LOCATION AND STABILITY OF THE TRIANGULAR POINTS IN THE TRIAXIAL ELLIPTIC RESTRICTED THREE-BODY PROBLEM

The main goal of the present paper is to evaluate the perturbed locations and investigate the linear stability of the triangular points. We studied the problem in the elliptic restricted three body problem frame of work. The problem is generalized in the sense that the two primaries are considered as triaxial bodies. It was found that the locations of these points are affected by the triaxiality coefficients of the primaries and the eccentricity of orbits. Also, the stability regions depend on the involved perturbations. We also studied the periodic orbits in the vicinity of the triangular points.

Charged thin-shell wormholes with quintessence effects

This work studies the theoretical construction of charged quintessence thin-shell wormholes using Israel thin-shell approach. The stability of these wormhole solutions is investigated by taking linear, logarithmic and Chaplygin gas models as a constituent of exotic matter at thin-shell. The presence of wormhole stability regions particularly relies on the physically justifiable values of charge and quintessence parameter. It is noted that the increasing value of charge seems as an effective component for stable regions while the rise in negativity of the quintessence parameter gives more stable wormhole configurations.

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

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.

Stability Regions of Fractional First Order Controllers Applied to Fractional Order Delay Systems

This paper focuses on the problem of stabilizing fractional order time delay systems by fractional first order controllers. A solution is proposed to find the set of all stability regions in the controller’s parameter space. The D-decomposition method is employed to find the real root boundary and complex root boundaries which are used to identify the stability regions. Illustrative examples are given to show the effectiveness of the proposed approach, and it is remarked that the stability region obtained for the fractional order controller is larger than the non-fractional controller.

Spatial and Temporal Variability of Rotational, Focal and Irregular Activity: Practical Implications for Mapping of Atrial Fibrillation

Background Charge density mapping of atrial fibrillation (AF) reveals dynamic patterns of localised rotational activation (LRA), irregular activation (LIA) and focal firing (FF). Their spatial stability, conduction characteristics and the optimal duration of mapping required to reveal these phenomena and has not been explored. Methods Bi-atrial mapping of AF propagation was undertaken and variability of activation patterns quantified up to a duration of 30-seconds(s). The frequency of each pattern was quantified at each vertex of the chamber over 2 separate 30s recordings prior to ablation and R2 calculated to quantify spatial stability. Regions with the highest frequency were identified at increasing time durations and compared to the result over 30s using Cohen’s kappa. Properties of regions with the most stable patterns were assessed during sinus rhythm and extrastimulus pacing. Results In twenty-one patients, 62 paired LA and RA maps were obtained. LIA was highly spatially stable with R2 between maps of 0.83(0.71-0.88) compared to 0.39(0.24-0.57) and 0.64(0.54-0.73) for LRA and FF, respectively. LIA was also most temporally stable with a kappa of >0.8 reached by 12s. LRA showed greatest variability with kappa>0.8 only after 22s. Regions of LIA were of normal voltage amplitude (1.09mv) but showed increased conduction heterogeneity during extrastimulus pacing (p=0.0480). Conclusion Irregular activation patterns characterised by changing wavefront direction are temporally and spatially stable in contrast with rotational patterns that are transient with least spatial stability. Focal activation appears of intermediate stability. Regions of LIA show increased heterogeneity following extrastimulus pacing and may represent fixed anatomical substrate.

Algebraic Stability Analysis of Particle Swarm Optimization Using Stochastic Lyapunov Functions and Quantifier Elimination

This paper adds to the discussion about theoretical aspects of particle swarm stability by proposing to employ stochastic Lyapunov functions and to determine the convergence set by quantifier elimination. We present a computational procedure and show that this approach leads to a reevaluation and extension of previously known stability regions for PSO using a Lyapunov approach under stagnation assumptions.