On the erugin and floquet-lyapunov theorems for countable systems of difference equations

Teplinskii, Yu. V.; Teplinskii, A. Yu.

doi:10.1007/bf02372054

Cited by 3 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%

“…In a relatively recent paper by Teplinski ȋ and Teplinski ȋ [35], Lyapunov transformations and discrete Floquet theory are extended to countable systems in l ∞ (N, R). It is proved that the countable time varying system can be represented by a countable time invariant system provided its finite-dimensional approximations can also be represented by time invariant systems.…”

Section: Introductionmentioning

confidence: 99%

“…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 . In a relatively recent paper by Teplinskiȋ and Teplinskiȋ [35], Lyapunov transformations and discrete Floquet theory are extended to countable systems in l ∞ (N, R). It is proved that the countable time varying system can be represented by a countable time invariant system provided its finite-dimensional approximations can also be represented by time invariant systems.…”

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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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Floquet Multipliers and the Stability of Periodic Linear Differential Equations: A Unified Algorithm and Its Computer Realization

Wu¹,

Xia²,

Zhao³

2023

jaac

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On the erugin and floquet-lyapunov theorems for countable systems of difference equations

Cited by 3 publications

References 1 publication

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

Floquet Multipliers and the Stability of Periodic Linear Differential Equations: A Unified Algorithm and Its Computer Realization

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