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Computing Lyapunov exponents of continuous dynamical systems: method of Lyapunov vectors
Chaos, Solitons & Fractals, 2005The paper proposes a new method for computing the Lyapunov characteristic exponents of general continuous dynamical systems. There are several methods for finding numerical values of Lyapunov's exponents, e.g., Wolf's algorithm and the QR method.
Lu, Jia +3 more
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2020
The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of ...
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The celebrated Lyapunov function method (or the direct Lyapunov method) introduced in the Ph.D. thesis of A. M. Lyapunov in 1892 is a simple effective tool for stability analysis of differential equations. The main advantage of this method lies in the fact that a decision on stability or instability can be made by means of a certain investigation of ...
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On the error in computing Lyapunov exponents by QR Methods
Numerische Mathematik, 2005The authors consider the nonautonomous linear system \[ x'= A(t)x,\quad t\geq 0,\quad x(0)= X_0. \] The study of the asymptotic behavior of the solution depends of the Lyapunov spectrum. If \(X\) is a fundamental matrix solution, then this spectrum is calculated using QR methods in two forms: discrete and continuous.
Luca Dieci, Erik S. Van Vleck
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Some Refinements of Lyapunov's Second Method
Canadian Journal of Mathematics, 1965Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated ...
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A Lyapunov analysis of accelerated methods in optimization
J. Mach. Learn. Res., 2021Summary: Accelerated optimization methods, such as Nesterov's accelerated gradient method, play a significant role in optimization. Several accelerated methods are provably optimal under standard oracle models. Such optimality results are obtained using a technique known as ``estimate sequences,'' which yields upper bounds on convergence properties ...
Ashia C. Wilson +2 more
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Suppressing chaos via Lyapunov–Krasovskii’s method
Chaos, Solitons & Fractals, 2006zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Kuang, JL, Meehan, PA, Leung, AYT
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1987
Let us consider a problem of mechanics defined by the system of ordinary differential equations (1.5.1) and let us investigate the stability of a particular solution q i 0 (t). For purposes of simplicity, we have omitted the influence of any parameters. The following variational equations are obtained by the well-known transformation (1.2) q
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Let us consider a problem of mechanics defined by the system of ordinary differential equations (1.5.1) and let us investigate the stability of a particular solution q i 0 (t). For purposes of simplicity, we have omitted the influence of any parameters. The following variational equations are obtained by the well-known transformation (1.2) q
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The status of synthesis using Lyapunov's method
Automatica, 1965This paper surveys the literature devoted to techniques for the synthesis of control systems based on Lyapunov's second method. The work is tutorial in nature and attempts to unify some of the scattered literature on the subject. A number of techniques are reviewed, illustrated with examples, and comments made regarding their applicability and ...
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1999
In Chapter 7 we discussed the first approach to realization of the Lyapunov direct method for FDE based on application of the Lyapunov functionals. In the second approach finite dimensional Lyapunov’s functions v(t, x): R × R n →R are used.
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In Chapter 7 we discussed the first approach to realization of the Lyapunov direct method for FDE based on application of the Lyapunov functionals. In the second approach finite dimensional Lyapunov’s functions v(t, x): R × R n →R are used.
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On the lyapunov's functionals method for systems with delays
Nonlinear Analysis: Theory, Methods & Applications, 1997The author analyzes various derivatives of functionals, needed to use Lyapunov theory. In particular, the notions (too lengthy to be quoted here) of invariant continuity and invariant differentiability are introduced. For the proofs of the few results given, the reader is referred elsewhere.
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