Floquet theory for linear differential equations with meromorphic solutions

Weikard, Rudi

doi:10.14232/ejqtde.2000.1.8

Cited by 8 publications

(3 citation statements)

References 6 publications

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“…Mathematicians have extended Floquet theory in different directions. We can classify the results of Floquet theory into some types: ODEs (almost Floquet systems [3], almost-periodic systems [4], periodic Euler-Bernoulli equations [5], delay differential equations [6], linear systems with meromorphic solutions [7]), PDEs (parabolic differential equations [8], periodic evolution problems [9]), DAEs [10,11], integro-differential equations [12], Volterra equations [13], discrete dynamical systems (countable systems [14]) and systems on time scales [15]. More details for the Floquet theory and applications, one can also refer to (monograph [16] and the works [17,18]).…”

Section: Introduction 1historymentioning

confidence: 99%

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Wu¹,

Xia²,

Zhao³

2022

Preprint

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Motivated by the work on the unified Floquet theory on time scales (DaCunha and Davis [1] J. Differential Equations, 2011), in this paper, we provide a unified algorithm to determine the stability of the second order periodic linear differential equations on periodic time scales. Our approach is based on calculating the value of A and B (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. Furthermore, a Matlab program is designed to realize our algorithm. We obtain an explicit expression of A (see Theorem 4.1) by the method of variation and approximation theory and an explicit expression of B by Liouville's formula. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of A (see Theorem 4.2). Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

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Section: Introduction 1historymentioning

confidence: 99%

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Wu¹,

Xia²,

Zhao³

2022

Preprint

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“…Without question, the study of periodic systems in general and Floquet theory in particular has been central to the differential equations theorist for some time. Researchers have explored these topics for ordinary differential equations [8,14,15,22,28,29,34,36], partial differential equations [7,9,15,25], differential-algebraic equations [13,26], and discrete dynamical systems [3,23,35]. Certainly [27] is a landmark paper in the area.…”

Section: Introductionmentioning

confidence: 99%

“…Not surprisingly, Floquet theory has wide ranging effects, including extensions from time varying linear systems to time varying nonlinear systems of differential equations of the form x ′ = f (t, x), where f (t, x) is smooth and ω-periodic in t. The paper by Shi [33] ensures the global existence of solutions and proves that this system is topologically equivalent to an autonomous system y ′ = g(y) via an ω-periodic transformation of variables. The theory has also been extended by Weikard [36] to nonautonomous linear systems of the forṁ z = A(x)z where A : C → C n×n is an ω-periodic function in the complex variable x, whose solutions are meromorphic. With the assumption that A(x) is bounded at the ends of the period strip, it is shown that there exists a fundamental solution of the form P (x)e Jx with a certain constant matrix J and function P which is rational in the variable e 2πix/ω .…”

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confidence: 99%

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

DaCunha¹,

Davis

2011

Journal of Differential Equations

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In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system's poles in the complex plane. We include several examples to show the utility of this theory. Contents1991 Mathematics Subject Classification. 34K13, 34C25, 39A11, 39A12.Proof. The fact that {γ i (t), w i (t)} n i=1 is a set of dynamic eigenpairs of R(t) follows immediately from Lemma 7.5. We now show each dynamic eigenvector w i satisfies the bound in (7.12). For an appropriately chosen nonsingular n × n constant matrix C, we define the Jordan form of the constant matrix Φ A (t 0 + p, t 0 ) by J := C −1 Φ A (t 0 + p, t 0 )C (7.13) =      J m1 (λ 1 ) J m2 (λ 2 ) . . .

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On a Theorem of Halphen and its Application to Integrable Systems

Gesztesy

Unterkofler

Weikard

2000

Journal of Mathematical Analysis and Applications

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We extend Halphen's theorem which characterizes solutions of certain nth-order differential equations with rational coefficients and meromorphic fundamental systems to a first-order n × n system of differential equations. As an application of this circle of ideas we consider stationary rational algebro-geometric solutions of the KdV hierarchy and illustrate some of the connections with completely integrable models of the Calogero-Moser type. In particular, our treatment recovers the complete characterization of the isospectral class of such rational KdV solutions in terms of a precise description of the Airault-McKean-Moser locus of their poles.

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Floquet theory for linear differential equations with meromorphic solutions

Cited by 8 publications

References 6 publications

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

On a Theorem of Halphen and its Application to Integrable Systems

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