scholarly journals THE EVOLUTION OF THE ELEMENTS AND THE STABILITY OF COMPLEX ATOMS. I. A NEW PERIODIC SYSTEM WHICH SHOWS A RELATION BETWEEN THE ABUNDANCE OF THE ELEMENTS AND THE STRUCTURE OF THE NUCLEI OF ATOMS.

1917 ◽  
Vol 39 (5) ◽  
pp. 856-879 ◽  
Author(s):  
William D. Harkins
Keyword(s):  
Periodic System ◽  
The Stability

The stability of heavy nuclei and the limit of the periodic system of elements

1970 ◽  
Vol 100 (1) ◽  
pp. 45-92 ◽  
Author(s):  
G.N. Flerov ◽  
V.A. Druin ◽  
A.A. Pleve
Keyword(s):  
Periodic System ◽  
Heavy Nuclei ◽  
The Stability

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Picard Iteration ◽  
Periodic Systems ◽  
Transition Matrices ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Stability Diagrams ◽  
The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

2017 ◽  
Author(s):  
Ashu Sharma ◽  
Subhash C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Periodic Systems ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Approximate Approach ◽  
Varying Coefficients ◽  
P Transformation ◽  
The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

The Stability of a Second Order Linear Periodic System

1969 ◽  
Vol 91 (2) ◽  
pp. 207-210 ◽  
Author(s):  
E. J. Davison
Keyword(s):  
Linear System ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Order Linear ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Linear Periodic System

Necessary and sufficient conditions are obtained for the stability of the following second order linear system: x˙=θ(t)x,θ(t)=θt+∑i=1lTi and θ(t) =A1,0<t<T1=A2,T1<t<T1+T2⋮=Al,∑i=1l−1Ti<t<∑i=1lTi in terms of the eigenvalues and elements of the matrices Ai, i = 1, 2…l. The conditions become very simple for the case that l = 2. An example of a pendulum with a vibrating support is included.

THE STABILITY AND ALMOST PERIODIC SOLUTION FOR GENERALIZED LOGISTIC ALMOST PERIODIC SYSTEM WITH DELAYS

2011 ◽  
Vol 04 (03) ◽  
pp. 313-328 ◽  
Author(s):  
XIANGLAI ZHUO
Keyword(s):  
Periodic Solution ◽  
Periodic System ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Computational Techniques ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Globally Asymptotically Stable ◽  
Discrete Delays ◽  
The Stability

The stability and almost periodic solution for generalized logistic almost periodic system with infinite and discrete delays is considered. Some sufficient conditions for the boundedness of the system are obtained to guarantee that the system is globally asymptotically stable. We also show that the almost periodic system has a unique globally asymptotically stable strictly positive almost periodic solution by using the almost periodic functional Hull theory and new computational techniques. Furthermore, some recent results are improved, and an open question is answered.

Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Peter M. B. Waswa ◽  
Sangram Redkar
Keyword(s):  
Periodic System ◽  
Constant Coefficient ◽  
Floquet Theory ◽  
Normal Forms ◽  
Time History ◽  
Mathieu Equation ◽  
Periodic Systems ◽  
State Augmentation ◽  
The Stability ◽  
Averaging Techniques

Abstract This article introduces a technique to accomplish reducibility of linear quasi-periodic systems into constant-coefficient linear systems. This is consistent with congruous proofs common in literature. Our methodology is based on Lyapunov–Floquet transformation, normal forms, and enabled by an intuitive state augmentation technique that annihilates the periodicity in a system. Unlike common approaches, the presented approach does not employ perturbation or averaging techniques and does not require a periodic system to be approximated from the quasi-periodic system. By considering the undamped and damped linear quasi-periodic Hill-Mathieu equation, we validate the accuracy of our approach by comparing the time-history behavior of the reduced linear constant-coefficient system with the numerically integrated results of the initial quasi-periodic system. The two outcomes are shown to be in exact agreement. Consequently, the approach presented here is demonstrated to be accurate and reliable. Moreover, we employ Floquet theory as part of our analysis to scrutinize the stability and bifurcation properties of the undamped and damped linear quasi-periodic system.

Controller Design of Output Feedback MIMO Networked Control Systems with Multiple Packet Transmission

2009 ◽  
Vol 16-19 ◽  
pp. 935-940
Author(s):  
Cheng Xiang Xie ◽  
Wei Li Hu
Keyword(s):  
Control Systems ◽  
Periodic System ◽  
Controller Design ◽  
Networked Control Systems ◽  
Sampling Period ◽  
Packet Transmission ◽  
Networked Control ◽  
Matrix Inequalities ◽  
Static Scheduling ◽  
The Stability

MIMO (mult-input multi-output) networked control systems (NCSs) with multiple packet transmission is considered. The network-induced delay is assumed to be time-varying and less than one sampling period. Suppose the data of sensor node is part of system outputs and is transferred with static scheduling strategy, then the NCS is modeled as uncertain periodic system. The uncertain periodic system theory is used for the stability sufficient condition. Then a control design approach is presented via bilinear matrix inequalities (BMIs) method. The simulation result shows the effectiveness of the proposed approach.

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