scholarly journals Initiation and directional control of period-1 rotation for a parametric pendulum

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160719 ◽  
Author(s):  
Santanu Das ◽  
Pankaj Wahi
Keyword(s):  
Initial Conditions ◽  
Linear Time ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Delayed Feedback Control ◽  
Initial Condition ◽  
Stable Direction ◽  
Control Gain ◽  
Anticlockwise Rotation ◽  
Parametric Pendulum

We study a time-delayed feedback control for initiating period-1 rotations of a vertically excited parametric pendulum from arbitrary initial conditions. The possibility of controlling the direction of rotation has also been explored. We start with a simple linear time-delayed control for which the control gain corresponding to the most stable period-1 rotation has been obtained using the Floquet theory. This control increases the basins of attraction of rotations, but they do not encompass the full initial condition space. We modify our control law by using a switched control gain that destabilizes all the oscillatory solutions, and the entire initial condition space becomes the basin of attraction of either the clockwise or the anticlockwise rotation. By a suitable modification of the switching condition, we can choose a preferential stable direction of rotation. Hence, we can initiate either clockwise or anticlockwise rotation for a parametric pendulum from arbitrary initial conditions. Performance of our controller in achieving this objective has been demonstrated for different sets of parameters to establish its effectiveness.

scholarly journals Robustness of Supercavitating Vehicles Based on Multistability Analysis

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi ◽  
Jun Guan
Keyword(s):  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Small Range ◽  
Practical Application ◽  
Stable States ◽  
Supercavitating Vehicles ◽  
Control Gain ◽  
Stable Periodic ◽  
Setting Parameters

Supercavity can increase speed of underwater vehicles greatly. However, external interferences always lead to instability of vehicles. This paper focuses on robustness of supercavitating vehicles. Based on a 4-dimensional dynamic model, the existence of multistability is verified in supercavitating system through simulation, and the robustness of vehicles varying with parameters is analyzed by basins of attraction. Results of the research disclose that the supercavitating system has three stable states in some regions of parameters space, namely, stable, periodic, and chaotic states, while in other regions it has various multistability, such as coexistence of two types of stable equilibrium points, coexistence of a limit cycle with a chaotic attractor, and coexistence of 1-periodic cycle with 2-periodic cycle. Provided that cavitation number varies within a small range, with increase of the feedback control gain of fin deflection angle, size of basin of attraction becomes smaller and robustness of the system becomes weaker. In practical application, robustness of supercavitating vehicles can be improved by setting parameters of system or adjusting initial launching conditions.

Multistability in a Simplified Underwater Supercavity System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750121 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi
Keyword(s):  
Feedback Control ◽  
Deflection Angle ◽  
Initial Conditions ◽  
Major Change ◽  
Basins Of Attraction ◽  
Cavitation Number ◽  
Stable Equilibrium Point ◽  
Periodic Attractor ◽  
Control Gain ◽  
The Stability

Supercavity can increase the velocity of underwater vehicles greatly, however the launching state of vehicle and systematic parameters often lead to unstable motion. To solve the problem, the effect of parameters and initial conditions on the stability of vehicles is studied. With two variable parameters, namely cavitation number and feedback control gain of fin deflection angle, a simple dynamic model of supercavity system is studied. The multistability is verified through simulation. Robustness of the system is also analyzed based on its basins of attraction. There are various coexisting attractors in a relatively large region of parameter space of the supercavity system, namely coexistence of a stable equilibrium point and a periodic attractor, coexistence of various periodic attractors, coexistence of a periodic attractor with a chaotic attractor and so on, which explain the effect of parameters and initial values on stability of vehicles qualitatively. In addition, without major change in cavitation number, there is a negative correlation between the robustness of the vehicle and feedback control gain of fin deflection angle. The robustness can be improved through optimization of parameters.

scholarly journals A New Circumscribed Self-Excited Spherical Strange Attractor

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ramesh Ramamoorthy ◽  
Sajjad Shaukat Jamal ◽  
Iqtadar Hussain ◽  
Mahtab Mehrabbeik ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Strange Attractor ◽  
Unique Feature ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Spherical Coordinates ◽  
Chaotic Flows ◽  
Pass Through ◽  
Third Dimension ◽  
The Basin Of Attraction

Studying new chaotic flows with specific characteristics has been an open-ended field of exploring nonlinear dynamics. Investigation of chaotic flows is an area of research that has been taken into consideration for many years; thus, it helps in a better understanding of the chaotic systems. In this paper, an original chaotic 3D system, which has not been investigated yet, is presented in spherical coordinates. A unique feature of the proposed system is that its velocity becomes zero for a specific value of the radius variable. Hence, the system’s attractor is expected to be stuck on one side of a plane in spherical coordinates and inside or outside a sphere in the corresponding Cartesian coordinates. It means that the attractor cannot pass through the sphere or even touch it. The introduced system owns two unstable equilibria and a self-excited strange attractor. The 1D and 2D system’s bifurcation diagrams concerning the alteration of two bifurcation parameters are plotted to investigate the system’s dynamical properties. Moreover, the system’s Lyapunov exponents in the corresponding period of bifurcation parameters are calculated. Then, two 2D basins of attraction for two different third dimension values are explored. Based on the basin of attraction, it can be found that the sphere has attraction itself, partially, and some initial conditions are led to the sphere, not to the strange attractor. Ultimately, the connecting curves of the proposed system are explored to find an informative 1D set in addition to the system’s equilibria.

scholarly journals Basins of Attraction for Two-Species Competitive Model with Quadratic Terms and the Singular Allee Effect

2015 ◽  
Vol 2015 ◽  
pp. 1-16 ◽  
Author(s):  
A. Brett ◽  
M. R. S. Kulenović
Keyword(s):  
Singular Point ◽  
Allee Effect ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Global Dynamics ◽  
System Of Difference Equations ◽  
Competitive Model ◽  
Basins Of Attractions

We consider the following system of difference equations:xn+1=xn2/B1xn2+C1yn2, yn+1=yn2/A2+B2xn2+C2yn2,  n=0, 1, …,  whereB1,C1,A2,B2,C2are positive constants andx0, y0≥0are initial conditions. This system has interesting dynamics and it can have up to seven equilibrium points as well as a singular point at(0,0), which always possesses a basin of attraction. We characterize the basins of attractions of all equilibrium points as well as the singular point at(0,0)and thus describe the global dynamics of this system. Since the singular point at(0,0)always possesses a basin of attraction this system exhibits Allee’s effect.

Is it a Right Assumption That B and Ge are Distributed Randomly After Growing a Strained HBT-Structure?

2002 ◽  
Vol 716 ◽  
Author(s):  
Victor I. Kol'dyaev
Keyword(s):  
Surface Segregation ◽  
Initial Conditions ◽  
Initial Condition ◽  
Basic Assumptions ◽  
Transient Diffusion ◽  
Segregation Process ◽  
Main Observation

AbstractIt is accepted that surface Ge atoms are considered to be responsible for the surface B segregation process. A set of original experiments is carried out. A main observation from the B and Ge profiles grown at different conditions shows that at certain conditions B is taking initiative and determine the Ge surface segregation process. basic assumptions are suggested to self-consistently explain these original experimental features and what is observed in the literature. These results have a strong implication for modeling the B diffusion in Si1-xGex where the initial conditions should be formulated accounting for the correlation in B and Ge distribution. A new assumption for the initial condition to be “all B atoms are captured by Ge” is regarded as a right one implicating that there is no any transient diffusion representing the B capturing kinetics.

scholarly journals The VIT Transform Approach to Discrete-Time Signals and Linear Time-Varying Systems

Eng ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 99-125
Author(s):  
Edward W. Kamen
Keyword(s):  
Discrete Time ◽  
Initial Conditions ◽  
Linear Time ◽  
Order Term ◽  
Initial Time ◽  
Time Varying ◽  
Varying Coefficients ◽  
Time Signals ◽  
Time Varying Systems ◽  
Time Varying Coefficients

A transform approach based on a variable initial time (VIT) formulation is developed for discrete-time signals and linear time-varying discrete-time systems or digital filters. The VIT transform is a formal power series in z−1, which converts functions given by linear time-varying difference equations into left polynomial fractions with variable coefficients, and with initial conditions incorporated into the framework. It is shown that the transform satisfies a number of properties that are analogous to those of the ordinary z-transform, and that it is possible to do scaling of z−i by time functions, which results in left-fraction forms for the transform of a large class of functions including sinusoids with general time-varying amplitudes and frequencies. Using the extended right Euclidean algorithm in a skew polynomial ring with time-varying coefficients, it is shown that a sum of left polynomial fractions can be written as a single fraction, which results in linear time-varying recursions for the inverse transform of the combined fraction. The extraction of a first-order term from a given polynomial fraction is carried out in terms of the evaluation of zi at time functions. In the application to linear time-varying systems, it is proved that the VIT transform of the system output is equal to the product of the VIT transform of the input and the VIT transform of the unit-pulse response function. For systems given by a time-varying moving average or an autoregressive model, the transform framework is used to determine the steady-state output response resulting from various signal inputs such as the step and cosine functions.

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

2002 ◽  
Vol 12 (06) ◽  
pp. 1333-1356 ◽  
Author(s):  
YOSHISUKE UEDA ◽  
HIROYUKI AMANO ◽  
RALPH H. ABRAHAM ◽  
H. BRUCE STEWART
Keyword(s):  
Power Systems ◽  
Power System ◽  
Initial Conditions ◽  
Electrical Power ◽  
Practical Importance ◽  
Intervention Strategy ◽  
Basins Of Attraction ◽  
Electric Power System ◽  
Point Of View ◽  
Electrical Power System

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

scholarly journals Application of Adjoint-Derived Forecast Sensitivities to the 24–25 January 2000 U.S. East Coast Snowstorm

2005 ◽  
Vol 133 (11) ◽  
pp. 3148-3175 ◽  
Author(s):  
Daryl T. Kleist ◽  
Michael C. Morgan
Keyword(s):  
Response Function ◽  
Initial Conditions ◽  
Forecast Error ◽  
Weather Prediction ◽  
Lower Troposphere ◽  
Vertical Motion ◽  
Initial Condition ◽  
Response Functions ◽  
Eastern United States ◽  
Surface Cyclone

Abstract The 24–25 January 2000 eastern United States snowstorm was noteworthy as operational numerical weather prediction (NWP) guidance was poor for lead times as short as 36 h. Despite improvements in the forecast of the surface cyclone position and intensity at 1200 UTC 25 January 2000 with decreasing lead time, NWP guidance placed the westward extent of the midtropospheric, frontogenetically forced precipitation shield too far to the east. To assess the influence of initial condition uncertainties on the forecast of this event, an adjoint model is used to evaluate forecast sensitivities for 36- and 48-h forecasts valid at 1200 UTC 25 January 2000 using as response functions the energy-weighted forecast error, lower-tropospheric circulation about a box surrounding the surface cyclone, 750-hPa frontogenesis, and vertical motion. The sensitivities with respect to the initial conditions for these response functions are in general very similar: geographically isolated, maximized in the middle and lower troposphere, and possessing an upshear vertical tilt. The sensitivities are maximized in a region of enhanced low-level baroclinicity in the vicinity of the surface cyclone’s precursor upper trough. However, differences in the phase and structure of the gradients for the four response functions are evident, which suggests that perturbations could be constructed to alter one response function but not necessarily the others. Gradients of the forecast error response function with respect to the initial conditions are used in an iterative procedure to construct initial condition perturbations that reduce the forecast error. These initial condition perturbations were small in terms of both spatial scale and magnitude. Those initial condition perturbations that were confined primarily to the midtroposphere grew rapidly into much larger amplitude upper-and-lower tropospheric perturbations. The perturbed forecasts were not only characterized by reduced final time forecast error, but also had a synoptic evolution that more closely followed analyses and observations.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

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