system together with a detection method from Extension theory to analyze the voltage signals. The measured bearing signals by acceleration sensor were introduced into the master and slave systems through a Chua’s Circuit. In a chaotic system minor differences can cause significant changes that generate dynamic errors, and extension matter-element models can be used to judge the bearing conditions. Extension theory can be used to establish classical and sectional domains using the dynamic errors of the fault conditions. The results obtained were compared with those from Discrete Fourier Transform analysis, Wavelet analysis and an integer order chaos system. The diagnostic ratio showed the fractional order master and slave chaos system calculations. The results show that the method presented in this paper is very suitable for monitoring the operational state of ball bearing system to be superior to the other methods. The diagnosis ratio was better and there were other significant advantages such as low cost and few.
ARTICLE | doi:10.20944/preprints201905.0230.v1
Subject: Social Sciences, Library And Information Sciences Keywords: Gini coefficient; online health community; medical service delivery; Lorenz curve; inequality of health service; differential Effect
Online: 20 May 2019 (03:13:23 CEST)
Inequality of health service for different specialty categories not only occurs in different areas inequality of health service for different specialty categories in the world, but also happens in the online service platform. In the online health community (OHC), health service was often of inequality for different specialty categories, including both online views and medical consultation for offline registered service. Moreover, how the factor city-level impacts the inequality of health service in OHC is still unknown. We designed a causal inference study with data on distributions of serviced patients and online views in over 100 distinct specialty categories on one largest OHC in China. To derive the causal effect of the city-levels (two levels inducing 1 and 0) on the Gini coefficient, we matched the focus cases in cities of rich healthcare resources with the potential control cities. For the Gini coefficient of serviced patients in over 100 specialty categories, the average treatment effect of level-1 cities is 0.470, which is 0.029 higher than that of the matched group. Similarly, for the Gini coefficient of online views, the average treatment effect of Level-1 cities is 0.573, which is 0.016 higher than that of the matched group. For each of the specialty categories, we also estimated the average treatment effect the specialty category’s Gini coefficient ( ) with the balanced covariates. The results support the argument that the total Gini coefficient of all the doctors in OHC shows that the inequality of health service is still very serious. This study contributes to the development of the theoretically grounded understanding of the causal effect of city-level on the inequality of health service in an online to offline healthcare service setting.
ARTICLE | doi:10.20944/preprints201810.0184.v1
Subject: Social Sciences, Sociology Keywords: technology adoption; Lorenz curves; Gini coefficient; local-national gap; Georgia; NIMBY; solar energy; community development; soft cost reduction
Online: 9 October 2018 (10:23:08 CEST)
Despite a global push in the development and implementation of widespread alternative energy use, significant disparities exist across given nation-states. These disparities reflect both technical and economic factors, as well as the social, political, and ecological gaps between how communities see energy development and national/global policy goals. Known as the “local-national gap,” many nations struggle with fostering meaningful conversations about the role of alternative energy technologies within communities. Mitigation of this problem first requires understanding the distribution of existing alternative energy technologies at the local level of policymaking. Using the State of Georgia, U.S.A. as a case study, we present a model for analyzing how existing adoption trends enable/limit conversation at the scale of local governance (i.e., county governments). Leveraging existing work on the Gini Coefficient as a metric for measuring energy inequity, we argue these tools can be applied to analyze where gaps exist in ongoing solar adoption trends. As we demonstrate, communities that adopt solar tend to be concentrated in a few counties, indicating existing conversations are limited to a circumscribed set of social networks. This information and the model we demonstrate can enable focused qualitative analyses of existing solar trends, not only amongst high-adoption areas but within communities where little to no adoption has occurred.
ARTICLE | doi:10.20944/preprints202203.0026.v1
Subject: Physical Sciences, Particle And Field Physics Keywords: Tc symmetry; CTP symmetry; Photon space; Elementary particles; Quantum wave functions; Modified Lorenz transformations; Left-handed neutrinos; Right-handed dark matters
Online: 1 March 2022 (15:01:52 CET)
The time symmetry of Tc = CT is redefined based on the 4-D Euclidean space. The time inversion symmetry of Tc = CT changes the signs of the charge (q) and absolute time (ct) of the particle. The P symmetry changes the signs of the space momenta and handedness of the particle. The Tc symmetry changes the signs of the time momentum (Pt=E/c) and charges. The handedness are classified as the left-handedness and right-handedness on the 3-D space. The charges (|q| = cDt) and energies (E= cDtDV) of the particles are originated from the upward warping (q>0) and downward warping (q<0) of the particles along the time axis. The evolution of our matter universe with Pt > 0 since the big bang can be interpreted based on the CTP symmetry with the partner antimatter universe with pt < 0. The photon space is called as the vacuum space. Then the E and M waves are the space fluctuations, and the gravitational G wave is the time fluctuations. And the E, M and G fields are newly explained. The zero E and M fields of the photon space indicates that the photon space and gamma ray are the 2EM waves. The rest mass energy of the particle is the 4-D space volume of the warped photon space. In the present work, the quantum wave function is considered as the 4-D vector with the space wave function and time axis wave function in the 4-D Euclidean space. The warped photon space corresponds to the squared wave function which is the probability density. It is concluded that, in the quantum mechanics, the imaginary number concept is introduced in the quantum wave functions instead of the time axis component of the 4-D wave function vector. The left-handed neutrino puzzle is explained from the handedness partner relation with the right-handed dark matters. Finally, the modified Lorentz transformations derived in the 4-D Euclidean space are compared with the Lorentz transformations of the special and general relativity theories in the 4-D Minkowski space.