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Stability Radii and Lyapunov Exponents
1990In the state space approach to stability of uncertain systems the concept of stability radius plays a central role. In this paper we use the classical concept of Lyapunov exponents, which describe the exponential growth behavior, in order to define a variety of stability and instability radii for families of linear systems ẋ = [A + u(t)]x, u(t) ∈ U ρ ,
Colonius, Fritz, Kliemann, Wolfgang
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Transient Angle Stability of Virtual Synchronous Generators Using Lyapunov’s Direct Method
IEEE Transactions on Smart Grid, 2019With an increasing number of distributed energy resources integrated into the power system, inverters need to take on the corresponding responsibility for the security and stability of the system.
Zhikang Shuai +4 more
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1999
The study of the stability of dynamical systems has a very rich history. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. A problem, which attracted a great deal of early interest was the problem of stability of the solar system, generalized under the title “the N-body stability problem.” One of ...
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The study of the stability of dynamical systems has a very rich history. Many famous mathematicians, physicists, and astronomers worked on axiomatizing the concepts of stability. A problem, which attracted a great deal of early interest was the problem of stability of the solar system, generalized under the title “the N-body stability problem.” One of ...
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Lyapunov's Stability Criteria for Plasmas
Journal of Mathematical Physics, 1963The orbit stability theory of Lyapunov has been adapted to the Vlasov-Boltzmann equation governing plasmas. Both linear and nonlinear stability are considered. The theory is characterized by a search for Lyapunov functions, whose existence implies stability in analogy with particles trapped in a potential well, as in the energy principle.
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2014
Basic concepts for the Lyapunov stability are introduced. Conditions are obtained for the stability of linear equations with constant, periodic, and general variable coefficients. Linearization and Lyapunov functions are used to deal with nonlinear stability problems.
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Basic concepts for the Lyapunov stability are introduced. Conditions are obtained for the stability of linear equations with constant, periodic, and general variable coefficients. Linearization and Lyapunov functions are used to deal with nonlinear stability problems.
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2012
The main intent of this chapter is to introduce the essential mathematical tools for stability analysis of continuous finite-dimensional dynamical systems. We begin with an overview of sufficient conditions to guarantee existence and uniqueness of the system solutions, followed by a collection of Lyapunov-based methods for studying stability of the ...
Eugene Lavretsky, Kevin A. Wise
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The main intent of this chapter is to introduce the essential mathematical tools for stability analysis of continuous finite-dimensional dynamical systems. We begin with an overview of sufficient conditions to guarantee existence and uniqueness of the system solutions, followed by a collection of Lyapunov-based methods for studying stability of the ...
Eugene Lavretsky, Kevin A. Wise
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Journal of Applied Mathematics and Mechanics, 2006
The stability of the stationary point of a Lyapunov system [Malkin IG, Some Problems in the Theory of Non-linear Oscillations. Moscow: Gostekhizd; 1956.], which describes the perturbed motion of a dynamical system with two degrees of freedom, is investigated.
A.L. Kunitsyn, V.N. Tkhai
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The stability of the stationary point of a Lyapunov system [Malkin IG, Some Problems in the Theory of Non-linear Oscillations. Moscow: Gostekhizd; 1956.], which describes the perturbed motion of a dynamical system with two degrees of freedom, is investigated.
A.L. Kunitsyn, V.N. Tkhai
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1995
The second or direct method of Lyapunov is entirely different from pole analysis in philosophy, nature, and detail, although there are a few overlaps for linear time-invariant systems.
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The second or direct method of Lyapunov is entirely different from pole analysis in philosophy, nature, and detail, although there are a few overlaps for linear time-invariant systems.
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Stability: Lyapunov, Linear Systems
2014The notion of stability allows to study the qualitative behavior of dynamical systems. In particular it allows to study the behavior of trajectories close to an equilibrium point or to a motion. The notion of stability that we discuss has been introduced in 1882 by the Russian mathematician A.M.
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2017
In the preceding section we introduced a classification of cellular automata based on attractors, their number and structure. In the present section we focus on the complexity of the dynamics. The two aspects are not independent, but differ slightly. We start with Devaney’s definition of chaos, and relate this definition to the Hurley classification ...
Karl-Peter Hadeler, Johannes Müller
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In the preceding section we introduced a classification of cellular automata based on attractors, their number and structure. In the present section we focus on the complexity of the dynamics. The two aspects are not independent, but differ slightly. We start with Devaney’s definition of chaos, and relate this definition to the Hurley classification ...
Karl-Peter Hadeler, Johannes Müller
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