INVESTIGATION OF THE GEOMETRY OF THE D-PARTITION OF ONE-DIMENSIONAL PLANE OF PARAMETER OF THE CHARACTERISTIC EQUATION OF A CONTINUOUS SYSTEM

2021 ◽  
Vol 4 ◽  
pp. 125-136
Author(s):  
Leonid Movchan ◽  
◽  
Sergey Movchan ◽  
Keyword(s):  
Imaginary Part ◽  
Characteristic Equation ◽  
Stability Region ◽  
Practical Interest ◽  
Stability Regions ◽  
Odd Order ◽  
Stability Intervals ◽  
Even Order ◽  
The Stability ◽  
Two Parameters

The paper considers two types of boundaries of the D-partition in the plane of one parameter of linear continuous systems given by the characteristic equation with real coefficients. The number of segments and intervals of stability of the X-partition curve is estimated. The maximum number of stability intervals is determined for different orders of polynomials of the equation of the boundary of the D-partition of the first kind (even order, odd order, one of even order, and the other of odd order). It is proved that the maximum number of stability intervals of a one-parameter family is different for all cases and depends on the ratio of the degrees of the polynomials of the equation of the D-partition curve. The derivative of the imaginary part of the expression of the investigated parameter at the initial point of the D-partition curve is obtained in an analytical form, the sign of which depends on the ratio of the coefficients of the characteristic equation and establishes the stability of the first interval of the real axis of the parameter plane. It is shown that for another type of the boundary of the D-partition in the plane of one parameter, there is only one interval of stability, the location of which, as for the previous type of the boundary of the stability region (BSR), is determined by the sign of the first derivative of the imaginary part of the expression of the parameter under study. Consider an example that illustrates the effectiveness of the proposed approach for constructing a BSR in a space of two parameters without using «Neimark hatching» and constructing special lines. In this case, a machine implementation of the construction of the stability region is provided. Considering that the problem of constructing the boundary of the stability region in the plane of two parameters is reduced to the problem of determining the BSR in the plane of one parameter, then the given estimates of the maximum number of stability regions in the plane of one parameter allow us to conclude about the number of maximum stability regions in the plane of two parameters, which are of practical interest. In this case, one of the parameters can enter nonlinearly into the coefficients of the characteristic equation.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (02) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

scholarly journals Convergence and Stability in Collocation Methods of Equationu′(t)=au(t)+bu([t])

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Han Yan ◽  
Shufang Ma ◽  
Yanbin Liu ◽  
Hongquan Sun
Keyword(s):  
Numerical Stability ◽  
Numerical Experiments ◽  
Stability Region ◽  
Collocation Methods ◽  
Stability Regions ◽  
Optimal Convergence ◽  
Analytic Stability ◽  
The Stability ◽  
Local Superconvergence ◽  
Global Superconvergence

This paper is concerned with the convergence, global superconvergence, local superconvergence, and stability of collocation methods foru′(t)=au(t)+bu([t]). The optimal convergence order and superconvergence order are obtained, and the stability regions for the collocation methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained, and some numerical experiments are given.

scholarly journals ORDER OF THE RUNGE-KUTTA METHOD AND EVOLUTION OF THE STABILITY REGION

2019 ◽  
Vol 5 (2) ◽  
pp. 64
Author(s):  
Hippolyte Séka ◽  
Kouassi Richard Assui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Absolute Stability ◽  
Stability Region ◽  
Kutta Method ◽  
Stability Regions ◽  
Runge Kutta Method ◽  
The Absolute ◽  
Runge Kutta ◽  
The Stability

In this article, we demonstrate through specific examples that the evolution of the size of the absolute stability regions of Runge–Kutta methods for ordinary differential equation does not depend on the order of methods.

scholarly journals Stability conditions for some distributed systems: buffered random access systems

1994 ◽  
Vol 26 (2) ◽  
pp. 498-515 ◽  
Author(s):  
Wojciech Szpankowski
Keyword(s):  
Stability Region ◽  
Sufficient Conditions ◽  
Random Access ◽  
Symmetric System ◽  
Slotted Aloha ◽  
Stability Regions ◽  
Novel Technique ◽  
Token Passing ◽  
The Stability ◽  
Necessary And Sufficient

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

scholarly journals Systematic survey of the dynamics of Uranus Trojans

2020 ◽  
Vol 633 ◽  
pp. A153 ◽  
Author(s):  
Lei Zhou ◽  
Li-Yong Zhou ◽  
Rudolf Dvorak ◽  
Jian Li
Keyword(s):  
Solar System ◽  
Stability Region ◽  
Large Fraction ◽  
Instability Strip ◽  
Stability Regions ◽  
Lagrange Points ◽  
Secular Resonances ◽  
Secondary Resonances ◽  
The Stability ◽  
Planetary Migration

Context. The discovered Uranus Trojan (UT) 2011 QF99 and several candidate UTs have been reported to be in unstable orbits. This implies that the stability region around the triangular Lagrange points L4 and L5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we constructed the dynamical maps on the (a0, i0) plane. The proper frequencies of UTs were determined precisely with a frequency analysis method that allows us to depict the resonance web via a semi-analytical method. We simulated radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. We find two main stability regions: a low-inclination (0° −14°) and a high-inclination regime (32° −59°). There is also an instability strip in each of these regions at 9° and 51°, respectively. These strips are supposed to be related with g − 2g5 + g7 = 0 and ν8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs is caused by the ν5 and ν7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration centre of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi-resonance has an ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive fast and slow migrations with migrating timescales of 1 and 10 Myr, respectively, most of which are in high inclination. Since low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10−3 and 9.07 × 10−5 of the original population for fast and slow migrations, respectively. The chaotic capture, just like depletion, results from secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive until today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i0 <  7.5°. However, the depletion of planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is favoured if any primordial UTs are detected.

scholarly journals A New Eighth Order Runge-Kutta Family Method

2019 ◽  
Vol 11 (2) ◽  
pp. 190
Author(s):  
Séka Hippolyte ◽  
Assui Kouassi Richard
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stability Region ◽  
Eighth Order ◽  
Stability Regions ◽  
New Family ◽  
Runge Kutta ◽  
The Stability

In this article, a new family of Runge-Kutta methods of 8^th order for solving ordinary differential equations is discovered and depends on the parameters b_8 and a_10;5. For b8 = 49/180 and a10;5 = 1/9, we find the Cooper-Verner method [1]. We show that the stability region depends only on coefficient a_10;5. We compare the stability regions according to the values of a_10;5 with respect to the stability region of Cooper-Verner.

The influence of the Stokes and Archimedes forces on the stability of a two-phase flow

2003 ◽  
Vol 3 ◽  
pp. 266-270
Author(s):  
B.H. Khudjuyerov ◽  
I.A. Chuliev
Keyword(s):  
Spectral Method ◽  
Stability Region ◽  
Gas Flow ◽  
Two Phase Flow ◽  
Solid Phase ◽  
Stability Characteristic ◽  
Phase Flow ◽  
Two Phase ◽  
The Stability

The problem of the stability of a two-phase flow is considered. The solution of the stability equations is performed by the spectral method using polynomials of Chebyshev. A decrease in the stability region gas flow with the addition of particles of the solid phase. The analysis influence on the stability characteristic of Stokes and Archimedes forces.

Enlarging the region of stability in robust model predictive controller based on dual-mode control

2021 ◽  
pp. 014233122110123
Author(s):  
Fatemeh Khani ◽  
Mohammad Haeri
Keyword(s):  
Stability Region ◽  
Linear Matrix ◽  
Input State ◽  
Dual Mode ◽  
Model Predictive Controller ◽  
Mode Control ◽  
Robust Model ◽  
Output Constraints ◽  
Predictive Controller ◽  
The Stability

Industrial processes are inherently nonlinear with input, state, and output constraints. A proper control system should handle these challenging control problems over a large operating region. The robust model predictive controller (RMPC) could be an linear matrix inequality (LMI)-based method that estimates stability region of the closed-loop system as an ellipsoid. This presentation, however, restricts confident application of the controller on systems with large operating regions. In this paper, a dual-mode control strategy is employed to enlarge the stability region in first place and then, trajectory reversing method (TRM) is employed to approximate the stability region more accurately. Finally, the effectiveness of the proposed scheme is illustrated on a continuous stirred tank reactor (CSTR) process.

scholarly journals Steady-state $$\dot{V}{\text{O}}_{2}$$ above MLSS: evidence that critical speed better represents maximal metabolic steady state in well-trained runners

2021 ◽  
Author(s):  
Rebekah J. Nixon ◽  
Sascha H. Kranen ◽  
Anni Vanhatalo ◽  
Andrew M. Jones
Keyword(s):  
Steady State ◽  
Critical Speed ◽  
Lactate Concentration ◽  
Practical Interest ◽  
Blood Lactate Concentration ◽  
Distance Runners ◽  
Severe Intensity ◽  
The Stability ◽  
Trained Runners ◽  
Over Time

AbstractThe metabolic boundary separating the heavy-intensity and severe-intensity exercise domains is of scientific and practical interest but there is controversy concerning whether the maximal lactate steady state (MLSS) or critical power (synonymous with critical speed, CS) better represents this boundary. We measured the running speeds at MLSS and CS and investigated their ability to discriminate speeds at which $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 was stable over time from speeds at which a steady-state $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 could not be established. Ten well-trained male distance runners completed 9–12 constant-speed treadmill tests, including 3–5 runs of up to 30-min duration for the assessment of MLSS and at least 4 runs performed to the limit of tolerance for assessment of CS. The running speeds at CS and MLSS were significantly different (16.4 ± 1.3 vs. 15.2 ± 0.9 km/h, respectively; P < 0.001). Blood lactate concentration was higher and increased with time at a speed 0.5 km/h higher than MLSS compared to MLSS (P < 0.01); however, pulmonary $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 did not change significantly between 10 and 30 min at either MLSS or MLSS + 0.5 km/h. In contrast, $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 increased significantly over time and reached $$\dot{V}{\text{O}}_{2\,\,\max }$$ V ˙ O 2 max at end-exercise at a speed ~ 0.4 km/h above CS (P < 0.05) but remained stable at a speed ~ 0.5 km/h below CS. The stability of $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 at a speed exceeding MLSS suggests that MLSS underestimates the maximal metabolic steady state. These results indicate that CS more closely represents the maximal metabolic steady state when the latter is appropriately defined according to the ability to stabilise pulmonary $$\dot{V}{\text{O}}_{2}$$ V ˙ O 2 .

scholarly journals The elastic stability of an annular plate

1924 ◽  
Vol 106 (737) ◽  
pp. 268-284 ◽  
Keyword(s):  
Annular Plate ◽  
Practical Interest ◽  
Middle Surface ◽  
Elastic Stability ◽  
Thin Plates ◽  
Thin Shells ◽  
Stability Equation ◽  
Inner Radius ◽  
Plane Plate ◽  
The Stability

1. Introduction and Summary. —This paper deals with the elastic stability of a circular annular plate under uniform shearing forces applied at its edges. Investigations of the stability of plane plates are altogether simpler than those necessary in the case of curved plates or shells. In the first place, as shown by Mr. R. V. Southwell, two of the three equations of stability relate to a mode of instability that is not of practical interest, and are entirely independent of the third equation which gives the ordinary mode of instability resulting in the familiar bending of the middle surface of the plate. Consequently with a plane plate there is only one equation of stability to be solved, as contrasted with the case of a shell where the three equations are dependent, and must all be solved. In the second place the theory of thin shells can be used with confidence in a plane plate problem, though a more laborious procedure is necessary to deal adequately with a shell. The only stability equation required for the annular plate is therefore deduced without trouble from the theory of thin shells, and its solution presents no difficulty in the case of uniform shearing forces. A numerical discussion is given of the stability of the plate under such forces, the “favourite type of distortion” and the stess that will produce it being obtained for plates with clamped edges in wich the ratio of the outer to the inner radius exceeds 3·2. To some extent to results have been checked by experiment, in which part of the work the viter is indebted to Prof. G. I. Taylor for his valuable help and advice. Distrtion of the type predicted by the theory took place in the two thin plates of rober different ratio of radii, which were used. The disposition of the loci of points which undergo maximum normal displace nt gives some idea of the appearance of the plate after distortion has taken pce. The points have been calculated for a plate in which the ratio of radii 4·18, and the loci are shown on a diagram, which may be compared with a potograph of a distorted plate in which this ratio is 4·3. The ratio of normal dplacements of points of the plate can be seen from contours drawn on the ne diagram. (See pp. 280, 281.)

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